Introduction to Quantum Information Processing

— Draft

Technical University of Clausthal SS 2001 – SS 2005

W. Lücke

Preface

Quantum information processing is one of the most fascinating and active fields of contemporary physics. Its central topic is the coherent control of quantum states in order to perform tasks — like quantum teleportation, absolutely secure data transmission and efficient factorization of large integers — that do not seem possible by means of classical systems alone. The vast possibilities of physical implementations are currently being extensively studied and evaluated. Various proof-of-principle experiments have already been performed. However, in the present note only some possiblities can be indicated. Main emphasis will be on quantum optical methods, indispensable for transmission of quantum information.

For more complete information on achivements and latest proposals concerning quantum information processing the Los Alamos preprint server

                http://xxx.lanl.gov/archive/quant-ph

is highly recommended.

Recommended Literature: (Alber et al., 2001; Bowmeester et al., 2000; Ekert et al., 2000; Nielsen and Chuang, 2001; Preskill, 01; Shannon, 1949; Bertlmann and Zeilinger, 2002; Audretsch, 2002; Bruß, 2003)

4

Contents

I Idealized Quantum Gates and Algorithms 9

1 Basics of Quantum Computation 11 1.1 Classical Logic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Quantum Computational Networks . . . . . . . . . . . . . . . . . . . 17

1.2.1 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.2 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . 24 1.2.3 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 QuantumAlgorithms 35

2.1 Quantum Data Base Search . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.1 Grover’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.2 Network for Grover’s Algorithm . . . . . . . . . . . . . . . . 36 2.1.3 Details and Generalization . . . . . . . . . . . . . . . . . . . . 37

2.2 Factoring Large Integers . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.2 The Quantum Fourier Transform . . . . . . . . . . . . . . . 42 2.2.3 Quantum Order Finding . . . . . . . . . . . . . . . . . . . . . 46

3 Physical Realizations of Quantum Gates 53

3.1 Quantum Optical Implementation . . . . . . . . . . . . . . . . . . . . 53 3.1.1 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.2 Photonic n-Qubit Systems . . . . . . . . . . . . . . . . . . . . 56 3.1.3 Nonlinear Optics Quantum Gates . . . . . . . . . . . . . . . . 60 3.1.4 Linear Optics Quantum Gates . . . . . . . . . . . . . . . . . . 62

3.2 Measurement-Based Quantum Computation . . . . . . . . . . . . . . 70 3.3 Cold Trapped Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . 74 3.3.2 Linear Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.3 Implementing Quantum Gates by Laser Pulses . . . . . . . . . 81 3.3.4 Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

II Fault Tolerant Quantum Information Processing 91

4 General Aspects of Quantum Information 93 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.1 Open Quantum Systems and Quantum Operations . . . . . . 95 4.2.2 Quantum Noise and Error Correction . . . . . . . . . . . . . . 103

4.3 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3.1 General Apects . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3.2 Classical Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.3 Quantum Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3.4 Reliable Quantum Computation . . . . . . . . . . . . . . . . . 118

4.4 Entanglement Assisted Channels . . . . . . . . . . . . . . . . . . . . 122 4.4.1 Quantum Dense Coding . . . . . . . . . . . . . . . . . . . . . 122 4.4.2 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . 123 4.4.3 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . 125 4.4.4 Quantum Cryptography . . . . . . . . . . . . . . . . . . . . . 126

5 Quantifying Quantum Information 129 5.1 Shannon Theory for Pedestrians . . . . . . . . . . . . . . . . . . . . . 129 5.2 Adaption to Quantum Communication . . . . . . . . . . . . . . . . . 133

5.2.1 Von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . . 133 5.2.2 Accessible Information . . . . . . . . . . . . . . . . . . . . . . 137 5.2.3 Distance Measures for Quantum States . . . . . . . . . . . . . 139 5.2.4 Schumacher Encoding . . . . . . . . . . . . . . . . . . . . . . 145 5.2.5 A la Nielsen/Chuang . . . . . . . . . . . . . . . . . . . . . . . 147 5.2.6 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Handling Entanglement 149

6.1 Detecting Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.1.1 Entanglement Witnesses . . . . . . . . . . . . . . . . . . . . . 149 6.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.1.3 Other Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.2 Local Operations and Classical Communication . . . . . . . . . . . . 158 6.2.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.2.2 Entanglement Dilution . . . . . . . . . . . . . . . . . . . . . . 164 6.2.3 Entanglement Distillation . . . . . . . . . . . . . . . . . . . . 164

6.3 Quantification of Entanglement . . . . . . . . . . . . . . . . . . . . . 164

A  167

A.1 Turing’s Halting Problem . . . . . . . . . . . . . . . . . . . . . . . . 167 A.2 Some Remarks on Quantum Teleportation . . . . . . . . . . . . . . . 168 A.3 Quantum Phase Estimation and Order Finding . . . . . . . . . . . . 169 A.4 Finite-Dimensional Quantum Kinematics . . . . . . . . . . . . . . . . 173

A.4.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . 173 CONTENTS 7

A.4.2 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 A.4.3 Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . 178

Bibliography 185 Index 202 8 CONTENTS

Part I

Chapter 1

Idealized Quantum Gates and Algorithms Basics of Quantum Computation

1.1 Classical Logic Circuits

The smallest entity of classical information theory (Shannon, 1949) is the bit (binary digit), i.e. the decision on a classical binary alternative. Usually bits are identified with the numbers 0 (for wrong) or 1 (for true) and typically correspond to the position of some simple switch. Every definite statement may be encoded into a sufficiently long but finite sequence (b_{1}, …, b_{n}) of bits.^{1} In this sense the essence of a calculations may be described as the transformations of a finite sequence of input bits (encoding the task) into a finite sequence of output bits (encoding the result). This suggests the following model for actual calculators:

1. An input register (array of switches) will be put into a state corresponding to the n_{1}-tuple (b_{1}, …, b_{n_{1}}) ∈ {0, 1}^{n_{1}} encoding the task.

2. A computational circuit, the elementary components of which are called gates,^{2} transforms (b_{1}, …, b_{n_{1}}) into an n_{2}-tuple (b′_{1}, …, b′_{n_{2}}) of bits encoding the result to be stored into an output register.

From the mathematical point of view it is only important which element of F_{n_{1},n_{2}}, denoting the set of all mappings from {0, 1}^{n_{1}} into {0, 1}^{n_{2}} , is implemented by the circuit. Therefore, computational circuits implementing the same mapping are called equivalent.

Every element of F_{n_{1},n_{2}} can be implemented by some assembly of gates listed in Table 1.1:

Lemma 1.1.1 For arbitrary positive integer n_{1}, n_{2} all elements of F_{n_{1},n_{2}} can be represented as compositions of tensor products of functions from Tabular 1.1.

Proof: See below.

DRAFT, June 26, 2009

1An important consequence of this fact is the halting problem (see Appendix A.1).

2For simple hardware implementations see (Pütz, 1971, pp. 244–252).

12 CHAPTER 1. BASICS OF QUANTUM COMPUTATION

Name  Symbol Class  Action
ID  F_{1,1}  b ↦ b
FANOUT  F_{1,2}  b ↦ (b, b)
[figure: NOT; F_{1,1}; b; ↦; 1 − b]
AND  &  F_{2,1} (b_{1}, b_{2}) ↦ b_{1}b_{2}
OR  ≥1  F_{2,1} (b_{1}, b_{2}) ↦ b_{1} + b_{2} − b_{1}b_{2}

Table 1.1: Elementary gates

Thus every classical logic circuit corresponds to a graph consisting of symbols from Tabular 1.1. For instance, the graph

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  & &  ≥ 1

corresponds to the mapping SWITCH =^{def} OR ∘ (AND ⊗ AND) ∘ (IDNOT ⊗ ⊗ID ⊗ ID) ∘ (ID ⊗ FANOUT ⊗ ID) , acting as

(b_{0}, s, b_{1}) ⟼ \{ \begin{matrix}b_{0} & if s = 0 , \\ b_{1} & if s = 1 . \end{matrix}

Another example is the graph

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; &; &; &; ≥ 1]

corresponding to

XOR =^{def} OR ∘ (AND ⊗ AND) ∘ (ID ⊗ FANOUT ⊗ ID)

∘(ID ⊗ (NOT ∘ AND) ⊗ ID) ∘ (FANOUT ⊗ FANOUT)

and acting as

(b_{1}, b_{2}) ⟼ b_{1} ⊕ b_{2} =^{def} \{ \begin{matrix}NOT(b_{2}) & if b_{1} = 1 \\ b_{2} & if b_{1} = 0 \end{matrix}
= b_{1} + b_{2} − 2b_{1}b_{2}
= b_{1} + b_{2} mod 2 .

Name Symbol Class  Action

[figure: CNOT; F_{2,2}; (b_{1}, b_{2}); ↦; (b_{1}, b_{1} ⊕ b_{2})]

[figure: TCNOT; F_{2,2}; (b_{1}, b_{2}); ↦; (b_{1} ⊕ b_{2}, b_{2})]

[figure: SWAP; F_{2,2}; (b_{1}, b_{2}); ↦; (b_{2}, b_{1})]

[figure: CSWAP^{3}; F_{3,3}; \begin{matrix}(0, b_{1}, b_{2}) \\ (1, b_{1}, b_{2}) \end{matrix}; \begin{matrix}↦ \\ ↦ \end{matrix}; \begin{matrix}(0, b_{1}, b_{2}) \\ (1, b_{2}, b_{1}) \end{matrix}]

[figure: CCNOT^{4}; F_{3,3}; (b_{1}, b_{2}, b_{3}); ↦; (b_{1}, b_{2}, b_{1}b_{2} ⊕ b_{3})]

14 CHAPTER 1. BASICS OF QUANTUM COMPUTATION

Table 1.2: Some reversible gates Of course, also for the gates listed in Table 1.2 there are equivalent networks, e.g.:^{5}

CNOT = (ID ⊗ XOR) ∘ (FANOUT ⊗ ID) ,  (1.1)
[figure] ≡ [figure] ,  (1.2)

[figure] ≡ [figure] .  (1.3)

[figure: ≡; (1.4)]

Now we are prepared for the

Proof of Lemma 1.1.1: Thanks to FANOUT and SWAP it is sufficient to proof the lemma for decision functions, i.e. for n_{2} = 1 . Obviously, then, the statement of the lemma holds for n_{1} = 1 , since the four elements of F_{1,1} are ID,

TRUE =^{def} OR ∘ (ID ⊗ NOT) ∘ FANOUT ,

and their compositions with NOT (applied last). Now, assume that the statement of the lemma has already been proved for n_{1} = n and consider an arbitrary f ∈ F_{n+1,1} . Then both f_{0} and f_{1} , where

f_{s}(b_{1}, …, b_{n}) =^{def} f(b_{1}, …, b_{n}, s) ,

can be represented as compositions of tensor products of functions from Tabular 1.1. There is a composition of FANOUTs and SWAPs acting as

(b_{1}, …, b_{n}, s) ⟼ (b_{1}, …, b_{n}, s, b_{1}, …, b_{n}).

Composing this with

SWITCH ∘ (f_{0} ⊗ ID ⊗ f_{1})

(to be applied last) gives f . This proves the statement of the lemma for n_{1} = n + 1 .

According to Lemma 1.1.1 we may perform arbitrarily complex computations by composing simple hardware components of very small variety. Of course, given f ∈ F_{n_{1},n_{2}} , there are infinitely many representations of f as composition of tensor products of elementary components. Therefore, the interesting problem arises how to simplify a given gate (logic circuit) without changing its action.^{6}

DRAFT, June 26, 2009

3The CSWAP gate is also called Fredkin gate. 4The CCNOT gate is also called Toffoli gate. 5See (Tucci, 2004) for more equivalences of classical and/or quantum networks. 6See (Lindner et al., 1999, Sect. 8.2.3) for n_{1} ≤ 6 , n_{2} = 1 and (Lee et al., 1999; Shende et al.,

2003) for quantum gates.

From the technological point of view it is also of interest that

NAND =^{def} NOT ∘ AND

is universal in the sense that it can replace NOT, AND, and OR as elementary gates:^{7}

NOT = NAND ∘ FANOUT ,
AND = NOT ∘ NAND ,
OR = NAND ∘ (NOT ⊗ NOT) .

In the same sense

NOR =^{def} NOT ∘ OR

is universal:

NOT = NOR ∘ FANOUT ,
AND = NOR ∘ (NOT ⊗ NOT) ,
OR = NOT ∘ NOR .

Alternatively, in order to minimize dissipation of energy (Landauer, 1961; Landauer, 1998; Plenio and Vitelli, 2001; Bub, 2001; Parker and Walker, 2003), one may execute all calculations using only reversible networks^{8} (Toffoli, 1980a):

Since^{9}

CCNOT_{3}(b_{1}, b_{2}, 1) = NAND(b_{1}, b_{2}) ∀ b_{1}, b_{2} ∈ {0, 1}

and

( \begin{matrix}CCNOT_{1}(b, 1, 0) \\ CCNOT_{3}(b, 1, 0) \end{matrix} ) = FANOUT(b) ∀ b ∈ \{0, 1\} ,

the CCNOT gate is universal for reversible classical computation in the following sense:

For every mapping ϕ ∈ F_{n_{1},n_{2}} there is a reversible n-bit network composed of only CCNOT gates,^{10} SWAP gates, and ID gates (wires) implementing a mapping f ∈ F_{n,n} (n ≥ n_{1}, n_{2}) fulfilling

( f_{1}(b_{1}, …, b_{n_{1}}, c_{n_{1}+1}, …, c_{n}) ; ⋮ ; f_{n_{2}}(b_{1}, …, b_{n_{1}}, c_{n_{1}+1}, …, c_{n}) ) = ϕ(b_{1}, …, b_{n_{1}}) ∀ b_{1}, …, b_{n_{1}} ∈ {0, 1}

for suitably chosen constant bits c_{n_{1}+1}, …, c_{n}.

DRAFT, June 26, 2009

7I.e., every classical logic circuit corresponds to a composition of tensor products of IDs, FANOUTs, and NANDs.

8Reversible classical networks (logic circuits) are those corresponding to bijections f ∈ F_{n,n} for some n ∈ ℕ.

9Toffoli called his gate the AND/NAND gate to indicate that also CCNOT_{3}(j, k, 0) = AND(j, k) holds. Correspondingly, he called CNOT the XOR/FANOUT gate.

10The CNOT gate cannot fulfill this purpose for classical computation.

Of course, it is a nontrivial task to optimize such networks.^{11}

Theorem 1.1.2 (Toffoli) For all n_{1}, n_{2} ∈ ℕ and for every ϕ ∈ F_{n_{1},n_{2}} there is some n ∈ (max {n_{1}, n_{2}} , …, n_{1} + n_{2}) and some bijection f ∈ F_{n,n} with

( f_{1}(b_{1}, …, b_{n_{1}}, 0, …, 0) ; ⋮ ; f_{n_{2}}(b_{1}, …, b_{n_{1}}, 0, …, 0) ) = ϕ(b_{1}, …, b_{n_{1}}) ∀ b_{1}, …, b_{n_{1}} ∈ {0, 1} .

Proof: See (Toffoli, 1980b, Theorem 4.1).

Exercise 1 Show that CSWAP acts as indicated and, therefore, is universal for classical reversible computation:

[figure: b_{1}; b_{2}; 0; b_{1}b_{2}; b; 1; 1 ⊕ b; 0; b; b; 0; b; 1]

Exercise 2 Show that the following networks act as indicated:^{12}

a) Multiplication by 2:

[figure: 0; a_{1}]
[figure: a_{1}; a_{2}]
[figure: a_{2}; ⋯; a_{3}]
⋮  ⋮
[figure: a_{n−1}; ⋯; a_{n}]
[figure: a_{n}; 0]

b) Adder^{13} for a, b ∈ {0, 1} :

+ ≡ [figure: 0; c_{out} = a c_{in} ⊕ b (a ⊕ c_{in}); a; a; b; b; c_{in}; a ⊕ b ⊕ c_{in}]

DRAFT, June 26, 2009

11See (Shende et al., 2003) and (Tsai and Kuo, 2001; Younes and Miller, 2004; Shende et al., 2006), in this connection.

12See also (Vedral et al., 1996; Draper, 2000; Tsai and Kuo, 2001; Cheng and Tseng, 2002). 13Note that a c_{in} ⊕ b (a ⊕ c_{in}) = 0 iff a + b + c_{in} ≤ 1 .

c) Adder for a, b ∈ {0, 1}^{n} :

0  ⋯  c_{0}
a_{1}  ⋯  + ^{a_{1}}
b_{1}  ⋯  ^{b_{1}}
0  ⋯  c_{1}  a_{1} ⊕ b_{1} ⊕ c_{1}
⋮  ⋮  ⋮
⋮  ⋮  ⋮
0  c_{n−2}  ⋯  a_{n−2} ⊕ b_{n−2} ⊕ c_{n−2}
a_{n−1}  +  ⋯  ^{a_{n−1}}
b_{n−1}  ⋯  ^{b_{n−1}}
0  c_{n−1}  ⋯  a_{n−1} ⊕ b_{n−1} ⊕ c_{n−1}
a_{n} +  ⋯  ^{a_{n}}
b_{n}  ⋯  ^{b_{n}}
0  ⋯  a_{n} ⊕ b_{n} ⊕ \underbrace{c_{n}}_{=0}

\underbrace{∑_{ν=1}^{n} a_{ν} 2^{n−ν}}_{x} + \underbrace{∑_{ν=1}^{n} b_{ν} 2^{n−ν}}_{y} = \underbrace{c_{0} 2^{n} + ∑_{ν=1}^{n} (a_{ν} ⊕ b_{ν} ⊕ c_{ν}) 2^{n−ν}}_{x+y} .

Exercise 3 Show that for every reversible classical 2-bit network there is an equivalent one composed only of CNOTs, TCNOTs and NOTs.^{14}

1.2 Quantum Computational Networks

1.2.1 Quantum Gates

If the computational registers are made smaller and smaller you will finally have to take into account the quantum behavior of the devices. Then a n-bit register has to be considered as an array of quantum mechanical systems to be ‘switched’ — for classical computation — into one of two selected (pure) states corresponding to two orthonormal state vectors, usually denoted |0⟩ and |1⟩. This way the n-bit information (b_{1}, …, b_{n}) will be encoded into the state vector

|b_{1}, …, b_{n}⟩ ≡ |b_{1}⟩ ⊗ ⋯ |b_{n}⟩  (1.5)

corresponding to the situation:

DRAFT, June 26, 2009

14Hint: Check the action of (ID ⊗ NOT) ∘ TCNOT on the ordered set of 2-bits.

‘Switch’ number ν being in the state corresponding to |b_{ν}⟩ for ν = 1, …, n.

Then reversible classical n-bit gates correspond to permutations of the 2^{n} states corresponding to the orthonormal computational basis

{|b⟩ : b ∈ {0, 1}^{n}}

of the registers state space. Since such permutations are special unitary transformations there is a chance to implement them by mathematically simple quantum mechanical evolution (governed by some Schrödinger equation). Of course the interpolating states do no longer correspond to some element of the computational basis even if this is the case for both the input state and the output state. Moreover, quantum mechanics allows coherent superpositions as input states,^{15} corresponding to complex linear combinations of the elements of the computational basis. Then the gate causes the transition

∑_{b∈\{0,1\}^{n}} \underbrace{λ_{b}}_{∈ℂ}|b⟩ ⟼ ∑_{b∈\{0,1\}^{n}} λ_{b} |f(b)⟩ ,

where f ∈ F_{n,n} is the mapping corresponding to the classical action of the gate. This means that, in a way, the gate is able to perform all the 2^{n} transitions

|b⟩ ⟼ |f(b)⟩ , b ∈ {0, 1}^{n} ,

simultaneously — thanks to quantum mechanical evolution. Of course, one would like to exploit this massive quantum parallelism for more efficient computation. Unfortunately quantum mechanics imposes severe restrictions:

1. Unknown coherent superpositions cannot be copied with arbitrary precision (Wootters and Zurek, 1982; Peres, 2002). Otherwise a device for superluminal communication could be constructed (Werner, 2001, Chapter 3).

2. Every measurement of an unknown state destroys most of the information carried by that state (quantum state collapse).

3. It is extremely difficult to correct errors caused by unwanted interaction with the environment.

Nevertheless quantum computational networks can be devised, at least in principle, which are much more efficient, for certain tasks, than classical computational networks. Their general structure is as follows:

• The information is usually processed on one and the same quantum register^{16} realized as an array of qubits,^{17} i.e. quantum mechanical systems with a preselected simple quantum alternative corresponding to orthonormal state vectors, usually denoted |0⟩ and |1⟩.

DRAFT, June 26, 2009

15See (Long and Sun, 2001) for an efficient preparation of these superpositions.

16Thanks to the SWAP gate this is not a necessity but this point of view simplifies the treatment. 17Usually qubits are treated as distinguishable, due to their localization in (essentially) disjoint

regions; see (Eckert et al., 2002) for a refined description.

• The possible (pure) states of such an n-qubit register correspond to the (normalized) complex linear combinations of the elements (1.5) of the computational basis.

• ‘Simple’ quantum computational steps are depicted in the network model by quantum gates with an equal number (≤ n) of quantum wires (horizontal lines) attached on both sides. These quantum wires represent the qubits on which the gate acts.

• Several quantum gates may be assembled as in the classical reversible case (without loopbacks, of course).

• The whole network itself is a (more complicated) n-qubit quantum gate acting corresponding to some unitary operator Û_{net} .

• The action of this operator on the initial state vector |b_{1}, …, b_{n}⟩ representing the task (encoded in the bit sequence (b_{1}, …, b_{n})) has to be checked — i.e. the output state Û_{net} |b_{1}, …, b_{n}⟩ has to be measured — to yield a result.

Even though only the probability of a certain outcome of a quantum computation is predictable (according to the rules of quantum mechanics) quantum computation may be very useful for problems of the type

“solution easy to check but difficult to find”.

This will be demonstrated by several quantum algorithms in Chapter 2.

Remarks:

1. We use the naive tensor product formalism of quantum mechanics to describe coupled systems. The latter is very problematic if the interaction of matter with radiation is to be described; see Section 6.2 of (Lücke, nlqo).

2. Of course, the network model described above is just the simplest model for quantum computing. Some possible generalizations, not to be discussed in this chapter, are:

• Computation with non-unitarily evolving mixed states (Tarasov, 2002).

• Quantum computation via application of sequences of one-qubit projective measurements to suitably prepared initial states (Raussendorf et al., 2002).

• Use of non-deterministic gates, i.e. those succeeding only with probability (considerably) less than 1 (Ralph et al., 2002; Bartlett et al., 2002).

3. We are not going to discuss oddities like^{18} “quantum computation even before its quantum input is defined” (Brukner et al., 2003) or “counterfactual computation” (Mitchison and Jozsa, 2001). For quantum programming we refer to (Bettelli et al., 2001).

DRAFT, June 26, 2009

18See 3.1.4, however.

Name Symbol Operator  Matrix  Action
Û gate  Û  Û  ( \begin{matrix}u_{11} & u_{12} \\ u_{21} & u_{22} \end{matrix} ) \begin{matrix}|0⟩ ↦ u_{11} |0⟩ + u_{21} |1⟩ \\ |1⟩ ↦ u_{12} |0⟩ + u_{22} |1⟩ \end{matrix}

Table 1.3: General one-qubit gate (\overline{u_{1j}} u_{1k} + \overline{u_{2j}} u_{2k} = δ_{jk})
Name  Symbol Operator  Matrix  Action
ID gate  ˆ1  ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )  |b⟩ ↦ |b⟩
[figure: NOT gate; ˆ¬; ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ); |0⟩ ⇌ |1⟩]

phase shift gate  δ  Ŝ_{δ}  ( \begin{matrix}1 & 0 \\ 0 & e^{iδ} \end{matrix} )  \begin{matrix}|0⟩ ↦ |0⟩ \\ |1⟩ ↦ e^{iδ} |1⟩ \end{matrix}

Hadamard gate  H  Û_{H}  \frac{1}{\sqrt{2}} ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} )  \begin{matrix}|0⟩ ↦ \frac{1}{\sqrt{2}} (|0⟩ + |1⟩) \\ |1⟩ ↦ \frac{1}{\sqrt{2}} (|0⟩ − |1⟩) \end{matrix}

Table 1.4: Special one-qubit gates

The elements |b_{1}, …, b_{n}⟩ of the computational basis of an n-qubit system are naturally ordered by the corresponding integers

I(b) =^{def} ∑_{ν=1}^{n} b_{ν} 2^{n−ν} ∀ b ∈ {0, 1}^{n} .  (1.6)

It is relative to this ordering that the actions of quantum gates are usually represented by unitary matrices as in Tables 1.3–1.5.

Remark: Note that every (complex) unitary 2 × 2 matrix corresponds to a spin rotation; see, e.g., Exercise 19 of (Lücke, tdst) and Sect. 4.2.1 of (Lücke, qft).

Note that a quantum gate may be used for classical computation iff the entries of its matrix take only values from {0, 1}. Whenever this is the case we use the same symbol and name for the quantum gate as for its classical analog. In this sense we have, e.g.,

⋀_{1}(ˆ¬) = CNOT :   ; ˆ¬  =  [figure] ,

Name  Symbol Matrix  Action

⋀_{n}\binom{}{Û} gate  [figure: ⋮; Û]  ( \begin{matrix}1l_{2^{n}} & 0 \\ 0 & Û \end{matrix} ) \begin{matrix}|b_{1}, …, b_{n}, c⟩ \\ ↦ \{ \begin{matrix}|b_{1}, … b_{n}⟩ ⊗ Û |c⟩ & if b_{1} = … = b_{n} = 1 \\ |b_{1}, …, b_{n}, c⟩ & else \end{matrix} \end{matrix}

Table 1.5: Special (n + 1)-qubit gates

⋀_{2}(ˆ¬) = CCNOT :  [figure: ˆ¬]  =  [figure] ,  (1.7)

i.e.

= ˆ¬ .  (1.8)

Obviously, the phase shift gate (for δ ≠ 0 mod π) and the Hadamard gate have no classical analog.

Sometimes it is more convenient to sketch the action of a gate as done in Figure 1.1 for the f-CNOT gate.

|b_{1}⟩  |b_{1}⟩
⋮ f ⋮
|b_{n}⟩  |b_{n}⟩
|c⟩  |c ⊕ f(b)⟩

Figure 1.1: f-CNOT gate for f ∈ F_{n,1}

Quantum computational networks are called equivalent if they implement the same mapping up to a phase factor.

Exercise 4

a) For arbitrary a ∈ {0, 1}^{n} , n ∈ ℕ, show that

_{}^{⋮ f_{a} ⋮}  ≡  [figure: N̂_{a_{1}}; N̂_{a_{1}}; ^{⋮  ⋮  ⋮}; N̂_{a_{n}}; N̂_{a_{n}}; ,]

where

f_{a}(b) =^{def} δ_{a,b} ∀ b ∈ {0, 1}^{n}

and

N̂_{b} =^{def} \{ \begin{matrix}ˆ¬ & if b = 0 , \\ ˆ1 & else . \end{matrix}

b) Show for arbitrary f, g ∈ F_{n,1} that

h = f ⊕ g ⟹  _{}^{⋮ h ⋮}  ≡  _{.}^{⋮ f  ⋮  g ⋮}

A first example showing the superiority of quantum networks is the Deutsch-Jozsa problem:

Assume you are given an (n + 1) qubit gate which is only known to be the f-CNOT gate of some f ∈ F_{n,1} that is either constant or balanced, i.e. fulfills

∑_{b∈{0,1}^{n}} (−1)^{f(b)} = 0 .

Find out by ‘asking’ this Deutsch-Jozsa oracle whether f is balanced or constant.

Note that in classical computation

|b_{1}⟩  |b_{1}⟩
⋮ f ⋮
|b_{n}⟩  |b_{n}⟩
|0⟩  |f(b)⟩

you may have to ask the Deutsch-Jozsa oracle 2^{n−1} + 1 times (in the worst case) to find the answer. Already for n = 60 that would take more than

\frac{2^{59}}{60 · 60 · 24 · 365 · 10^{9}} years > 18 years

to get the answer if the oracle is asked at a frequency of 1 GHz. In quantum computation, however, we may take advantage of coherent superpositions:

[figure: ^{|b_{1}⟩  |b_{1}⟩}; ^{⋮ f ⋮}; ^{|b_{n}⟩  |b_{n}⟩}; \frac{1}{\sqrt{2}} (|0⟩ − |1⟩); \begin{matrix}(−1) & ^{f(b)} \\ √ & \overline{2} \end{matrix} (|0⟩ − |1⟩) ,; ^{|0⟩ H}; ⋮  f  ⋮ \} ∼ ∑_{b∈\{0,1\}^{n}}(−1)^{f(b)} |b⟩; ^{|0⟩ H}; |1⟩; H; H; |1⟩ .]

Since

\begin{aligned}Û_{H} |b⟩ & = & \frac{1}{\sqrt{2}} (|0⟩ + (−1)^{b} |1⟩) \\ & = & \frac{1}{\sqrt{2}} ∑_{b′∈\{0,1\}} (−1)^{b′b} |b′⟩ ∀ b ∈ \{0, 1\} \end{aligned}

and hence

Û_{H}^{⊗n} |b⟩ = 2^{−n/2} ∑_{b′∈{0,1}^{n}} (−1)^{b′·b} |b′⟩ ∀ b ∈ {0, 1}^{n}  (1.9)

we have

Û_{H}^{⊗(n+1)}∘f-CNOT∘Û_{H}^{⊗(n+1)} |0, …, 0, 1⟩ = 2^{−n} ∑_{b,b′∈{0,1}^{n}} (−1)^{f(b)+b′·b} |b′, 1⟩ . (1.10)

For the Deutsch-Jozsa oracle this means

Û_{H}^{⊗(n+1)} ∘ f-CNOT ∘ Û_{H}^{⊗(n+1)} |0, …, 0, 1⟩ = \{ \begin{matrix}∼ |0, …, 0, 1⟩ & if f is constant, \\ ⊥ |0, …, 0, 1⟩ & else. \end{matrix}

Therefore, the following quantum gate has to be used only^{19} once in order to solve the Deutsch-Jozsa problem (Cleve et al., 1998, Sect. 3):

\begin{matrix}|0⟩ & H & & H \\ ⋮ & ⋮ & f & ⋮ \\ |0⟩ & H & & H \\ |1⟩ & H &  & H & |1⟩ \end{matrix} \} Ψ = \{ \begin{matrix}∼ |0, …, 0⟩ & if f is constant, \\ ⊥ |0, …, 0⟩ & else. \end{matrix}

DRAFT, June 26, 2009

19Actually, since there is always some tiny probability for getting the wrong answer, the quantum test should be repeated a few times. For a physical realization of the algorithm see (Gulde et al.,

Remark: (1.10) holds for every f ∈ F_{2,1} and can therefore be applied also to the Bernstein-Vazirani oracle, i.e. the f-CNOT gate with f(b) = a · b for some a ∈ {0, 1}^{n}. Then Ψ becomes

2^{−n} ∑_{b,b′∈{0,1}^{n}} (−1)^{(a+b′)·b} |b′⟩ = |a⟩ .

1.2.2 Quantum Teleportation

Obviously, the Bell network

[figure: H]

acts according to^{20}

|0, 0⟩ ⟼ Ψ_{0,0} =^{def} Φ_{+} =^{def} \frac{|0, 0⟩ + |1, 1⟩}{\sqrt{2}} = (Û_{0,0} ⊗ ˆ1) Ψ_{0,0} ,
|0, 1⟩ ⟼ Ψ_{0,1} =^{def} Ψ_{+} =^{def} \frac{|0, 1⟩ + |1, 0⟩}{\sqrt{2}} = (Û_{0,1} ⊗ ˆ1) Ψ_{0,0} ,
|1, 0⟩ ⟼ Ψ_{1,0} =^{def} Φ_{−} =^{def} \frac{|0, 0⟩ − |1, 1⟩}{\sqrt{2}} = (Û_{1,0} ⊗ ˆ1) Ψ_{0,0} ,
|1, 1⟩ ⟼ Ψ_{1,1} =^{def} Ψ_{−} =^{def} \frac{|0, 1⟩ − |1, 0⟩}{\sqrt{2}} = (Û_{1,1} ⊗ ˆ1) Ψ_{0,0} ,

where

Û_{0,0} =^{def} + |0⟩⟨0| + |1⟩⟨1| = ˆ1 ,
Û_{0,1} =^{def} + |1⟩⟨0| + |0⟩⟨1| = ˆ¬ ,
Û_{1,0} =^{def} + |0⟩⟨0| − |1⟩⟨1| = Ŝ_{π} ,
Û_{1,1} =^{def} − |1⟩⟨0| + |0⟩⟨1| = Ŝ_{π} ˆ¬ .

This indicates the possibility of quantum dense coding:^{21}

DRAFT, June 26, 2009

Ψ_{−} = \frac{1}{\sqrt{2}} (\frac{Ψ}{∥Ψ∥} ⊗ \frac{Ψ_{⊥}}{∥Ψ_{⊥}∥} − \frac{Ψ_{⊥}}{∥Ψ_{⊥}∥} ⊗ \frac{Ψ}{∥Ψ∥}) ∀ Ψ ∈ ℂ^{2} \ \{0\} .

20The states on the r.h.s are usually called Bell states. They exhibit maximal correlation between the two qubits. We use the notation ( \begin{matrix}α \\ β \end{matrix} )_{⊥} =^{def} ( \begin{matrix}−\overline{β} \\ +\overline{α} \end{matrix} ). Thus

21See (Mermin, 2002) for an interesting discussion of dense coding.

Bob prepares the entangled^{22} state Ψ_{0,0} by applying the Bell network to the easily available state |0, 0⟩ and sends his first qubit to Alice (arbitrarily far away). Now Alice may transfer a 2-bit message to Bob by applying one of the operators Û_{0,0} , Û_{0,1} , Û_{1,0} , Û_{1,1} to this single qubit and sending it back to Bob. Bob can ‘read’ this message by performing a Bell measurement, i.e. checking whether the state of the 2-qubit system is Ψ_{0,0} , Ψ_{0,1} , Ψ_{1,0} , or Ψ_{1,1} .

The whole procedure is described by the following network action:

|b_{1}⟩  ^{|b_{1}⟩}
|b_{2}⟩  ^{|b_{2}⟩}
[figure: |0⟩; H; π; H; |b_{1}⟩]
[figure: |0⟩; |b_{2}⟩ .]

Remarks:

1. As pointed out in (Mermin, 2002), the network

[figure: H; π; H]  ≡  [figure]

— although unsuitable for dense coding — has the same effect on the considered special input.

2. We use abbreviations like

[figure: Â; for; Â]

without explicit definition.

Moreover, checking the special cases Ψ ∈ {|0⟩ , |1⟩}, we see that the teleportation network

DRAFT, June 26, 2009

22Entanglement of vector states Ψ means

⟨Ψ| Â ⊗ B̂ |Ψ⟩ ≠^{i.g.} ⟨Ψ| Â ⊗ ˆ1 |Ψ⟩ ⟨Ψ| ˆ1 ⊗ B̂ |Ψ⟩ ,

i.e. (non-classical) correlations between the subsystems. See (Brukner et al., 2001) in this context.

H
[figure: H]

acts on ψ ⊗ |0, 0⟩ in the following way:

ψ ⊗ |0, 0⟩ ⟼ \frac{1}{2} ∑_{b∈\{0,1\}^{2}} |b⟩ ⊗ Û_{b}^{−1} ψ .

(for every one-qubit state vector ψ). This indicates the possibility of quantum teleportation:^{23}

Qubits 2 and 3 are prepared in the state Φ_{0,0} (indicated by the Bell subnetwork on the left acting on |0, 0⟩). Then qubit 2 is sent to Alice and qubit 3 to Bob (far apart). Since, now, Alice and Bob share an entangled pair of qubits, Alice may teleport the unknown state ψ of qubit 1 to Bob in the following way:

Alice performs a Bell measurement on the system formed by qubits 1 and 2 and sends Bob the classical 2-bit information b if the result is Ψ_{b} (corresponding to the output |b⟩ of the teleportation network for qubits 1 and 2). After receiving this information Bob transforms the (collapsed) state of qubit 3 into ψ by applying Û_{b}.

Note that the actions taken by Alice and Bob, sharing the entangled pair, have the same effect on qubits 1–3 as the following post selection scheme:^{24}

[figure: H; ⌢ /]
[figure: ⌢ /]
ˆ¬  π

In this sense the following scheme describes teleportation of entanglement, also called

DRAFT, June 26, 2009

23The possibility of teleportation, further discussed in Appendix A.2, was first pointed out in (Bennet et al., 1993). Generalization to n-qubit states is straightforward (Díaz-Caro, 2005). Concerning the experimental realization of quantum teleportation see (Giacomini et al., 2002).

24The symbol ⌢ / represents an ideal test (projective measurement) whether the corresponding qubit is in state |0⟩ or |1⟩. It is instructive to study the Bell measurement as a two-step process, in this connection (Groisman, 2008).

entanglement swapping:^{25}

|0⟩  H
[figure: |0⟩; H; ⌢ /]
[figure: |0⟩; H; ⌢ /]
|0⟩  ˆ¬  π

This possibility is very important for creating entanglement for teleportation over very large distances.

Exercise 5 Show that the entanglement swapping scheme prepares the subsystem formed by qubits 1 and 4 (arbitrarily far apart) in the state Ψ_{0,0} .

1.2.3 Universality

A 2-qubit gate corresponding to the unitary operator Û^{(2)} is called universal if for every quantum network there is an equivalent one composed only of one-qubit gates and 2-qubit gates corresponding to Û^{(2)} .

Ordering the computational basis vectors

|I(b)⟩_{n} =^{def} |b⟩ ∀ b ∈ {0, 1}^{n}  (1.11)

as |0⟩_{n} , |1⟩_{n} , … |2^{n} − 1⟩_{n} then λ_{n−1}(ˆ¬) just interchanges the last two of these vectors, the latter being |1, …, 1, 0⟩ and |1, …, 1, 1⟩. Also cyclic permutations of the computational basis vectors can be achieved by suitable composition of Λ_{ν}(ˆ¬) gates, as the following exercise shows.

Exercise 6 Show that the n-qubit network

⋯
[figure: ⋯]
⋮ ⋮ ⋮
[figure: ⋯]
[figure: ⋯]
[figure: ⋯]
[figure: ⋯; ˆ¬]

DRAFT, June 26, 2009

25See (Bowmeester et al., 2000, Sect. 3.10) and (Gisin and Gisin, 2002).

acts according to

|x⟩_{n} ⟼ |(x + 1) mod 2^{n}⟩_{n} ∀ x ∈ {0, …, 2^{n} − 1} ,

where

|∑_{ν=1}^{n} b_{ν} 2^{ν−1}⟩_{n} =^{def} |b⟩ ∀ b ∈ {0, 1}^{n} .  (1.12)

Therefore:

For every quantum gate that has a classical analogue there is an equivalent quantum network composed only of Λ_{ν}(ˆ¬) (and ID) gates.

This together with the following theorem shows that the CNOT gate is universal if the following holds for every ν ∈ ℕ :

For every Û ∈ U(2) there is a network composed only of single qubit gates and CNOT gates (and ID gates) that is equivalent to the Λ_{ν}(Û) gate.

(1.13)

Theorem 1.2.1 Let 2 < N ∈ ℕ. Then every unitary N × N-matrix may be represented as a product of permutation matrices and N × N-matrices of the form

( \begin{matrix}ˆ1 & 0 \\ 0 & Û \end{matrix} ) , Û ∈ U(2) .

Outline of proof:^{26} Given 2 < N ∈ ℕ, choose some orthonormal basis {e_{1}, …, e_{N}} of ℂ^{N} . Then, for arbitrary

Û = ( \begin{matrix}u_{11} & u_{12} \\ u_{21} & u_{22} \end{matrix} ) ∈ U(2) ,

N′ ∈ {2, …, N}, and ν ∈ {1, …, N} define

Û_{N′} e_{ν} =^{def} \{ \begin{matrix}u_{11} e_{N′−1} + u_{21} e_{N′} & for ν = N′ − 1 , \\ u_{12} e_{N′−1} + u_{22} e_{N′} & for ν = N′ , \\ e_{ν} & else . \end{matrix}

Then, for arbitrary normed

z = ∑_{ν=1}^{N} z^{ν} e_{ν} ∈ ℂ^{N} ,

N′ ∈ {2, …, N}, and ζ ∈ ℂ with

|ζ|^{2} + ∑_{ν=N′+1}^{N} |z^{ν}|^{2} = 1

DRAFT, June 26, 2009

26Compare (Reck et al., 1994; Diţă, 2001).

1.2. QUANTUM COMPUTATIONAL NETWORKS 29

we have

Û_{N′}^{(N′)} ( 0 ; ⋮ ; 0 ; 0 ; ζ ; z^{N′+1} ; ⋮ ; z^{N} ) = ( 0 ; ⋮ ; 0 ; ζ′ ; z^{N′} ; z^{N′+1} ; ⋮ ; z^{N} )

for suitable Û^{(N′)} ∈ U(2) and ζ′ ∈ ℂ. Thus, by iteration, we see that there are Û^{(N)}, …, Û^{(2)} ∈ U(2) with

z = Û_{2}^{(2)} ⋯ Û_{N}^{(N)}e_{N}

and hence

\binom{}{Û_{N}^{(N)}}^{−1} ⋯ \binom{}{Û_{2}^{(2)}}^{−1} z = e_{N} .

Identifying z with the last column of an arbitrarily given unitary N × N-matrix ˆ Û we get

(\binom{}{Û_{N}^{(N)}}^{−1} ⋯ \binom{}{Û_{2}^{(2)}}^{−1} ˆ Û)_{ν,N} = δ_{ν,N} .

Thanks to unitarity, the latter also implies

(\binom{}{Û_{N}^{(N)}}^{−1} ⋯ \binom{}{Û_{2}^{(2)}}^{−1} ˆ Û)_{N,ν} = δ_{N,ν} .

Iteration of this argument, if necessary, proves the theorem.

That (1.13) holds for ν = 1 is a simple consequence of the following lemma (Barenco et al., 1995):

Lemma 1.2.2 For every^{27} Û ∈ SU(2) there are Â, B̂, Ĉ ∈ SU(2) with

ÂB̂Ĉ = ˆ1 , Âˆ¬B̂ ˆ¬Ĉ = Û .

Proof: Let Û ∈SU(2) . Then one may easily show (see Exercise 28 of (Lücke, eine)) that there are angles ψ, θ, ϕ with

Û = R̂_{3}(ψ) R̂_{2}(θ) R̂_{3}(ϕ) ,

where

R̂_{3}(ψ) =^{def} ( \begin{matrix}e^{+i \frac{ψ}{2}} & 0 \\ 0 & e^{−i \frac{ψ}{2}} \end{matrix} ) , R̂_{2}(θ) =^{def} ( \begin{matrix}+ cos \frac{θ}{2} & + sin \frac{θ}{2} \\ − sin \frac{θ}{2} & + cos \frac{θ}{2} \end{matrix} ) .

With the definitions

Â =^{def} R̂_{3}(ψ) R̂_{2}(\frac{θ}{2}) , B̂ =^{def} R̂_{2}(−\frac{θ}{2}) R̂_{3}(−\frac{ψ + ϕ}{2}) , Ĉ =^{def} R_{3}(\frac{ϕ − ψ}{2} )

DRAFT, June 26, 2009

27As usual, we denote by U(2) the set of all (complex) unitary 2 × 2-matrices and by SU(2) the set of all Û ∈U(2) with det Û = 1 . this gives

Â B̂ Ĉ = R̂_{3}(ψ) R̂_{2}(\frac{θ}{2}) R̂_{2}(−\frac{θ}{2}) R̂_{3}(−\frac{ψ+ϕ}{2}) R_{3}(\frac{ϕ−ψ}{2} )
= R̂_{3}(ψ) R̂_{3}(−ψ)
= ˆ1

and

Âˆ¬B̂ ˆ¬Ĉ = R̂_{3}(ψ) R̂_{2}(\frac{θ}{2}) ˆ¬ R̂_{2}(−\frac{θ}{2}) R̂_{3}(−\frac{ψ+ϕ}{2}) ˆ¬ R_{3}(\frac{ϕ−ψ}{2} )
= R̂_{3}(ψ) R̂_{2}(\frac{θ}{2}) (ˆ¬ R̂_{2}(−\frac{θ}{2}) ˆ¬) (ˆ¬ R̂_{3}(−\frac{ψ+ϕ}{2}) ˆ¬) R_{3}(\frac{ϕ−ψ}{2} )
= R̂_{3}(ψ) R̂_{2}(\frac{θ}{2}) ˆ R̂_{2}(\frac{θ}{2}) R̂_{3}(\frac{ψ+ϕ}{2}) R_{3}(\frac{ϕ−ψ}{2} )
= R̂_{3}(ψ) R̂_{2}(θ) R̂_{3}(ϕ)
= Û .

Remark: Note that ψ, θ, ϕ in the above proof correspond to the well-known Euler angles; see Sect. 2.1.1.3 of (Lücke, mech) and — for generalization — also (D’Alessandro, 2001).

Because of

[figure: |0⟩; |0⟩]
[figure: |b⟩; Ĉ; B̂; Â; ÂB̂Ĉ |b⟩]

and

[figure: |1⟩; |1⟩]
[figure: |b⟩; Ĉ; B̂; Â; Âˆ¬B̂ ˆ¬Ĉ |b⟩]

this has the following consequence:

Corollary 1.2.3 For Â, B̂, Ĉ, Û according to Lemma 1.2.2 we have:

 ; Û  ≡  [figure: Ĉ; B̂; Â] .

Corollary 1.2.3 together with

δ ; Â ≡ ; Â′ , Â′ = e^{iδ}Â ,

shows that that (1.13) holds for ν = 1 , indeed.

1.2. QUANTUM COMPUTATIONAL NETWORKS 31

Exercise 7 Show that

[figure] ≡ [figure: H; H; H; H]

and^{28}

[figure] ≡ [figure: H; H; H; H; .]

Moreover, because of

[figure: |0⟩; |0⟩]
[figure: |0⟩; |0⟩]
|b⟩ Ĉ  B̂  Â |b⟩
[figure: |0⟩; |0⟩]
[figure: |1⟩; |1⟩]
|b⟩ Ĉ  B̂  Â ^{B̂Ĉ |b⟩}
[figure: |1⟩; |1⟩]
[figure: |0⟩; |0⟩]
|b⟩ Ĉ  B̂  Â ^{ÂB̂ |b⟩}
[figure: |1⟩; |1⟩]
[figure: |1⟩; |1⟩]
|b⟩ Ĉ  B̂  Â ^{ÂĈ |b⟩}

we have (Sleator and Weinfurter, 1994):

[figure: V̂^{2}]  ≡  [figure: ^{V̂  V̂^{∗}  V̂}]

This Sleator-Weinfurter construction may be generalized for ⋀_{n}(V̂^{2}) with arbitrary n ∈ ℕ :^{29}

DRAFT, June 26, 2009

28Recall (1.3). 29Of course, one should look for more efficient implementations; see, e.g. Exercise 8 and (Aho

and Svore, 2003; Vatan and Williams, 2004; Shende et al., 2004). For a nice introduction into the general theory of computational complexity see (Mertens, 2002).

[figure: ⋮; V̂^{2}]  ≡  [figure: ^{⋮  ⋮  ⋮}; ^{V̂  V̂^{∗}  V̂}]

Since

U(2) = {V̂^{2} : V̂ ∈ U(2)}

this that (1.13) holds for every ν ∈ ℕ, hence:

The CNOT gate is universal.^{30}

Exercise 8 Show that the following network acts as indicated:

[figure: |b_{1}⟩; ⋯; ⋯; |b_{1}⟩]
[figure: |b_{2}⟩; ⋯; ⋯; |b_{2}⟩]
[figure: |b_{3}⟩; ⋯; ⋯; |b_{3}⟩]
[figure: |b_{4}⟩; ⋯; ⋯; |b_{3}⟩]
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
[figure: |b_{n−1}⟩; ⋯; ⋯; |b_{n−1}⟩]
[figure: |b_{n}⟩; ⋯; ⋯; |b_{n}⟩]
[figure: |0⟩; ⋯; ⋯; |0⟩]
[figure: |0⟩; ⋯; ⋯; |0⟩]
[figure: |0⟩; ⋯; ⋯; |0⟩]
⋮ ⋮ ⋯ ⋮ ⋮ ⋮ ⋮ ⋯ ⋮ ⋮
[figure: |0⟩; ⋯; ⋯; |0⟩]
[figure: |0⟩; ⋯; ⋯; |0⟩]
[figure: |0⟩; ⋯; ⋯; |0⟩]
|b_{n+1}⟩  ^{⋯}  ^{Û}  ^{⋯}  ψ

ψ = \{ \begin{matrix}Û |b_{n+1}⟩ & if b_{1} = … = b_{n} = 1 , \\ |b_{n+1}⟩ & else. \end{matrix}

DRAFT, June 26, 2009

30Therefore, if CNOT is implementable together with all one-qubit gates and projective measurements w.r.t. the computational basis, all observables can actually be measured.

1.2. QUANTUM COMPUTATIONAL NETWORKS 33 Exercise 9 Show that^{31}

[figure] ≡ _{H  π  H}^{}

and that

; π ≡ π ;

acts according to

|j, k⟩ ⟼ (−1)^{δ_{j,1}δ_{k,1}} |j, k⟩ .

DRAFT, June 26, 2009

31This equivalence is exploited in most suggestions for physical realization of CNOT.

34 CHAPTER 1. BASICS OF QUANTUM COMPUTATION

QuantumAlgorithms^{1}

So far, we have only discovered a few techniques which can produce speed up versus classical algorithms. It is not clear yet whether the reason for this is that we do not have enough intuition to discover more techniques, or that there are only a few problems for which quantum computers can significantly speed up the solution.

(Shor, 2000)

2.1 Quantum Data Base Search

2.1.1 Grover’s Algorithm

Let us assume that the b ∈ {0, 1}^{n} are the indices of the entries of some unstructured data base. Moreover let us assume we are given a search machine that provides an implementation of the f_{a}-CNOT gate, where

f_{a}(b) = \{ \begin{matrix}1 & if b = a \\ 0 & else \end{matrix} ∀ b ∈ \{0, 1\}^{n} ,

when fed with a unique characterization of some entry indexed by a . Now consider the following problem:

Find a with probability ≥ 50% by testing the behavior of the f_{a}-CNOT gate.

In classical computing (recall 1.2.2) one has to test f_{a}-CNOT at least 2^{n−1}-times^{2} in order to find a with probability of 50% . A substantial speedup,^{3} exploiting quantum

DRAFT, June 26, 2009

1Algorithms are general, step-by-step procedures for solving general problems.

2Moreover, expectation value for the necessery number of tests for finding a is

\overline{N} = ∑_{ν=1}^{2^{n}} \frac{ν}{N} = \frac{2^{n} + 1}{2} .

3See (Aaronson and Gottesman, 2004), however.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 2θ; θ; |a⟩; Ĝ_{a} Φ_{0}^{(n)}; ⊥ |a⟩; Φ_{0}^{(n)}]
Figure 2.1: Action of Ĝ on Φ_{0}^{(n)} .

parallelism, was suggested in (Grover, 1996). The basic ingredients of Grover’s algorithm are the initial state

Φ_{0}^{(n)} =^{def} Û_{H}^{⊗n} |0, …, 0⟩ = 2^{−n/2} ∑_{b∈{0,1}^{n}} |b⟩ = 2^{−n/2} Φ_{1,1}  (2.1)

and the unitary operator

Ĝ_{a} =^{def} −R̂_{Φ_{0}^{(n)}} R̂_{|a⟩} ,  (2.2)

where

R̂_{Ψ} =^{def} ˆ1 − 2 P̂_{Ψ} ∀ Ψ ∈ H_{n} .

Since both reflections R̂_{|a⟩} and R̂_{Φ_{0}^{(n)}} leave the |a⟩-Φ_{0}^{(n)}-plane invariant, Ĝ_{a} acts as a rotation in this plane. To determine this rotation it suffices to check its effect on Φ_{0}^{(n)} .

As explained in Figure 2.1 this action is a rotation by the angle 2 θ towards |a⟩, where π/2 − θ is the angle between Φ_{0}^{(n)} and |a⟩. Therefore:

Applying Ĝ_{a} an appropriate number of times to Φ_{0}^{(n)} and testing the result with respect to the computational basis solves the posed problem.

2.1.2 Network for Grover’s Algorithm

Exercise 10 Show that the following (n + 1)-qubit networks act as indicated:^{4}

Ψ{ _{|b⟩ H  H |b⟩,}^{⋮  f_{a}  ⋮} }δ_{0b}Ψ + δ_{1b}R̂_{|a⟩}Ψ

Ψ\{ \begin{aligned}& H & & H \\ & ⋮ & f_{0} & ⋮ \\ & H & & H \\ |b⟩ & H &  & H & π & |b⟩. \end{aligned} \}δ_{0b}Ψ − δ_{1b}R̂_{Φ_{0}^{(n)}}Ψ

Using the obvious notation and generalization of ⋀_{1}(Û) for n-qubit unitary operators Û , we get from Exercise 10

_{}^{⋮ Ĝ_{a} ⋮}  ≡  \begin{aligned}& & H & & H \\ ⋮ & f_{a} & ⋮ & f_{0} & ⋮ \\ & & H & & H \\ H &  & &  & H & π \end{aligned}

and therefore:

|0⟩ H  ⋯
⋮ ⋮ Ĝ_{a} ^{⋮ ⋮} Ĝ_{a}
|0⟩ H  ⋯
[figure: |1⟩; ⋯; |1⟩.] _{μ-times}

} Ĝ_{a}^{μ}Φ_{0}^{(n)}

2.1.3 Details and Generalization

For actual application of Grover’s one has to know the number of times G_{a} should be applied. This can be read off from^{5}

G_{a}^{μ} Φ_{0}^{(n)} = sin((2μ + 1) θ) |a⟩ + cos((2μ + 1) θ)\frac{1}{\sqrt{2^{n} − 1}} ∑_{b≠a} |b⟩ ,  (2.3)

where

θ =^{def} arcsin (\frac{1}{\sqrt{2^{n}}} ) .  (2.4)

Since

θ ≈ 2^{−n/2} for 2^{−n/2} ≪ 1

DRAFT, June 26, 2009

5Formula (2.3) was presented first in (Boyer et al., 1998).

we have

G_{a}^{μ} Φ_{0}^{(n)} ≈ |a⟩ for μ = [\frac{π}{4} \sqrt{2^{n}}] and 2^{n} ≫ 1 .

In this sense Grover’s algorithm provides a quadratic speedup compared to classical computation.

Let us now consider the case that there are exactly t data base entries,^{6} indexed by a_{1}, …, a_{t} ∈ {0, 1}^{n}, meeting the search criteria and that the search engine, therefore, provides an implementation of the (f_{a_{1}} + … f_{a_{t}})-CNOT gate. In order to find at least one of these a_{ν} we just have to replace Ĝ_{a} by

Ĝ_{a_{1},...,a_{t}} =^{def} −R̂_{Φ_{0}^{(n)}} R̂_{|a_{1}⟩} ⋯ R̂_{|a_{t}⟩} ,

which may be implemented as described in 2.1.2 with the f_{a}-CNOT gate replaced by the (f_{a_{1}} + … f_{a_{t}})-CNOT gate. Correspondingly, (2.3)/(2.4) have to be replaced by

Ĝ_{a_{1},...,a_{t}}^{μ} Φ_{0}^{(n)} = sin((2μ + 1) θ_{t})\frac{|a_{1}⟩ + … |a_{t}⟩}{\sqrt{t}}+cos((2μ + 1) θ_{t})\frac{1}{\sqrt{2^{n} − t}} ∑_{b∉\{a_{1},...,a_{t}\}} |b⟩

(2.5)

and

θ_{t} =^{def} arcsin (\frac{t}{\sqrt{2^{n}}} ) .  (2.6)

Choosing μ such that sin^{2}((2μ + 1) θ_{t}) is close to 1 we get a state that is essential a superposition of only those states of the computational basis which correspond to data base entries meeting the search criteria. Performing a test we select one solution at random.

Unfortunately, we only know how to choose μ if we know t. If t is unknown we find a solution after an expected number of O(\sqrt{\frac{2^{n}−1}{t}}) applications of the described procedure with suitably chosen μ′s (Boyer et al., 1998, Sect. 4).

For interesting modifications of Grover’s algorithm see (Ambainis, 2005; Korepin and Grover, 2006; Tulsi et al., 2005) and references given there. For a few-qubit experimental implementation of the algorithm see (Walther et al., 2005) and references given there. For application to robots see (Dong et al., 2006).

Final Remark: Presumably Grover’s algorithm will not be useful for searching a standard database, because transferring the database to quantum memory would require too much effort (Deutsch and Ekert, 1998).

DRAFT, June 26, 2009

6For instance, we may be interested in only a subset of bit-values (Grover and Radhakrishnan,

2.2. FACTORING LARGE INTEGERS 39

2.2 Factoring Large Integers

2.2.1 Basics

The greatest common divisor gcd(n_{0}, n_{1}) for given n_{0}, n_{1} ∈ ℤ may be efficiently determined via Euclid’s algorithm:^{7}

Defining

n_{ν+2} =^{def} \{ \begin{matrix}n_{ν} − n_{ν+1}⌊\frac{n_{ν}}{n_{ν+1}} ⌋ & if n_{ν+1} ≠ 0 \\ 0 & else \end{matrix}  (2.7)

successively for ν = 0, 1, 2, … we get

gcd(n_{0}, n_{1}) = n_{s} for s = sup {ν ∈ ℕ : n_{ν} ≠ 0} < ∞ .  (2.8)

Thus, factoring a given product N = p_{1}p_{2} of unknown large integers p_{1}, p_{2} is a task of the type

“A solution is easy to check but extremely difficult to find.”

Classical encryption schemes rely on this fact.

The most popular public-key encryption algorithm^{8} is RSA, named after its three inventors Ron Rivest, Adi Shamir, and Leonhard Adleman. It works as follows (Rivest et al., 1978):

• Messages and keys are represented by natural numbers corresponding to binary strings.

• Messages M are encoded as

C = M^{e} mod N ,

where N > M and e are two public keys created in the following way:

1. Two large prime numbers p and q of comparable size are randomly chosen^{9} and kept secret. Only their product

N = p · q

is publicly announced.

2. e is chosen as a large random number having 1 as largest common divisor with (p − 1) · (q − 1) — to be checked by Euclid’s algorithm.

DRAFT, June 26, 2009

7As usual, we use the notation

⌊x⌋ =^{def} sup {n ∈ ℤ : n ≤ x} ∀ x ∈ ℝ .

Thus, for n_{ν+1} ≠ 0 , n_{ν+2} is the remainder of the integer division of n_{ν} by n_{ν+1} . For details concerning Euclid’s algorithm see Section 2.2.3.

8See (Singh, 2002; Kahn, 1967) for the history of classical cryptography and (Schneier, 1996) for applications.

9See (Rivest et al., 1978, Section VII.B) how to find large prime numbers without testing primality by factorization.

40 CHAPTER 2. QUANTUMALGORITHMS

• The message may be decrypted in the form

M = C^{d} mod N .

where d ∈ {1, …, (p − 1) · (q − 1) − 1} is the private key to be determined from^{10}

e · d = 1 mod (p − 1) · (q − 1) .

Of course, d has to be kept secret as well as the prime numbers p, q which then may be forgotten.

The task of factorizing N in the form

N = p · q with p, q ∈ {2, …, N − 1}

is essentially solved if a factorization

n_{+} · n_{−} = 0 (modN)  (2.9)

of an integer multiple of N is found that fulfills the conditions

n_{±} ≠ 0 (modN)  (2.10)

Then^{11}

gcd (n_{±}, N) ∈ {2, …, N − 1}  (2.11)

and these factors may be efficiently determined using Euclid’s algorithm.

Outline of proof for (2.11): Obviously, every prime factor of N must be a factor of either n_{+} or n_{−} (or both) and neither n_{+} nor n_{−} can be the product of all these (not necessarily pairwise different) factors.

Finding a factorization of type (2.9),(2.10) is facilitated by the following theorem.

Theorem 2.2.1 (Euler’s Theorem) Let x, N ∈ ℕ. If x and N are coprime, i.e. if gcd(x, N) = 1 , then

x^{φ(N)} = 1 mod N

holds, where Euler’s φ function is defined as^{12}

φ(N) =^{def} |{y ∈ ℕ : y < N , gcd(y, N) = 1}| .

DRAFT, June 26, 2009

10The essential point is that

(p − 1) · (q − 1) = φ(p · q) ,

where φ denotes the Euler function introduced in Theorem 2.2.1, below. d may be determined modulo (p − 1) · (q − 1) running Euclid’s algorithm (2.7) for n_{0} = (p − 1) · (q − 1) and n_{1} = e and resubstituting iteratively the expressions for n_{s} = 1 , s given by (2.8), using (2.7) to yield the representation 1 = e · x + y · N with certain integers x, y.

11As usual, we denote by gcd(n_{1}, n_{2}) the greatest common divisor of two integers n_{1}, n_{2} .

Proof: See, e.g., (Schroeder, 1997, Sect. 8.3) or (Nielsen and Chuang, 2001, Theorem A4.9).

By Euler’s theorem, for N ∈ ℕ and

x ∈ {2, …, N − 1} , gcd (x, N) = 1  (2.12)

the function

f(ν) =^{def} x^{ν} ∀ν ∈ ℤ  (2.13)

has a minimal period

r =^{def} inf {a ∈ ℕ : x^{a} = 1 (modN)} ,  (2.14)

called the order^{13} of x modulo N . If r is even and

x^{r/2} ≠ −1 (modN)  (2.15)

then the conditions (2.9) and (2.10) are fulfilled for n_{±} = x^{r/2} ± 1 .

Outline of proof: (2.9) is a consequence of (2.14) and

(x^{r/2} + 1)(x^{r/2} − 1) = x^{r} − 1 .

(2.10) follows from (2.15) and the corresponding property

x^{r/2} ≠ +1 (modN)

implied by (2.14).

Summarizing, we have the following factoring algorithm:

1. Randomly choose some x ∈ {2, …, N − 1}.

2. If gcd(x, N) ≠ 1 then x is already a nontrivial factor of N .

3. If (2.12) holds, determine the order r of x modulo N .

4. If r is odd or x^{r/2} = −1 (modN), restart the algorithm.

5. If r is even and x^{r/2} ≠ −1 (modN), determine the factors gcd (x^{r/2} ± 1, N) using Euler’s algorithm.

DRAFT, June 26, 2009

12By |M| we denote the number of elements of a finite set M .

13The numbers x ∈ {1, …, N − 1} form a group w.r.t. multiplication modulo N . Every element x of this group generates a cyclic subgroup of order r.

Thanks to the following theorem (and Euler’s algorithm) the efficiency of this factoring algorithm depends solely on the available techniques for determining (2.14).

Theorem 2.2.2 Let m be the number of different prime factors of the positive integer N and let x ∈ {1, …, N − 1} be randomly chosen. If gcd(x, N) = 1 , then the (conditional) probability for (2.14) being even and x^{r/2} ≠ −1 (modN) is not smaller than 1 − 2^{−m}.

Proof: See, e.g., (Nielsen and Chuang, 2001, Theorem A.4.13).

In order to gain exponential speed up for the determination of (2.14), Peter W. Shor suggested the following (Shor, 1994):

Instead of calculating x^{a} (modN) for a = 1, 2, … until the result is 1 (modN), transform the state

\frac{1}{2^{L}} ∑_{a=0}^{2^{2L}−1} |a⟩_{2L} ⊗ |0⟩_{L} ,

where

L =^{def} min {l ∈ ℕ : N ≤ 2^{l}} ,

into the state^{14}

\frac{1}{2^{L}} ∑_{a=0}^{2^{2L}−1} |a⟩_{2L} ⊗ |x^{a} (modN)⟩_{L}

(exploiting quantum parallelism) and evaluate the latter by means of the quantum Fourier transform applied to the first 2L qubits..

2.2.2 The Quantum Fourier Transform

In order to plot, over the interval [−Ω, +Ω] , the Fourier transform

f̃(ω) = \frac{1}{\sqrt{2π}} ∫_{0}^{T_{0}} f(t) e^{i ω t} dt

of a (sufficiently well-behaved) signal restricted to the time interval [0, T_{0}] it is sufficient to have the discrete values

f̃(k \frac{2π}{T_{0}} ) , k ∈ ℤ , |k \frac{2π}{T_{0}} | ≤ Ω ,

DRAFT, June 26, 2009

14An efficient implementation is described in (Vedral et al., 1996).

2.2. FACTORING LARGE INTEGERS 43 if T_{0} is large enough (depending on the required precision). In order to determine these values approximately it is sufficient to know the sampling values

f (j \frac{T_{0}}{N} ) , j ∈ \{0, 1, …, N − 1\} ,

for sufficiently large N ∈ ℕ (depending on T_{0} and Ω):

f̃(k \frac{2π}{T_{0}} ) ≈ \frac{1}{\sqrt{2π}} ∑_{j=0}^{N−1} f (j \frac{T_{0}}{N} ) e^{i k \frac{2π}{N} j} \frac{T_{0}}{N} for |k \frac{2π}{T_{0}} | ≤ Ω .

Remark: In order to estimate the quality of this approximation note that

∑_{j=0}^{N−1} f (j \frac{T_{0}}{N} ) e^{i k \frac{2π}{N} j} = ∫_{0}^{T_{0}} Δ_{T_{0}/N}(t) f(t) e^{i ω t} dt ,

where

Δ_{T_{0}/N}(t) =^{def} ∑_{ν∈ℤ} δ (t − ν \frac{T_{0}}{N} )

and hence^{15}

Δ̃_{T_{0}/N}(ω) = \sqrt{2π} \frac{N}{T_{0}} ∑_{μ∈ℤ} δ(ω − μ π \underbrace{\frac{2N}{T_{0}}}_{Nyquist-frequency}) .

Hence, using the so-called discrete Fourier transform

\{x_{j}\}_{j∈\{0,...,N−1\}} ⟼ \{x̃_{k} =^{def} \frac{1}{\sqrt{N}} ∑_{j=0}^{N−1} x_{j} e^{i k \frac{2π}{N} j}\}_{k∈\{0,...,N−1\}}  (2.16)

we get

f̃(k \frac{2π}{T_{0}} ) ≈ \frac{T_{0}}{\sqrt{2π N}} x̃_{k( mod N)} if |k \frac{2π}{T_{0}} | ≤ Ω

for the sampling values

x_{j} = f (j \frac{T_{0}}{N} ) , j ∈ \{0, …, N − 1\} .

Since

∑_{k=0}^{N−1} e^{i k \frac{2π}{N} m} = \frac{1 − e^{i 2π m}}{1 − e^{i \frac{2π}{N} m}} = 0 ∀ m ∈ \{1, …, N − 1\} ,  (2.17)

DRAFT, June 26, 2009

15See, e.g., (Lücke, musi, Anhang A.1).

44 CHAPTER 2. QUANTUMALGORITHMS the inverse of the transformation (2.16) is

\left\{{ \atop x_{k}}\right\}_{k∈\{0,...,N−1\}} ⟼ \{x_{j} =^{def} \frac{1}{\sqrt{N}} ∑_{k=0}^{N−1} x̃_{k} e^{−i k \frac{2π}{N} j}\}_{j∈\{0,...,N−1\}} .  (2.18)

Especially for N = 2^{n} we define the quantum Fourier transform F̂_{n} by

F̂_{n} ∑_{b∈{0,1}^{n}} x(b) |b⟩ =^{def} ∑_{b∈{0,1}^{n}} x̃(b) |b⟩ ∀ (x_{0}, …, x_{2^{n}−1}) ∈ ℂ^{2^{n}} ,  (2.19)

where^{16}

\begin{matrix}x(b) & =^{def} & x_{I(b)} \\ x̃(b) & =^{def} & x̃_{I(b)} \end{matrix} \} ∀ b ∈ \{0, 1\}^{n} .

Obviously, F̂_{n} is a linear operator on ℂ^{2^{n}} and, therefore,

(x(a) = δ_{a,b} ∀ a ∈ \{0, 1\}^{n}) ⟹_{(2.16)} (x̃(a) = \frac{1}{\sqrt{2^{n}}} e^{i I(a) \frac{2π}{2^{n}} I(b)} ∀ a ∈ \{0, 1\}^{n})

implies^{17}

\sqrt{2^{n}} F̂_{n} |b⟩ = ∑_{a∈\{0,1\}^{n}} e^{i I(a) \frac{2π}{2^{n}} I(b)} |a⟩  (2.20)
= ∑_{a∈\{0,1\}^{n}} ⨂_{ν=1}^{n} (e^{i 2^{n−ν}a_{ν} \frac{2π}{2^{n}} I(b)} |a_{ν}⟩) ∀ b ∈ \{0, 1\}^{n} .

The latter implies^{18}

F̂_{n} |b⟩ = \frac{1}{\sqrt{2^{n}}} ⨂_{ν=1}^{n} (|0⟩ + e^{i 2π 2^{−ν} I(b)} |1⟩) ∀ b ∈ \{0, 1\}^{n} .  (2.21)

Thanks to

e^{i 2π b_{μ} 2^{n−μ−ν}} = 1 for μ ≤ n − ν

we may rewrite this as

F̂_{n} |b⟩ = \frac{1}{\sqrt{2^{n}}} ⨂_{ν=1}^{n} (|0⟩ + e^{i 2π ∑_{μ=n−ν+1}^{n} b_{μ} 2^{n−μ−ν}} |1⟩) ∀ b ∈ \{0, 1\}^{n}  (2.22)

DRAFT, June 26, 2009

16Recall the definition of I(b) in (1.6).

17Note the ordering of

⊗_{ν=1}^{n}χ_{ν} =^{def} χ_{1} ⊗ … ⊗ χ_{n} .

18An import special case is F̂_{n} |0⟩ = Û_{H}^{⊗n} |0⟩.

or as

F̂_{n} |b⟩ = \frac{1}{\sqrt{2^{n}}} ⨂_{ν=1}^{n} (|0⟩ + (∏_{α=1}^{ν} e^{i 2π 2^{−(α−1)} b_{(n−ν)+α}}) |1⟩) ∀ b ∈ \{0, 1\}^{n} (2.23)

showing, by the way, that F̂_{n} is isometric.

Exercise 11 Using (2.23), show that the n-qubit network

H ŝ_{2} ⋯ ŝ_{n−1} ŝ_{n}
⋯  H ⋯ ŝ_{n−2} ŝ_{n−1} ⋯
⋮  ⋮  ⇕
[figure: ⋯; H; ŝ_{2}]
[figure: ⋯; H]

with

ŝ_{ν} =^{def} ( \begin{matrix}1 & 0 \\ 0 & e^{i 2π/2^{ν}} \end{matrix} ) , \begin{matrix}|b_{1}⟩ & & |b_{n}⟩ \\ ⋮ & ⇕ & ⋮ \\ |b_{n}⟩ & & |b_{1}⟩ \end{matrix} ,

implements the quantum Fourier transform.^{19} Moreover, using Corollary 1.2.3, show that

ŝ_{ν} ; ≡ [figure: Ĉ_{ν}; Ĉ_{ν}^{−1}; δ_{ν}]

holds for δ_{ν} = −π/2^{ν} and Ĉ_{ν} = R̂_{3}(δ_{ν}) .

Remarks:

1. If the crossings are ignored then the above implementation of the quantum Fourier transform uses n/2 SWAP gates, n Hadamard gates and n^{2}/2 Λ_{1}(ŝ_{ν}) gates.

2. Since

x̃_{I(a)} =_{(2.19)} ∑_{b∈{0,1}^{n}} x_{I(b)} ⟨a | F̂_{n} b⟩
= ∑_{b∈{0,1}^{n}} x_{I(b)} ⟨F̂_{n}^{−1} a | b⟩ ∀ a ∈ {0, 1}^{n} ,

DRAFT, June 26, 2009

19Note that

\frac{1}{\sqrt{2}} (|0⟩ + e^{i π b} |1⟩) = Û_{H} |b⟩ ∀ b ∈ \{0, 1\} .

the above network implementation for F̂_{n} yields a nice factorization of the matrix corresponding to the discrete Fourier transformation. This factorization is the core of the radix-2 version of the fast Fourier transform (FFT) algorithm.^{20}

Exercise 12 Prove the identities^{21}

\sqrt{2} F_{n′}(0, b_{2}, …, b_{n}) = +F_{n′−1}(b_{2}, …, b_{n′}, 0, b_{n′+1}, …, b_{n})
+e^{i 2π∑_{ν=2}^{n′} b_{ν} 2^{−ν}} F_{n′−1}(b_{2}, …, b_{n′}, 1, b_{n′+1}, …, b_{n}) ,
\sqrt{2} F_{n′}(1, b_{2}, …, b_{n}) = +F_{n′−1}(b_{2}, …, b_{n′}, 0, b_{n′+1}, …, b_{n})
−e^{i 2π∑_{ν=2}^{n′} b_{ν} 2^{−ν}} F_{n′−1}(b_{2}, …, b_{n′}, 1, b_{n′+1}, …, b_{n})

for the partial discrete Fourier transform F_{n′} defined by

(F_{n′} x) (b) =^{def} ∑_{b′∈{0,1}^{n′}} x(b′_{1}, …, b′_{n′}, b_{n′+1}, …, b_{n}) exp(+i 2π ∑_{ν,ν′=1}^{n′} b_{ν} b_{ν′} 2^{n′−ν−ν′})

for n′ < n ∈ ℕ and

(F_{n} x) (b) =^{def} x̃(b) , (F_{0} x) (b) =^{def} x(b) .

2.2.3 Quantum Order Finding

As already mentioned at the end of 2.2.1, to find the order (2.14) of a given integer x ∈ {2, …, N − 1} , Shor suggested to exploit the effectively implementable^{22} state

Ψ_{Shor}  =^{def}  \frac{1}{2^{L}} ∑_{a=0}^{2^{2L}−1} (F̂_{2L} |a⟩_{2L}) ⊗ |x^{a} (modN)⟩_{L}

=_{(2.19),(2.16)} \frac{1}{2^{2L}} ∑_{a,c=0}^{2^{2L}−1} e^{i a \frac{2π}{2^{2L}} c} |c⟩_{2L} ⊗ |x^{a} (modN)⟩_{L} .

where

L =^{def} min {l ∈ ℕ : N ≤ 2^{l}} .

The essential point is the following:

DRAFT, June 26, 2009

20See, e.g., (Brigham, 1974; Nussbaumer, 1982).

21Let us point out that

e^{i 2π∑_{ν=2}^{n′} b_{ν} 2^{−ν}} = e^{i π \frac{I(b_{2},...,b_{n′})}{2^{n′−1}}} ∀ b_{2}, …, b_{n′} ∈ \{0, 1\} .

22See also (Coppersmith, 1994), in this connection.

x^{a} (modN) = x^{a′} (modN) ⟺ a′ = a mod r .

Therefore,

p(a, c) =^{def} |⟨|c⟩_{2L} ⊗ |x^{a} (modN)⟩_{L} | Ψ_{Shor}⟩|^{2}

= \frac{1}{2^{4L}} | ∑_{a′∈M_{a}} e^{i a′ \frac{2π}{2^{2L}} c}|^{2} ∀ a, c ∈ \{0, …, 2^{2L} − 1\} ,

where

M_{a} =^{def} {a′ ∈ {0, …, 2^{2L} − 1}: a′ = a mod r} ,

is negligible unless all the phases

2π \frac{b r c}{2^{2L}}(mod2π) ∈ [0, 2π)

with

b ∈ {0, …, ⌊(2^{2L} − 1 − min M_{a}) /r⌋} ,

are predominantly almost the same, i.e. (assuming L sufficiently large) unless

[0, 2π) ∋ 2π \frac{r c}{2^{2L}} (mod2π) = O (\frac{r}{2^{2L}} ) .

The latter means that^{23}

|\frac{c}{2^{2L}} − \frac{d}{r}| = O (\frac{1}{2^{2L}} )

holds for some integer d.

More precisely, one can show:^{24}

A projective measurement of Ψ_{Shor} w.r.t. the computational basis is likely to find the first 2L-qubit register in a state |c⟩_{2L} with

|\frac{c}{2^{2L}} − \frac{d}{r}| ≤ \frac{1}{2 r^{2}} , gcd(d, r) = 1  (2.24)

being fulfilled for integer d.

DRAFT, June 26, 2009

23If, by chance, r divides 2^{2L} then b runs from 0 to \frac{2^{2L}}{r} − 1 and, therefore, an adaption of (2.17) shows that p(a, c) vanishes exactly unless \frac{r c}{2^{2L}} = d holds for some integer d.

24See (Shor, 1997, Section 5) for details and Appendix A.3 for an improved search algorithm. Note that r ≤ φ(N) < N .

48 CHAPTER 2. QUANTUMALGORITHMS

If we have found a fraction c/2^{2L} fulfilling (2.24) for some (unknown) integer d with gcd(c, d) = 1 , then r may be efficiently determined using the continued fraction algorithm:^{25}

The motivation for the definition (2.7) — given n_{0}, n_{1} ∈ ℕ — is the observation that

\frac{n_{ν}}{n_{ν+1}} = ⌊\frac{n_{ν}}{n_{ν+1}} ⌋ + \frac{1}{n_{ν+1}} (\underbrace{n_{ν} − n_{ν+1} ⌊\frac{n_{ν}}{n_{ν+1}} ⌋}_{remainder}) ∀ n_{ν}, n_{ν+1} ∈ ℕ

since, then, we have

\frac{n_{0}}{n_{1}} = ⌊\frac{n_{0}}{n_{1}} ⌋ + \frac{1}{(\frac{n_{1}}{n_{2}} )}

= ⌊\frac{n_{0}}{n_{1}} ⌋ + \frac{1}{⌊\frac{n_{1}}{n_{2}} ⌋ + _{\frac{1}{( \frac{n_{2}}{n_{3}} )}}}

= ⌊\frac{n_{0}}{n_{1}} ⌋ + \frac{1}{⌊\frac{n_{1}}{n_{2}} ⌋ + _{\frac{1}{⌊ \frac{n_{2}}{n_{3}} ⌋+ \frac{1}{(\frac{n_{3}}{n_{4}} )}}}}

etc.,

hence the continued fraction expansion^{26}

\frac{n_{0}}{n_{1}} = ⌊\frac{n_{0}}{n_{1}} ⌋ + ∑_{μ=1}^{ν−2} \frac{1  |}{| ⌊\frac{n_{μ}}{n_{μ+1}} ⌋} + \frac{1 |}{| \frac{n_{ν−1}}{n_{ν}}} ∀ ν ∈ \{1, …, s\}  (2.25)

with s given by (2.8).

Remark: Formally, i.e. without specification of the a_{μ} and b_{μ}, finite continued fractions^{27}

b_{0} + \frac{a_{1}|}{| b_{1}} + \frac{a_{2}|}{| b_{2}} + … + \frac{a_{ν} |}{| b_{ν}} ,

are recursively defined by

b_{0} + \frac{a_{1}|}{| b_{1}} =^{def} b_{0} + \frac{a_{1}}{b_{1}}

and

b_{0} + \frac{a_{1}|}{| b_{1}} + \frac{a_{2}|}{| b_{2}} + …\frac{a_{ν+1}|}{| b_{ν+1}} =^{def} b_{0} + \frac{a_{1}|}{| b_{1}} + \frac{a_{2}|}{| b_{2}} + …\frac{a_{ν} |}{| b_{ν} + \frac{a_{ν+1}}{b_{ν+1}}} .

DRAFT, June 26, 2009

25Otherwise the prescription will yield an integer r′ that fails the test x^{r′} = 1(modN) almost certainly. Then the whole procedure has to be repeated.

26Note that the continued fraction expansion does not change if n_{0} and n_{1} are replaced by n′_{0} = p n_{0} and n′_{1} = p n_{1} , where p ∈ ℕ.

27See, e.g., (Perron, 1954; Perron, 1957) or (Brezinski, 1991) for the general theory of continued fractions. See also (Baladi and Vallee, 2003).

Due to

n_{ν+1} ≠ 0 ⟹_{(2.7)} n_{ν+2} < n_{ν+2} < n_{ν+1}

s must be finite and fulfill the equation ⌊\frac{n_{s−1}}{n_{s}} ⌋ = \frac{n_{s−1}}{n_{s}} . Since

gcd (n_{ν−1}, n_{ν}) ≠ 0 ⟺ gcd (n_{ν+1}, n_{ν})

we see that gcd(n_{0}, n_{1}) divides n_{s} and that n_{s}, dividing n_{s−1} , also divides n_{0} and n_{1} . Hence (2.8) holds, indeed.

Given c, L ∈ ℕ fulfilling (2.24), the set of possible fractions \frac{d}{c} is strongly restricted by the following lemma.

Lemma 2.2.3 Given n_{0}, n_{1}, d, r ∈ ℕ fulfilling

|\frac{n_{0}}{n_{1}} − \frac{d}{r}| < \frac{1}{2 r^{2}} ,

using definition (2.7) and (2.8), we have^{28}

\frac{d}{r} = ⌊\frac{n_{0}}{n_{1}} ⌋ + ∑_{μ=1}^{t} \frac{1 |}{| ⌊\frac{n_{μ}}{n_{μ+1}} ⌋}

for some t ∈ {1, …, s − 1} .

Proof: See (Nielsen and Chuang, 2001, Theorem A4.16).

Therefore, using (2.7) for

n_{0} = c , n_{1} = 2^{2L}

and determining — for t = 1, 2, …, s — the numbers A_{t}, B_{t} ∈ ℕ characterized by

⌊\frac{n_{0}}{n_{1}} ⌋ + ∑_{μ=1}^{t} \frac{1 |}{| ⌊\frac{n_{μ}}{n_{μ+1}} ⌋} = \frac{A_{t}}{B_{t}} gcd(A_{t}, B_{t}) = 1 ,

we have

r = B_{t} for some t ≤ s.

DRAFT, June 26, 2009

28Note, however, that

a_{t} = 1 ⟹ ∑_{μ=1}^{t} \frac{1 |}{| a_{μ}} = ∑_{μ=1}^{t−1} \frac{1 |}{| a_{μ}} + \frac{1 |}{| a_{t−1} + 1} ∀ a_{0}, …, a_{t} ∈ ℕ .

50 CHAPTER 2. QUANTUMALGORITHMS Also these A_{t}, B_{t} can be efficiently determined via Euclid’s algorithm:

If

b_{0} + \frac{a_{1}|}{| b_{1}} + \frac{a_{2}|}{| b_{2}} + … + \frac{a_{n}|}{| b_{n}} ,

is well-defined then

b_{0} + ∑_{μ=1}^{ν+1} \frac{a_{μ}|}{| b_{μ}} = b_{0} + ∑_{μ=1}^{ν−1} \frac{a_{μ}|}{| b_{μ}} + \frac{a_{ν} b_{ν+1} |}{| b_{ν} b_{ν+1} + a_{ν+1}} ∀ ν = 1, …, n − 1

and, therefore,

b_{0} + \frac{a_{1}|}{| b_{1}} + \frac{a_{2}|}{| b_{2}} + … + \frac{a_{ν} |}{| b_{ν}} = \frac{A_{ν}(a_{1}, …, a_{ν}; b_{0}, …, b_{ν})}{B_{ν}(a_{1}, …, a_{ν}; b_{0}, …, b_{ν})} ∀ ν = 1, …, n ,

if the A_{ν} and B_{ν} are recursively defined by^{29}

\begin{matrix}A_{ν} & =^{def} & b_{ν} A_{ν−1} + a_{ν} A_{ν−2} \\ B_{ν} & =^{def} & b_{ν} B_{ν−1} + a_{ν} B_{ν−2} \end{matrix} \} for ν = 1, 2, …, n ,

where

A_{−1} =^{def} 1 , A_{0} =^{def} b_{0} , B_{−1} =^{def} 0 , B_{0} =^{def} 1 .

Fortunately, these definitions also imply

\begin{matrix}a_{μ} = 1 ∀ μ ∈ \{1, …, n\} \\ b_{μ} ∈ ℕ ∀ μ ∈ \{0, …, n\} \end{matrix} \} ⟹ gcd (A_{ν}, B_{ν}) = 1 ∀ ν ∈ \{1, …, n\} .

Outline of proof:

\begin{aligned}A_{ν+1} B_{ν} & = & b_{ν+1} A_{ν} B_{ν} + A_{ν−1} B_{ν} \\ & = & b_{ν+1} A_{ν} B_{ν} + A_{ν−1} B_{ν−2} + b_{ν} A_{ν−1} B_{ν−1} \end{aligned}

and the corresponding equation with A, B interchanged imply

A_{ν+1} B_{ν} − B_{ν+1} A_{ν} = − (A_{ν} B_{ν−1} − B_{ν} A_{ν−1}) .

By induction, starting from

A_{0} B_{−1} − B_{0} A_{−1} = −1 ,

this gives

A_{ν} B_{ν−1} − B_{ν} A_{ν−1} = (−1)^{ν+1} ∀ ν ∈ {0, …, n} .  (2.26)

In case

A_{ν} = d_{ν} c_{ν} , B_{ν} = e_{ν} c_{ν}

the positive integer c_{ν} would divide the l.h.s. of (2.26), hence also the r.h.s. The latter, however, is only possible for c_{ν} = 1 .

DRAFT, June 26, 2009

29Thus

(b_{ν} b_{ν+1} + a_{ν+1}) A_{ν−1} + (a_{ν} b_{ν+1}) A_{ν−2} = b_{ν+1}A_{ν} + a_{ν+1}A_{ν−1}

and similarly for the B_{μ}.

\underline{Example^{30}N = 899 , L = 10 :} If, for instance, the projective measurement gives |267137⟩_{2L} for the first 2L qubits, then we get^{31}

\begin{aligned}\frac{c}{2^{2L}} & = & [0, 3, 1, 12, 2, 1, 1, 1, 26, 1, 1, 22, 2] \\ & =^{def} & 0 + \frac{1 |}{| 3} + \frac{1 |}{| 1} + \frac{1 |}{| 12} + \frac{1 |}{| 2} + \frac{1 |}{| 1} + \frac{1 |}{| 1} + \frac{1 |}{| 1} + \frac{1 |}{| 26} + \frac{1 |}{| 1} + \frac{1 |}{| 1} + \frac{1 |}{| 22} + \frac{1 |}{| 2} \end{aligned}

and succeed with t = 7 :

[0, 3, 1, 12, 2, 1, 1, 1] = \frac{107}{420} = \frac{d}{r} ,

i.e. testing r = 420 , we get^{32} the factors

gcd(11^{210} + 1, 899) = 29 , gcd(11^{210} − 1, 899) = 31 = \frac{899}{29} .

If, for instance, we unfortunately measure c = 801411 then we get

\frac{801411}{2^{20}} = [0, 1, 3, 4, 7, 1, 80, 1, 1, 7, 6]

with the subconvergent

[0, 3, 4, 7, 1] = \frac{107}{140} (= \frac{321}{420}) .

In this case, although the condition

|\frac{c}{2^{2L}} − \frac{d}{r}| ≤ \frac{1}{2 r^{2}}

(but not gcd(d, r) = 1) is fulfilled (with d = 321) r = 420 will presumably not be detected and the whole procedure will be repeated, maybe with a different random value for x.

DRAFT, June 26, 2009

30Compare (Rosé et al., 2004, Section III).

31In Maple, after the command ”with(numtheory);” this result will be produced by the command ”convert(267137/2^{20}, cfrag);” and the subconvergent [0, 3, 1, 12, 2, 1, 1, 1] can be evalu-

ated by the command ”nthconver([0,3,1,12,2,1,1,1],7);”.

32E.g., by the Maple commands ”gcd(11^{2}10,899);” and ”gcd(11^{2}10,899);”.

52 CHAPTER 2. QUANTUMALGORITHMS

Physical Realizations of Quantum Gates^{1}

A set of necessary conditions to be fulfilled for the physical implementation of quantum computation is given by DiVincenzo’s checklist (DiVincenzo, 2000; Perez-Delgado and Kok, 2009):

1. Qubits have to be well characterized and scalable.^{2}

2. The standard states |0, …, 0⟩ must be preparable.^{3}

3. The duration of a gate operation must be much smaller than the decoherence time.

4. A universal set of gates must be implementable.

5. The qubits must be measurable in order to be able to ‘read out the result’ of a quantum computation.

Remark: For quantum communication the following two requirements have to be added:

• It must be possible to convert static qubits into flying qubits (typically photons).

• It must be possible to protect flying qubits against decoherence.

3.1 Quantum Optical Implementations

Optical systems currently constitute the only realistic proposal for long-distance quantum communication and underly implementations of quantum cryptography.

(Knill et al., 2001)

DRAFT, June 26, 2009

1See the special volume Fortschr. Phys. 48 (2000) No. 9–11. 2Thus, 1-photon realizations of n-qubit systems should be considered as qudits with d = 2^{n}

rather than n-qubit systems proper.

3This is still difficult for n-photon systems.

3.1.1 Photons

In the Coulomb gauge the free electromagnetic field E(x, t) , B(x, t) is given in the form

B(x, t) = curl A(x, t) , E(x, t) = −\frac{∂}{∂t}A(x, t) ,
A(x, t) = A^{(+)}(x, t) + (A^{(+)}(x, t))^{∗}

by some complex vector potential

A^{(+)}(x, t) = ∫ (∑_{j=1}^{2} ϵ_{j}(k) f̌_{j}(k)) e^{−i(c|k|t−k·x)} \frac{dk}{\sqrt{2 |k|}} ,

where, for every k ≠ 0 , the vectors ϵ_{j}(k) form a

right handed orthonormal basis \{ϵ_{1}(k), ϵ_{2}(k), \frac{k}{|k|}\}

and thus guarantee div A^{(+)} = 0 .

Remark: We use SI conventions; see Apendix A.3.3 of (Lücke, edyn).

In the Heisenberg picture of the quantized theory the classical fields E(x, t) and B(x, t) have to be replaced by corresponding observables on the state space H_{field} , i.e. by operator-valued (generalized) functions Ê(x, t) and B̂(x, t) to be interpreted in the following way:

If Φ ∈ H_{field} is a sufficiently well-behaved (and ∥Φ∥ = 1) then

⟨Φ | Ê(x, t) Φ⟩ resp. ⟨Φ | B̂(x, t) Φ⟩

is the expectation value for E(x, t) resp. B(x, t) in the Heisenberg state (corresponding to) Φ .

Up to unitary equivalence these observables are given by

B̂(x, t) = curl Â(x, t) , Ê(x, t) = −\frac{∂}{∂t}Â(x, t) ,
Â(x, t) = Â^{(+)}(x, t) + (Â^{(+)}(x, t))^{†} ,

where^{4}

Â^{(+)}(x, t) =^{def} (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∫ (∑_{j=1}^{2} ϵ_{j}(k) â_{j}(k)) e^{−i(c|k|t−k·x)} \frac{dk}{\sqrt{2 |k|}}  (3.1)

and the â_{j}(k) are annihilation operators^{5} fulfilling the commutation relations^{6}

[â_{j}(k), â_{j′}(k′)]_{−} = 0 , [â_{j}(k), (â_{j′}(k′))^{†}]_{−} = δ_{jj′} δ(k − k′)  (3.2)

on a suitable dense subspace D_{0} of the Hilbert space H containing a cyclic normalized vacuum state vector Ω characterized (up to a constant phase factor) by

â_{j}(k) Ω = 0  (3.3)

(in the distributional sense). This also fixes the inner product on H.

The operators â of the form

â = ∑_{j=1,2} ∫ â_{j}(k) (f̌_{j}(k))^{∗} dk , f̌_{1} , f̌_{2} ∈ L^{2}(ℝ^{3})  (3.4)

with^{7}

[â , â^{†}]_{−} = ˆ1  (3.5)

characterize modes of the quantized electromagnetic field corresponding to the classical complex vector potentials

A^{(+)}(x, t) =  ⟨(Â^{(+)}(x, t))^{†} Ω | â^{†} Ω⟩
=  ⟨Ω | [Â^{(+)}(x, t) , â^{†}]_{−} Ω⟩

=_{(3.4),(3.1),(3.2)} (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∫ (∑_{j=1}^{2} ϵ_{j}(k) f̌_{j}(k)) e^{−i(c|k|t−k·x)} \frac{dk}{\sqrt{2 |k|}} .

(3.6)

DRAFT, June 26, 2009

4Thanks to the special choice of the factor in front of the integral we get the desired expression

Ĥ = \frac{1}{2} ∫ (ϵ_{0} :Ê(x, t) · Ê(x, t): + \frac{1}{μ_{0}} :B̂(x, t) · B̂(x, t):) dx

for the Hamilton operator, characterized (up to an additive constant) by

\frac{∂}{∂t} Â^{(+)}(x, t) = \frac{i}{ℏ} [Ĥ , Â^{(+)}(x, t)]_{−} .

5See (Mizrahi and Dodonov, 2002), however.

6Of course, the notation ^{†} includes the requirement

⟨Φ | â_{j}(k) Φ′⟩ = ⟨(â_{j}(k))^{†} Φ | Φ′⟩ .

7Condition (3.5) is equivalent to ∥â^{†} Ω∥ = 1 .

With these modes the subspaces H^{(n)} ⊂ H of n-photon state vectors may be defined recursively by

H^{(0)} =^{def} {λ Ω : λ ∈ ℂ}

and

H^{(n+1)} =^{def} {â^{†} Φ^{(n)} : Φ^{(n)} ∈ H^{(n)} , â mode} for n = 0, 1, 2, ….

Physical characterization of n-photon states: n (ideal) detectors but no more can be made fire by an incoming n-photon state.

Modes â_{ν} , â_{μ} are called orthogonal, iff the states â_{ν}^{†} Ω , â_{μ}^{†} Ω are orthogonal, i.e. iff [â_{ν} , â_{μ}]_{−} = 0 .

Fortunately, already classical electrodynamics tells us how photons are affected by passive linear optical components:

The change of the mode â of a photon, caused by a passive linear optical component, is such that the corresponding complex vector potential changes as predicted by classical electrodynamics.

3.1.2 Photonic n-Qubit Systems

Single-Photon Realization

For every n ∈ ℕ, using sufficiently many beam splitters a single-photon state â^{†} Ω can be changed into a coherent superposition^{8}

∑_{b∈{0,1}^{n}} λ_{b} â_{b}^{†} Ω

of (essentially) orthogonal 1-photon states

|b⟩ =^{def} a_{b}^{†} Ω

which may be chosen as elements of the computational basis of a (simulated) n-qubit system.

Remarks:

1. Since the single qubits of the 1-photon realization cannot exist independently from each other we should better call the system a qu2^{n}it system.

DRAFT, June 26, 2009

8For the special case n = 3 an example is illustrated in Figure 3.1.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; ^{^{\frac{1}{\sqrt{8}}} |0, 0, 1⟩}; ^{^{\frac{1}{\sqrt{8}}} |0, 0, 0⟩}; ^{^{\frac{1}{\sqrt{8}}} |0, 1, 0⟩}; ^{^{\frac{1}{\sqrt{8}}} |0, 1, 1⟩}; ^{^{\frac{1}{\sqrt{8}}} |1, 0, 0⟩}; ^{^{\frac{1}{\sqrt{8}}} |1, 0, 1⟩}; ^{^{\frac{1}{\sqrt{8}}} |1, 1, 0⟩}; ^{^{\frac{1}{\sqrt{8}}} |1, 1, 1⟩}]

Figure 3.1: Preparation of Φ_{0}^{(3)} in a single-photon realization.

2. Cascades of beam splitters may also be used to implement approximate mea-

surement of the number of photons:^{9}

E.g., replace the single-photon input in Figure 3.1 by a 2-photon state and direct the output rays into separate ideal detectors. Then the probability that exactly two of these detectors ‘fire’ is 15/16 (≈ 94 %).

For such a choice of computational bases all unitary transformations can be (essentially) effected by linear optical components. Thanks to Theorem 1.2.1 it is sufficient to show this for n = 1 :

Assume, for instance, that that |0⟩ and |1⟩ describe horizontally polarized (almost monochromatic) photons. Then the 1-qubit gates may be implemented by linear optical elements corresponding to Jones matrices Û in the following way:

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; λ_{1} |0⟩; λ_{2} |1⟩; D̂\frac{π}{2}; Û; χ; χ′; D̂_{− \frac{π}{2}}; λ′_{1} |0⟩; λ′_{2} |1⟩]

All unitary transformations of the polarization state of a photon (with almost sharp momentum) can be (almost accurately) performed by proper use of only λ/2-blades and polarization-dependent phase shifters.

DRAFT, June 26, 2009

9See (Bartlett et al., 2002) for details. See also (Haderka et al., 2004; Waks et al., 2003).

General remark: Instead of performing the n-qubit transformations Û_{(n)} one may choose a fixed n-qubit transformation V̂_{(n)} perform the n-qubit transformations Û_{(n)} V̂_{(n)} and interprete the result w.r.t. the \underline{new computational bases} \{|b⟩′ =^{def} V̂_{(n)} |b⟩ : b ∈ \{0, 1\}^{n}\} . Usually, in practice, this freedom is tacitly made use of.

Exercise 13 Consider the single-photon realization of a 2-qubit system with:

â_{I(0,0)} ˆ= \{ \begin{matrix}upper path and \\ vertical polarization , \end{matrix}
â_{I(0,1)} ˆ= \{ \begin{matrix}upper path and \\ horizontal polarization , \end{matrix}
â_{I(1,0)} ˆ= \{ \begin{matrix}lower path and \\ vertical polarization , \end{matrix}
â_{I(1,1)} ˆ= \{ \begin{matrix}lower path and \\ horizontal polarization . \end{matrix}

a) Show that the CNOT gate may be implemented by placing a 90^{∘} polarization rotator into the lower path.

b) Show that the TCNOT gate may be implemented by applying a polarization beam splitter — reflecting the vertically polarized components and transmitting the horizontally polarized components — in the following way:

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; α_{10} |1, 0⟩ + α_{01} |1, 1⟩ .; α_{00} |0, 0⟩ + α_{01} |0, 1⟩; α_{10} |1, 0⟩ + α_{11} |1, 1⟩; α_{00} |0, 0⟩ + α_{11} |0, 1⟩]

The single-photon ‘realization’ has a serious disadvantage spoiling the eventual speed-up of quantum computation:

The number of optical devices required for the single-photon simulation of n-qubit systems grows exponentially with n (Cerf et al., 1998; Kwiat et al., 2000).

Therefore, we will consider only many-photon realizations in the following.

Multi-Photon Realization

For given n ∈ ℕ we may choose a fixed^{10} set

{â_{ν,j} : ν ∈ {1, …, n} , j ∈ {0, 1}}

of 2^{n} pairwise orthogonal modes and consider the n-photon states

|b⟩ =^{def} â_{1,b_{1}}^{†} ⋯ â_{n,b_{n}}^{†} Ω ∀ b ∈ {0, 1}^{n}

as elements of the computational basis of a true n-qubit system.^{11}

Here, while the 1-qubit gates may still be easily implemented by linear optical components, the physical realization of universal 2-qubit gates is quite a technological challenge.^{12}

Fock Realization

Another possibility is to choose some fixed set {â_{1}, …, â_{n}} of pairwise orthogonal modes and consider the states

|b⟩ = |b⟩^{F} ∀ b ∈ {0, 1}^{n} ,  (3.7)

where

|ν_{1}, …, ν_{n}⟩^{F} =^{def} \frac{1}{\sqrt{ν_{1}! ⋯ ν_{n}!}} (â_{1}^{†})^{ν_{1}} ⋯ (â_{n}^{†})^{ν_{n}} Ω ∀ ν_{1}, …, ν_{n} ∈ ℤ_{+} ,  (3.8)

as elements of the computational basis of the n-qubit system.

The states â_{b}^{†} Ω of the single-photon realization for n-qubit systems form the subset

{|b⟩^{F} : b_{1} + … + b_{n} = 1}

of the computational basis of the Fock realization for 2^{n}-qubit systems using the 2^{n} modes

â_{I(b)+1} = â_{b} , b ∈ {0, 1}^{n} .

Obviously, for n=1 the Fock realization does not coincide with the single-photon realization. Actually, 1-qubit gates like the Hadamard gate cannot be implemented by linear optical components, since |0⟩^{F} represents the vacuum state. Nevertheless, the Fock realization has certain advantages as, e.g., those to be discussed in 3.1.4.

DRAFT, June 26, 2009

10Recall the above general remark, however.

11Of course, for n = 1 the n-photon realization coincides with the single-photon realization. 12Concerning recent progress in detector technologies see (Rosenberg et al., 2005).

3.1.3 Nonlinear Optics Quantum Gates

For both the n-photon and the Fock realization of n-qubit systems the (universal) CPHASE gate Λ_{1}(Ŝ_{π})

; π ≡ [figure: H; H]

is nonlinear in the sense that — contrary to the action of linear optics components — the modes are are not transformed independently of each other.

If a non-linear sign gate NS is available,^{13} i.e. an optical one-way gate acting according to^{14}

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then the CPHASE gate can be easily implemented by proper use of linear optical components like the Hadamard beam splitters (with deflecting mirrors) characterized in Figure 3.2.

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Figure 3.2: Hadamard beam splitter.

One such implementation,^{15} suggested in (Ralph et al., 2002), is sketched in Figure 3.3. The essential idea is exploiting two-photon interference (see Section 5.2.3 of (Lücke, nlqo)) at two Hadamard beam splitters forming a balanced Mach-Zehnder interferometer as sketched in Figure 3.4.

One would like to realize the necessary NS gates by means of optical nonlinearities. Unfortunately, sufficiently strong nonlinearities of crystals are accompanied by too strong absorption and therefore are not suitable. A way out may eventually be

DRAFT, June 26, 2009

13See (Sanaka et al., 2003) for an experimental realization.

14The gate is non-linear in the sense that its action cannot be reduced to a linear transformation of the modes â.

15The modes â_{ν,μ} are assumed to differ only by vertical translation and to describe photons with (almost) sharp momenta.

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Figure 3.3: Optical implementation of a CPHASE gate.

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Figure 3.4: Action of a balanced Mach-Zehnder interferometer.

provided by electromagnetically induced transparency^{16} (EIT), discussed in Section 8.3.2 of (Lücke, nlqo).

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indeterministic^{17} NS_{x}-gates, i.e. gates acting according to

(α ˆ1 + β â^{†} + γ (â^{†})^{2}) Ω ⟼ (α ˆ1 + β â^{†} + x γ (â^{†})^{2}) Ω

if successful, can be implemented^{18} as sketched in Figure 3.5. According to (Knill et al., 2001, Fig. 1) these gates act successfully iff a single-photon state is detected

DRAFT, June 26, 2009

16See (Ottaviani et al., 2006) and concluding discussion of (Munro et al., 2005a). See also (Munro et al., 2005b) concerning the use of ‘weak’ cross Kerr nonlinearities.

17A gate is called indeterministic if it acts correctly with nonzero probability < 1 not depending on the input state and if, after its action, it is known whether the action was correct or not. An indeterministic gate is called near-deterministic if its probability of success is near to 1.

18See also (Rudolph and Pan, 2001; Ralph et al., 2002; Hofmann and Takeuchi, 2002) and especially (Gilchrist and Milburn, 2002) for other possibilities.

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Figure 3.5: An indeterministic NS_{x}-gate

by D_{1} and the vacuum state is detected by D_{2} . For

( \begin{matrix}ϑ_{1} \\ ϑ_{2} \\ ϑ_{3} \end{matrix} ) = ( \begin{matrix}+22.5^{∘} \\ 65.5302^{∘} \\ −22.5^{∘} \end{matrix} ) and ( \begin{matrix}φ_{1} \\ φ_{2} \\ φ_{3} \\ φ_{4} \end{matrix} ) = ( \begin{matrix}0 \\ 0 \\ 0 \\ 180^{∘} \end{matrix} )

we have \underline{x=-1} and the probability of success is 0.25. For

( \begin{matrix}ϑ_{1} \\ ϑ_{2} \\ ϑ_{3} \end{matrix} ) = ( \begin{matrix}+36, 53^{∘} \\ 62.25^{∘} \\ −36.53^{∘} \end{matrix} ) and ( \begin{matrix}φ_{1} \\ φ_{2} \\ φ_{3} \\ φ_{4} \end{matrix} ) = ( \begin{matrix}88, 24^{∘} \\ −66, 52^{∘} \\ −11, 25^{∘} \\ 102, 24^{∘} \end{matrix} )

we have \underline{x=i} and the probability of success is 0.18082.

3.1.4 Linear Optics Quantum Gates^{19}

Indeterministic Gates Fortunately, using photonic memory and quantum teleportation, (deterministic) CPHASE gates may by implemented using indeterministic ones:

DRAFT, June 26, 2009

19See (Dowling et al., 2004).

The essential observation, due to (Gottesman and Chuang, 1999), is the equivalence

[figure: \underline{B}; ⌢ /; Φ_{+} \{; ⌢ /; π; Φ_{+} \{; π; π; \overline{B}; ⌢ /; ⌢ /]  ≡  _{π  ,}^{}(3.9)

where

\underline{B}  ≡  [figure: H]

is the inverse of the Bell network and Φ_{+} the state defined in 1.2.2. This equivalence is obvious from the discussion of quantum teleportation in 1.2.2 and, thanks to

[figure: π]  ≡  [figure: ^{π  π}]

implies the equivalence^{20}

[figure: \underline{B}; ⌢ /; Φ_{+} \{; ⌢ /; π; π; Φ_{+} \{; π; π; π; \overline{B}; ⌢ /; ⌢ /]  ≡  _{π  .}^{}

Therefore:

If the auxiliary 4-qubit state

Φ_{B}^{(4)} =^{def} (ˆ1 ⊗ CPHASE ⊗ ˆ1)(Φ_{+} ⊗ Φ_{+})

is available then the (deterministic) CNOT gate can be replaced by appropriately modified double quantum teleportation.

DRAFT, June 26, 2009

20See also (Brukner et al., 2003) in this connection.

If Φ_{B}^{(4)} can be stored for later use^{21} then it is sufficient to have a nondeterministic gate producing Φ_{B}^{(4)} with nonzero probability from standard input. One such possibility, obviously, is to replace the deterministic CNOT gates in the above characterization of Φ_{B}^{(4)} by nondeterministic ones, if the latter are available.

In principle (Kim et al., 2001) , (deterministic) teleportation is possible by exploiting sum frequency generation of both type I and type II for a complete Bell measurement (see, e.g., Section ref*nlqo-S-TWM of (Lücke, nlqo) for the explanation of sum frequency generation). However, also indeterministic teleportation by only partial Bell measurement — as, e.g. sketched in Figure 3.6 and explained by Exercise 14 — may be useful for improving the probability of success, at least , of an indeterministic CPHASE gate.

....................................................................  ....................................................................  .............................................................................. ............................................................ .......... .......... .......... . ............................................................ .......... .......... .......... . .......... .......... .......... .......... ................................................... .......... .......... .......... .......... ................................................... ................................. .................................. ................................. .................................. .................................

Figure 3.6: Partial Bell measurement.

Exercise 14 Consider a Hadamard beam splitter for the orthogonal modes â_{1} â_{2} , i.a a beam splitter acting as

â_{1} ↦ \frac{1}{\sqrt{2}} (â_{1} + â_{2}) , â_{2} ↦ \frac{1}{\sqrt{2}} (â_{1} − â_{2}) .

Show that the beam splitter, applied to the Bell states

Φ_{±} = \frac{1}{\sqrt{2}} (Ω ± â_{1}^{†} â_{2}^{†} Ω) , Ψ^{±} = \frac{1}{\sqrt{2}} (â_{1}^{†} ± â_{2}^{†}) Ω

in the corresponding Fock representation as sketched in Figure 3.6, acts as follows:

Φ_{±} ⟼ \frac{1}{\sqrt{2}}(|0, 0⟩^{F} ± \frac{1}{2} (|2, 0⟩^{F} − |0, 2⟩^{F})) ,
Ψ_{01} = Ψ_{+} ⟼ |1, 0⟩^{F} ,
Ψ_{11} = Ψ_{−} ⟼ |0, 1⟩^{F} .

According to (3.9) and Exercise 14 a so-called CZ_{\frac{1}{4}} gate, i.e. an indeterministic CPHASE gate working with probability of success 1/4 , can be implemented in the Fock realization as sketched in Figure 3.7 This gate succeeds if the (photon number

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.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; π; π; π; Ψ_{01} {; Ψ_{01} {; D′_{1}; D′_{2}; D_{2}; D_{1}]

Figure 3.7: Implementation of a CZ_{\frac{1}{4}} gate

resolving) detectors D_{1}, …, D′_{2} indicate that the common state of the first two qubits as well as that of the last two qubits is a single-photon state.^{22} The ancillary states

Ψ_{01} = \frac{|0, 1⟩ + |1, 0⟩}{\sqrt{2}} = \frac{â_{1}^{†} + â_{2}^{†}}{\sqrt{2}} Ω

may be easily prepared using a single-photon source and a Hadamard beam splitter:^{23}

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Near-Deterministic Quantum Gates

The construction sketched in Figure 3.7 can be generalized in the following way using 4n ancillary qubits^{24} instead of only 4 (Knill et al., 2000; Knill et al., 2001):

DRAFT, June 26, 2009

21See (Pittman and Franson, 2002) in this connection.

22The dotted vertical lines indicate that the effect of the corresponding CPHASE gates on the output qubits can be achieved via LOCC (Local Operations and Classical Communication).

23Note that the single-photon state Ψ_{01} is entangled when interpreted as Fock realization of a 2-qubit state.

24The case n = 2 is of special interest (Nielsen, 2004).

Let â_{0}, …, â_{2n} be pairwise orthonormal modes describing photons with (almost) sharp momenta and prepare the first 2n ancillary qubits in the state

Φ_{tele}^{(n)} =^{def} \frac{1}{\sqrt{n + 1}} ∑_{j=0}^{n} (∏_{ν=1}^{j} â_{ν}^{†})( ∏_{μ=n+j+1}^{2n} â_{μ}^{†}) Ω ,

(instead of Ψ_{01}). Let F̂_{(n)} be the linear operator on H_{â_{0},...,â_{2n}} characterized by F̂_{(n)} Ω = Ω and

F̂_{(n)} â F̂_{(n)}^{−1} = \{ \begin{matrix}â & for â ⊥ \{â_{0}, …, â_{n}\} , \\ \frac{1}{\sqrt{n+1}} ∑_{j=0}^{n} e^{−i k \frac{2π}{n+1} j} â_{j} & for â = â_{k} , k ∈ \{0, …, n\}, \end{matrix}  (3.10)

where — as in Section 1.2.4 of (Lücke, nlqo) — we denote by H_{â_{0},...,â_{2n}} the smallest closed subspace of H_{field} that contains Ω and is invariant under â_{0}^{†}, …, â_{2n}^{†}. Then, as explained in connection with the single-photon simulation of n-qubit systems, the transformation

â_{ν} ↦ F̂_{(n)} â_{ν} F̂_{(n)}^{−1} ∀ ν ∈ {0, …, 2n}  (3.11)

can be implemented using by linear optics.

Remark: For n + 1 = 2^{m} and^{25}

|k⟩_{m} = â_{k}^{†} Ω ∀ k ∈ {0, …, 2^{m} − 1}

the corresponding state transformation

|k⟩_{m} ⟼ F̂_{(n)} |k⟩_{m} = \frac{1}{\sqrt{n + 1}} ∑_{j=0}^{n} e^{+i k \frac{2π}{n+1} j} |j⟩_{m} ∀ k ∈ \{0, …, 2^{m} − 1\}

simulates the m-qubit quantum Fourier transform.

Exercise 15 Using the Campbell-Hausdorff formula^{26}

eÂB̂e^{−Â} = exp(adÂ) B̂ ,  (3.12)

where

adÂ Ĉ =^{def} [Â, Ĉ]_{−} ,

show the following:

a)

e^{−i â_{j}^{†} â_{j} φ} â_{k} e^{+i â_{j}^{†} â_{j} φ} = e^{+i φ δ_{j,k}} â_{k} ∀ j, j ∈ {0, …, 2n} , φ ∈ ℝ .

DRAFT, June 26, 2009

25Recall (1.12).

26Compare Footnote 50.

3.1. QUANTUM OPTICAL IMPLEMENTATION 67

b)

\begin{matrix}e^{−i(â_{1}^{†} â_{2}+â_{2}^{†} â_{1})θ} â_{1} e^{+i(â_{1}^{†} â_{2}+â_{2}^{†}â_{1})θ} & = & cos θ â_{1} + i sin θ â_{2} \\ e^{−i(â_{1}^{†} â_{2}+â_{2}^{†} â_{1})θ} â_{2} e^{+i(â_{1}^{†} â_{2}+â_{2}^{†}â_{1})θ} & = & i sin θ â_{1} + cos θ â_{2} \end{matrix} \} ∀ θ ∈ ℝ .

c) F̂_{(n)} is a unitary operator on H_{â_{0},...,â_{2n}}.

Since

\begin{aligned}∏_{ν=1}^{j} (F̂_{(n)} â_{ν} F̂_{(n)}^{−1}) & =_{(3.10)} & ∏_{ν=1}^{j} (∑_{l_{ν}=0}^{n} e^{+i ν \frac{2π}{n+1} l_{ν}} â_{l_{ν}}^{†}) \\ & = & ∑_{l_{1},...,l_{j}=0}^{n} ∏_{ν=1}^{j} (e^{+i ν \frac{2π}{n+1} l_{ν}} â_{l_{ν}}^{†}) ∀ j ∈ \{1, …, n\} \end{aligned}

and

∏_{ν=0}^{j} (F̂_{(n)} â_{ν} F̂_{(n)}^{−1}) = ∑_{l_{0},...,l_{j}=0}^{n} ∏_{ν=0}^{j} e^{+i ν \frac{2π}{n+1} l_{ν}} â_{l_{ν}}^{†}
= ∑_{l_{1},...,l_{j+1}=0}^{n} ∏_{ν=1}^{j+1} e^{−i \frac{2π}{n+1} l_{ν}} e^{+i ν \frac{2π}{n+1} l_{ν}} â_{l_{ν}}^{†} ∀ j ∈ \{0, …, n − 1\}
= ∑_{l′_{0},...,l′_{j}=0}^{n} ∏_{ν=1}^{j+1} e^{−i \frac{2π}{n+1} l′_{ν}} e^{+i ν \frac{2π}{n+1} l′_{ν}} â_{l′_{ν}}^{†} ∀ j ∈ \{0, …, n − 1\} ,

we get F̂_{(n)} (α ˆ1 + β â_{0}^{†}) Φ_{tele}^{(n)}

= \frac{1}{\sqrt{n + 1}} (α ( ∏_{ν=n+1}^{2n} â_{ν}^{†})Ω + β F̂_{(n)}(∏_{ν=0}^{n} â_{ν}^{†})Ω)

+α ∑_{N_{0},...,N_{n}=0_{0<N_{0}+...+Nn≤n}}^{n} λ_{N_{0},...,N_{n}} (∏_{j=0}^{n} (â_{j}^{†})^{N_{j}}) ( ∏_{ν=n+1+N_{0}+...+N_{n}}^{2n} â_{ν}^{†}) Ω

+β ∑_{N_{0},...,N_{n}=0_{0<N_{0}+...+Nn≤n}}^{n} λ_{N_{0},...,N_{n}} (∏_{j=0}^{n} (e^{−i \frac{2π}{n+1} j}â_{j}^{†})^{N_{j}}) ( ∏_{ν=n+N_{0}+...+N_{n}}^{2n} â_{ν}^{†}) Ω

(3.13)

with suitable λ_{N_{0},...,N_{n}} ∈ ℝ, not depending on α, β ∈ ℂ. Now, if the number N_{ν} of photons of mode â_{ν} in the state F̂_{(n)} (α ˆ1 + β â_{0}^{†}) F_{(n)}Φ_{tele}^{(n)} is checked for all ν ∈ {0, …, n} by projective measurement then:

• If

0 < N_{0} + … + N_{n} ≤ n  (3.14)

then the state is projected onto

Â_{N_{0},...,N_{n}} (α ˆ1 + β (∏_{j=1}^{n} e^{−i \frac{2π}{n+1} j N_{j}}) â_{n+N_{0}+...+N_{n}}^{†}) B̂_{N_{0},...,N_{n}} Ω ,

where

\begin{matrix}Â_{N_{0},...,N_{n}} & =^{def} & λ_{N_{0},...,N_{n}} ∏_{ν=0}^{n} (â_{ν}^{†})^{N_{ν}} , \\ B̂_{N_{0},...,N_{n}} & =^{def} & ∏_{ν=n+1+N_{0}+...+N_{n}}^{2n} â_{ν}^{†} . \end{matrix}

• The probability for (3.14) — thanks to unitarity of F̂_{(n)} — is \frac{n}{n + 1} .

Since the phase factor ∏_{j=1}^{n} e^{−i \frac{2π}{n+1} j N_{j}} is fixed by the measurement result, this shows:

With probability arbitrarily close to 1 the state (α ˆ1 + β â_{0}^{†}) Ω with un-

known α , β ∈ ℂ can be teleported^{27} into (α ˆ1 + β â_{n+N_{0}+...+N_{n}}^{†}) Ω with random N_{0}, …, N_{n} fulfilling (3.14) resulting from — typically destructive — measurement of the corresponding photon numbers in the state F̂_{(n)} (α ˆ1 + β â_{0}^{†}) F_{(n)}Φ_{tele}^{(n)} .

In order to implement a CZ_{n^{2}/(n+1)^{2}} gate, i.e. an indeterministic CPHASE gate working with probability of success n^{2}/(n + 1)^{2} , 4n + 2 pairwise orthogonal modes â_{0}, …, â_{2n}, b̂_{0}, …, b̂_{2n} are needed. â_{0} resp. b̂_{0} is used for the Fock realization of the first resp. second input qubit of the gate. The ancillary Fock qubits corresponding to the modes â_{1}, …, â_{2n}, b̂_{1}, …, b̂_{2n} have to be prepared in the 4n-qubit state

Ψ̌_{anc}^{(n)} =^{def} ∏_{ν,μ=n+1}^{2n} Ŝ_{π}^{(ν,μ)} Φ_{tele}^{(n)} ⊗ Ψ_{tele}^{(n)} ,

where Ψ_{tele}^{(n)} is defined similarly to Φ_{tele}^{(n)} with modes â_{ν} replaced by modes b̂_{ν} and Ŝ_{π}^{(ν,μ)} means action of the CPHASE gate on the pair of Fock qubits corresponding to the modes â_{ν}, b̂_{μ}. Defining F̂′_{(n)} resp. Â′_{N_{0},...,N_{n}} resp. B̂′_{N_{0},...,N_{n}} similarly to F̂_{(n)} resp. Â_{N_{0},...,N_{n}} resp. B̂_{N_{0},...,N_{n}} with â-modes replaced by b̂-modes and

t̂_{α,β}(N) =^{def} (α ˆ1 + β (∏_{j=1}^{n} e^{−i \frac{2π}{n+1} j N_{j}}) â_{n+N_{0}+...+N_{n}}^{†}) ,

t̂′_{α,β}(N) =^{def} (α ˆ1 + β (∏_{j=1}^{n} e^{−i \frac{2π}{n+1} j N_{j}}) b̂_{n+N_{0}+...+N_{n}}^{†})

(3.15)

DRAFT, June 26, 2009

27See (Fattal et al., 2003) for an experimental demonstration of the basic version.

we get

F̂_{(n)} F̂′_{(n)} ((α ˆ1 + β â_{0}^{†})(α′ ˆ1 + β′ b̂_{0}^{†}) Ψ̌_{anc}^{(n)})
= ∏_{ν,μ=n+1}^{2n} Ŝ_{π}^{(ν,μ)} (F̂_{(n)} (α ˆ1 + β â_{0}^{†}) Φ_{tele}^{(n)}) ⊗ (F̂′_{(n)} (α′ ˆ1 + β′ b̂_{0}^{†}) Ψ_{tele}^{(n)})

and hence, by (3.13):

F̂_{(n)} F̂′_{(n)} ((α ˆ1 + β â_{0}^{†})(α′ ˆ1 + β′ b̂_{0}^{†}) Ψ̌_{anc}^{(n)})

= ∏_{ν,μ=n+1}^{2n} Ŝ_{π}^{(ν,μ)} ( (\frac{α}{\sqrt{n + 1}} ( ∏_{ν=n+1}^{2n} â_{ν}^{†})Ω + \frac{β}{\sqrt{n + 1}} F̂_{(n)}(∏_{ν=0}^{n} â_{ν}^{†})Ω

+ ∑_{N_{0},...,N_{n}=0_{0<N_{0}+...+Nn≤n}}^{n} λ_{N_{0},...,N_{n}} Â_{N_{0},...,N_{n}} t̂_{α,β}(N) B̂_{N_{0},...,N_{n}} Ω)

⊗ (\frac{α′}{\sqrt{n + 1}} ( ∏_{ν=n+1}^{2n} b̂_{ν}^{†})Ω + \frac{β′}{\sqrt{n + 1}} F̂′_{(n)}(∏_{ν=0}^{n} b̂_{ν}^{†})Ω

+ ∑_{N′_{0},...,N′_{n}=0_{0<N′_{0}+...+N′_{n}≤n}}^{n} λ_{N′_{0},...,N′_{n}} Â′_{N′_{0},...,N′_{n}} t̂′_{α′,β′}(N′) B̂′_{N′_{0},...,N′_{n}} Ω) )

Therefore, if the numbers N_{0}, …, N_{n}, N′_{0}, …, N′_{n} of photons in the modes â_{0}, …, â_{n}, b̂_{0}, …, b̂_{n} are checked for the state F̂_{(n)} F̂′_{(n)} ((α ˆ1 + β â_{0}^{†})(α′ ˆ1 + β′ b̂_{0}^{†}) Ψ̌_{anc}^{(n)}) by projective measurement then:

• If

0 < N_{0} + … + N_{n} ≤ n and 0 < N′_{0} + … + N′_{n} ≤ n  (3.16)

then the state is projected onto a state with the factor

Ψ_{out} =^{def} ∏_{ν,μ=n+1}^{2n} Ŝ_{π}^{(ν,μ)} t_{α,β}(N) t′_{α′,β′}(N′) Ω
= Ŝ_{π}^{(n+N_{0}+...+N_{n},n+N′_{0}+...+N′_{n})} t_{α,β}(N) t′_{α′,β′}(N′) Ω

which may be easily transformed, using only linear optical components, into the desired output

Ŝ_{π}^{(0,0)} (α ˆ1 + β â^{†})(α′ ˆ1 + β′ b̂^{†}) Ω .

• The probability for (3.16) is n^{2}/ (n + 1)^{2} .

Especially for n = 2 , according to (Knill et al., 2001, Supplementary Information, Fig. 4), the ancillary state Φ_{tele}^{(n)} can be prepared (indeterministically) using only linear optics as sketched in Figure 3.8.

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+45^{∘}  −45^{∘} NS_{i} NS_{i} 0^{∘} 0^{∘} ϑ  0^{∘} +45^{∘} 90^{∘}  −45^{∘} 90^{∘}  123 4

Figure 3.8: Preparation of Φ_{tele}^{(2)}

3.2 Measurement-Based Schemes for Quantum Computation^{28}

The implementation of near deterministic CPHASE gates just described shows some interesting deviations from the conventional network model:

• Many ancillary qubits are used in addition to the input-qubits.

• The total state of all the qubits is suitably prepared using easily implementable deterministic gates and indeterministic gates, the latter acting only on the ancillary qubits.

• Then the desired output state can be transported onto the output-qubits using only (photon number) measurements and applying some easily performable final correction depending on the measurement results.

Meanwhile it turned out that for every n-qubit gate (n ∈ ℕ) and standard input state the corresponding output state can be efficiently produced,^{29} e.g., as follows (Childs et al., 2005; Walther et al., 2005; Nielsen, 2005):

1. Prepare a n×m-qubit cluster state^{30} with sufficiently large m in the following way:

DRAFT, June 26, 2009

28See also (Browne and Rudolph, 2005; Rudolph and Virmani, 2005; Varnava et al., 2006; Raussendorf, 2005) and (Lim et al., 2005).

29Concerning the preparation of explitly known states see (Kaye and Mosca, 2004).

30We use the notion cluster state in a more general sense than originally introduced in (Raussendorf and Briegel, 2001; Raussendorf, 2003).

3.2. MEASUREMENT-BASED QUANTUM COMPUTATION 71

(a) Prepare n × m qubits in the states

|+⟩_{νμ} =^{def} \frac{|0⟩_{νμ} + |1⟩_{νμ}}{\sqrt{2}} , (ν, μ) ∈ \{1, … n\} × \{1, … m\} ,

and imagine them arranged in a n × m-matrix scheme:

\begin{matrix}^{|+⟩_{11}} & ^{|+⟩_{12}} & ⋯ & ^{|+⟩_{1m}} \\ ^{|+⟩_{21}} & ^{|+⟩_{22}} & ⋯ & ^{|+⟩_{2m}} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ ^{|+⟩_{n1}} & ^{|+⟩_{n2}} & ⋯ & ^{|+⟩_{nm}} \end{matrix}

(b) Join all horizontally neighbouring qubit states and certain vertically neighbouring qubit states, depending on the effective network action desired, by lines respecting the rule that every state attached to a vertical line should have only one neighbour attached to a vertical line. Writing

_{νμ} instead of |+⟩_{νμ}

we may get, e.g.

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as an example for n = 3 , m = 5 .

(c) For every (horizontal or vertical) line (bond) apply a CPHASE gate to the pair of qubits connected by the line.

2. Once the cluster state is prepared certain projective single qubit measurements are performed on the qubits corresponding to the first row (qubits 11, …, n1).

3. Once the qubits of the ν-th column have been tested certain projective single qubit measurements, depending on the outcome of the previous measurements^{31} (and the final output desired), are performed on the qubits corresponding to the (ν + 1)-th column.

4. When the measurements on the m − 1-th column are performed the n-qubit system corresponding to the m-th column is left in the desired output state up to known single-qubit transformations depending on the results of the previous measurements.

5. The deviation from the desired output state may be either corrected or may be taken account of by appropriate change of the computational basis.

DRAFT, June 26, 2009

31Measurements the outcome of which determine the choice of subsequent measurements are called feed-forwardable.

A fascinating aspect of such measurement-based schemes for quantum computation is that the universal CPHASE gate is needed only for preparing the cluster state and for this purpose an indeterministic implementation of the phase gate is sufficient (Nielsen, 2004; Nielsen and Dawson, 2004; Chen et al., 2006):^{32}

Every cluster state can be prepared by successively applying single-bond and/or double-bond operations:

\underline{I. Single-bond operations:} E.g. for the case sketched in Figure 3.9 the two teleportations (measurements of N_{0}, N_{1}, N_{2} resp. N′_{0}, N′_{1}, N′_{2}) are attempted one after the other.

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Figure 3.9: Trying to add qubit C with a single bond.

The teleportation that does not affect A is tried first. If it does not succeed one has to restart with a newly prepared qubit C. If it does succeed then the second teleportation is tried. If the latter also succeeds C is added with a bond to A. If it does not succeed then a projective singlequbit measurement w.r.t. the computational basis has been performed on A and this qubit has to be removed from the cluster and single-qubit correction have to be performed on the qubits connected to A (for the sketched special case only one) depending on the measurement result indicated by the corresponding detectors. As soon as the size of the cluster has been changed by this procedure we say that a single-bond operation has been performed. Thus:

A single-bond operation adds resp. removes a qubit with probability 2/3 resp. 1/3.

\underline{II. Double-bond operations:} E.g. for the case sketched in Figure 3.10 one first try to connect D to C by a single-bond operation. If this adds D to the cluster we apply CZ_{4}/9 to the pair B, D. If the corresponding teleportation not affecting B fails then D has to be removed from the cluster and we have to restart with a newly prepared qubit D. If the first teleportation succeeds also the second teleportation is tried. If the latter succeeds D is connected to both C and B if not then B and D

DRAFT, June 26, 2009

32Moreover, such schemes circumvent the problem of programmable deterministic quantum gate arrays (Nielsen and Chuang, 1997).

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Figure 3.10: Trying to add qubit D with two bonds.

have to be removed from the cluster. As soon as either C or B has been removed or D has been connected to both C and D we say that a double-bond operation has been performed. Thus:

A double-bond operation adds resp. removes a qubit with probability \frac{2}{3} · \frac{2}{3} resp. \frac{1}{3} + \frac{2}{3} · \frac{1}{3} .

In order to create the cluster, after every double-bond operation that removed a qubit a single-bond operation can be applied. Then:

On average, 2N successive operations add at least

⌊(\frac{2}{3} − \frac{1}{3}) N + (\frac{2}{3} · \frac{2}{3} − \frac{2}{3} · \frac{1}{3}) N⌋ = ⌊\frac{2N}{9} ⌋

qubits to the cluster.

Easier implementable indeterministic CPHASE gates^{33} with lower probabilities of success are sufficient if the cluster is built from microclusters in which a single qubits are connected to several dangling qubits in order to allow for multiple gluing attemps.^{34}

3.3 Cold Trapped Ions^{35}

Of all the proposed technologies for quantum information processing devices, arguably one of the most promising and certainly one of the most popular is trapped ions.

(James, 2000)

DRAFT, June 26, 2009

33See, e.g., (Gasparoni et al., 2004; Zhao et al., 2005).

34See end of (Nielsen, 2004).

35See (Bužek and Šašura, 2002; Ghosh, 1995; Wunderlich and Balzer, 2003).

3.3.1 General Considerations

One of the earliest proposals to fulfill DiVincenzo’s requirements for quantum computation is the following:

1. • Qubits are identified with ions of some specified kind being in a superposition of two specified (at least) metastable energy eigenstates |g⟩ resp. |e⟩ representing the computational basis states |0⟩ resp. |1⟩. Typically, |g⟩ is the ground state.

• The ions are bound to specified places inside an ion trap and their collective oscillation is cooled to the quantum mechanical ground state.

2. The states |g_{1}, …, g_{n}⟩ ≡ |0, …, 0⟩ may be prepared by applying Laser radiation tuned to the transition of |e⟩ into a rapidly decaying higher energy eigenstate.

3. The decoherence time is of the order 10^{−1} seconds while the duration of gate operations is of the order 10^{−14} seconds.^{36}

4. • 1-qubit rotations are implemented by laser pulses — of appropriate duration and phase — tuned to the transition between |g⟩ and |e⟩.

• The controlled sign gate Λ_{1}\binom{}{Ŝ_{π}} is implemented by a suitable sequence of laser pulses exploiting one of the collective translational modes as data bus in the following way:

(i) A first pulse on the control qubit (Ion j_{1}) acts according to

|g_{j_{1}}⟩|0⟩_{osc} ⟼  |g_{j_{1}}⟩|0⟩_{osc} ,
|e_{j_{1}}⟩|0⟩_{osc} ⟼ −i |g_{j_{1}}⟩|1⟩_{osc} . (3.17)

(ii) A second laser pulse on the target qubit (Ion j_{2}) — tuned to the transition of |g⟩ into an exited energy eigenstate \left|{ \atop ě}\right⟩ different from |e⟩ — acts for b ∈ {0, 1} according to

|g_{j_{2}}⟩|b⟩_{osc} ⟼ (−1)^{b} |g_{j_{2}}⟩|b⟩_{osc} ,
|e_{j_{2}}⟩|b⟩_{osc} ⟼  |e_{j_{2}}⟩|b⟩_{osc} . (3.18)

(iii) A third pulse of the same type as used in (i) acts according to

|g_{j_{1}}⟩|0⟩_{osc} ⟼  |g_{j_{1}}⟩|0⟩_{osc} ,
|g_{j_{1}}⟩|1⟩_{osc} ⟼ −i |e_{j_{1}}⟩|0⟩_{osc} . (3.19)

These laser pulses have no effect if the control qubit (Ion j_{1}) is originally in the state |g_{j_{1}}⟩. On the other hand, if the control qubit (Ion j_{1}) is originally in the state |e_{j_{1}}⟩ then the action is as follows:

DRAFT, June 26, 2009

36See, e.g. (Nielsen and Chuang, 2001, Fig. 7.1) for a comparison with other implementations.

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 x y z

Figure 3.11: A linear Paul trap

– The first pulse transfers this information into the data bus by exciting it to |1⟩_{osc} and applies −i σ̂_{1} to the control qubit.

– Then the second pulse multiplies the |g_{j_{2}}⟩-component of the target qubit’s state by −1.

– Finally, the third pulse returns the data bus into its ground state and acts on the |g_{j_{2}}⟩-component of the control qubit’s state by −i σ̂_{1} , once more.

Obviously, the resulting action is that of a Λ_{1}\binom{}{Ŝ_{π}} gate.

5. Measurement of the final state of the computation may be done by irradiating by continuous laser radiation tuned to the transition of |g⟩ into a rapidly decaying higher energy eigenstate. Then only those ions ‘found’ in the ground state show strong fluorescence.

All this will be explained in more detail — for the special case^{37} of ^{40}Ca^{+} ions in a linear Paul trap — in the subsequent sections.

3.3.2 Linear Paul Trap

The linear Paul trap used by the Innsbruck group is sketched in Figure 3.11. There are four rod electrodes, namely along

C_{1} =^{def} \{(x, y, z) ∈ ℝ : y = −r_{0} , z = 0 , x ∈ \{−\frac{x_{0}}{2} , +\frac{x_{0}}{2} \}\} ,
C_{2} =^{def} \{(x, y, z) ∈ ℝ : z = +r_{0} , y = 0 , x ∈ \{−\frac{x_{0}}{2} , +\frac{x_{0}}{2} \}\} ,
C_{3} =^{def} \{(x, y, z) ∈ ℝ : y = +r_{0} , z = 0 , x ∈ \{−\frac{x_{0}}{2} , +\frac{x_{0}}{2} \}\} ,
C_{4} =^{def} \{(x, y, z) ∈ ℝ : z = −r_{0} , y = 0 , x ∈ \{−\frac{x_{0}}{2} , +\frac{x_{0}}{2} \}\} ,

DRAFT, June 26, 2009

  37See http://heart-c704.uibk.ac.at/quantumcomputation.html

and two ring electrodes (end caps). The electric potential of the rod electrodes along C_{1} and C_{3} is^{38} Φ(t) while the rod electrodes C_{2} and C_{4} are grounded. Both ring electrodes have the constant electric potential Φ_{ring} > 0 .

Trapping in Radial direction

If the rod electrodes were infinitely long and infinitely thin then they would contribute the electric potential

Φ_{ideal}(x, y, z, t) = \frac{Φ_{0}}{2} (1 + \frac{y^{2} − z^{2}}{r_{0}^{2}} ) if Φ(t) = Φ_{0} = const

since this fulfills the Laplace equation as well as the boundary conditions along the rods. If

Φ(t) = Φ_{0} cos(Ω t) , Ω ≈ 16 − 18 MHz (radio frequency) ,

then the quasi stationary approximation

Φ_{⊥}(x, y, z, t) ≈ \frac{Φ_{0} cos(Ω t)}{2} (1 + \frac{y^{2} − z^{2}}{r_{0}^{2}} )
= Φ_{0} cos(Ω t) \frac{y^{2} − z^{2}}{2 r_{0}^{2}} +\frac{1}{2} Φ_{0} cos(Ω t)  (3.20)

can be used near the center of the trap — also for finite rods, if sufficiently long.

The evolution equations of a particle of mass m and electric charge q in the electric potential (3.20) are^{39}

ÿ(t) + \frac{q Φ_{0}}{m r_{0}^{2}} cos(Ω t) y(t) = 0 ,  (3.21)
z̈(t) − \frac{q Φ_{0}}{m r_{0}^{2}} cos(Ω t) z(t) = 0 ,  (3.22)
ẍ(t) = 0 .

With

b =^{def} \frac{2 q Φ_{0}}{m r_{0}^{2} Ω^{2}} , ζ =^{def} \frac{Ω t}{2}  (3.23)

equations (3.21) and (3.22) become equivalent to the special cases

(\frac{d}{dζ} )^{2} y(ζ) + 2 b cos(2 ζ) y(ζ) = 0 ,  (3.24)
(\frac{d}{dζ} )^{2} z(ζ) − 2 b cos(2 ζ) z(ζ) = 0  (3.25)

DRAFT, June 26, 2009

38Note that static electric potentials cannot have minima in regions free of electric charge.

39In the Innsbruck experiment: Φ_{0} ≈ 300 − 800 V , \frac{Ω}{2π} , r_{0} ≈ 1.2 mm .

of the Mathieu differential equation

(\frac{d}{dζ} )^{2} y(ζ) + (a + 2 b cos(2 ζ))y(ζ) = 0 .  (3.26)

The general solution of (3.26) is^{40}

y(ζ) = ∑_{n∈ℤ}C_{2n} (λ_{+} e^{+μ ζ} e^{+2i n ζ} + λ_{−} e^{−μ ζ} e^{−2i n ζ}) ,

with general integration constants λ_{±} (to be adapted to the initial conditions) and certain constants C_{2n} and μ depending an a, b. We are interested in stable solutions, only, and therefore have to require

μ = i β , β ∈ ℝ .

Then

y(ζ) = ∑_{n∈ℤ}C_{2n} (λ_{1} cos((2n + β) ζ) + λ_{2} sin((2n + β) ζ)) ,  (3.27)

where

λ_{1} =^{def} λ_{+} + λ_{−} , λ_{2} = i (λ_{+} − λ_{−}) .

Inserting this into (3.26) gives the recursion formula

C_{2n+2} − \frac{a − (2n + β)}{b} C_{2n} + C_{2n−2} = 0 .  (3.28)

Defining

G_{2n} =^{def} \frac{C_{2n}}{C_{0}} , A =^{def} λ_{1} C_{0} , B =^{def} λ_{2} C_{0}

and exploiting the well-known theorems for sin and cos we can rewrite (3.27) in the equivalent form

y(ζ) = Y^{+}(ζ) + δ_{y}(ζ) ,

where:

Y^{±}(ζ) =^{def} A cos(β ζ) ± B sin(β ζ) ,
δ_{y}(ζ) =^{def} ∑_{n=1}^{∞} (Y^{+}(ζ) (G_{2n} + G_{−2n}) cos(2nβζ) − Y^{−}(ζ) (G_{2n} − G_{−2n}) sin(2nβζ)) .

For sufficiently small^{41} |b| one may show:

1. Solution of (3.28) gives a stable solution (3.27).

2. The micromotion δ_{y}(ζ) is negligible.

3. The corresponding statements hold also for z(ζ) .

In the following the micromotion will be neglected.

DRAFT, June 26, 2009

40See (Ghosh, 1995, Section 2.3).

41Note that b ∼ q/m.

Trapping in Axial Direction

The positive potential of the ring electrodes in Figure 3.11 servers to confine positive ions also in axial direction. The additional (time-independent) electric potential contributed by these electrodes is

Φ_{∥}(x, y, z) ≈ ξ Φ_{ring} (\frac{x}{x_{0}} )^{2} for x/x_{0} ≪ 1 ,

where ξ is some geometric factor of order 1 characterizing the contribution of Φ_{ring} on the center of the ring electrodes.

In the Innsbruck experiment we have

Φ_{ring} ≈ 2000 V , x_{0} ≈ 5 mm

and

q Φ_{∥}(x, y, z) ≈ \frac{m}{2} ω_{x}^{2} x^{2}

with

\frac{ω_{x}}{2π} ≈ 500 − 700 kHz for ^{40}Ca^{+} .

Since ω_{x} is definitely smaller than β ζ ≈ 1, 4 − 2 MHz , in this experiment, it is sufficient — for what follows — to consider only the motion along the x-axis in the potential contributed by the ring electrodes and the repulsive Coulomb potential of the ions. The total mechanical potential for N identical ions then is (approximately)

V̌ (x_{1}, …, x_{N}) = \frac{m}{2} ω_{x}^{2} ∑_{j=1}^{N} x_{j}^{2} + \frac{q^{2}}{4π ϵ_{0}} ∑_{j,k=1_{j<k}}^{N} \frac{1}{|x_{j} − x_{k}|} .  (3.29)

Mean Values of the Ion Positions

For j ∈ {1, …, N} , let x̌_{j} be the mean value of the j-th ion’s x-coordinate. Obviously, for

x =^{def} (x_{1}, …, x_{N}) = x̌ =^{def} (x̌_{1}, …, x̌_{N})

the potential of the N-ion system has to be minimal:

(\frac{∂}{∂x_{j}} V̌ (x_{1}, …, x_{N}))_{|_{x=x̌}} = 0 ∀ j ∈ \{1, …, N\} .  (3.30)

Without loss of generality we may assume

x̌_{1} < x̌_{2} < … < x̌_{N} .  (3.31)

under this condition (3.30) is equivalent to

X_{j} − ∑_{k,j=1_{k<j}}^{N} \frac{1}{(X_{k} − X_{j})^{2}} + ∑_{k,j=1_{k>j}}^{N} \frac{1}{(X_{k} − X_{j})^{2}} = 0 ∀ j ∈ \{1, …, N\} ,  (3.32)

where^{42}

X_{j} =^{def} x̌_{j}/γ , γ =^{def} (\frac{q^{2}}{4πϵ_{0} m ω_{x}^{2}} )^{\frac{1}{3}} .  (3.33)

For N ≤ 3 the solutions are easily determined:

\begin{matrix}N = 1 : & X_{1} = 0 , \\ N = 2 : & X_{1} = − ^{3} \sqrt{\frac{1}{4}} , X_{2} = + ^{3} \sqrt{\frac{1}{4}} , \\ N = 3 : & X_{1} = − ^{3} \sqrt{\frac{5}{4}} , X_{2} = 0 , X_{3} = + ^{3} \sqrt{\frac{5}{4}} . \end{matrix}

For N > 3 the X_{j} have to be determined numerically.

Of course, the distance between the ions is minimal at the center of the trap. Numerical calculations show that

Δx̌_{min} ≈ \frac{2.018}{N^{0.559}} γ

(James, 1998). Therefore, in the Innsbruck experiment, the overlap of the ions’ wave functions is negligible^{43} and the ions are individually addressable by laser beams.

Collective Oscillations

Near its minimum the potential (3.29) may be approximated as

V̌ (x_{1}, …, x_{N}) ≈ V (q_{1}, …, q_{N}) =^{def} \frac{m}{2} ω_{x}^{2} ∑_{j,k=1}^{N} V_{jk} q_{j}q_{k} ,

where

q_{j} =^{def} x_{j} − x̌_{j} ∀ j ∈ {1, …, N} ,  (3.34)
V_{jk} =^{def} \frac{1}{m ω_{x}^{2}} (\frac{∂}{∂x_{j}} \frac{∂}{∂x_{k}} V (x_{1}, …, x_{N}))_{|_{x=x̌}} ∀ j, k ∈ \{1, …, N\} .

Explicitly, by (3.29) and (3.33), we have

V_{jk} = V_{kj} = \{ \begin{matrix}1 + ∑_{l=1_{l≠j}}^{N} \frac{2}{|X_{l} − X_{j}|^{3}} & for j = k , \\ −\frac{2}{|X_{k} − X_{j}|^{3}} & for j ≠ k. \end{matrix}  (3.35)

DRAFT, June 26, 2009

42For ^{40}Ca^{+} and \frac{ω_{x}}{2π} ≈ 700 kHz: γ ≈ 4.85 μm .

43Recall Footnote 17 of Chapter 1.

80 CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES The corresponding system of evolution equations is

q̈_{j}(t) + ω_{x}^{2} ∑_{k=1}^{n} V_{jk} q_{k}(t) = 0 .  (3.36)

Since the matrix (V_{jk}) is positive and symmetric, there is an orthonormal system of eigenvectors

C_{l} = ( C_{l1} ; ⋮ ; C_{lN} ) , l ∈ {1, …, N} ,

of this matrix in ℝ^{N} with positive eigenvalues:

( \begin{matrix}V_{11} & … & V_{1N} \\ ⋮ & & ⋮ \\ v_{N1} & … & V_{NN} \end{matrix} ) ( C_{l1} ; ⋮ ; C_{lN} ) = (\frac{ω̌_{l}}{ω_{x}} )^{2} ( C_{l1} ; ⋮ ; C_{lN} ) l ∈ \{1, …, N\} .  (3.37)

Hence, every solution of (3.36) is a superposition

q_{j}(t) = ∑_{l=1}^{N} (λ_{l}^{+} q_{j}^{(+l)}(t) + λ_{l}^{−} q_{j}^{(−l)}(t))  (3.38)

of the special collective motions (eigenmodes)

q_{j}^{(±l)}(t) =^{def} C_{lj} e^{∓i ω̌_{l} t} .  (3.39)

The first two eigenvectors may always be chosen as

C_{1} = \frac{1}{\sqrt{N}} ( 1 ⋮ 1 )  with ω̌_{1} = ω_{x} ,
C_{2} = \frac{1}{\sqrt{X_{1}^{2} + … + X_{N}^{2}}} ( X_{1} ; ⋮ ; X_{N} ) with ω̌_{2} = \sqrt{3} ω_{x} .

(3.40)

∑_{k=1}^{N} V_{jk} X_{k} =_{(3.35)} X_{j} + ∑_{j≠l=1}^{N} \frac{2 X_{l}}{|X_{l} − X_{j}|^{3}} − ∑_{j≠l=1}^{N} \frac{2 X_{j}}{|X_{l} − X_{j}|^{3}}
= X_{j} + 2 ∑_{j≠l=1}^{N} \frac{X_{j} − X_{l}}{|X_{l} − X_{j}|^{3}}
=_{(3.31)} X_{j} + ∑_{l=1_{l<j}}^{N} \frac{2}{|X_{l} − X_{j}|^{2}} − ∑_{l=1_{l>j}}^{N} \frac{2}{|X_{l} − X_{j}|^{2}}
=_{(3.32)} 3 X_{j} .

Proof: (3.35) directly implies ∑_{k=1}^{N} V_{jk} = 1 and, therefore, (3.37) for l = 1 with

ω̌_{1} = ω_{x}. On the other hand, we have

The latter implies (3.37) for l = 2 with ω̌_{2} = \sqrt{3} ω_{x}.

Then (q_{1}^{(1)}, …, q_{N}^{(1)}) is called the center of mass mode and (q_{1}^{(2)}, …, q_{N}^{(2)}) the breathing mode.

3.3.3 Implementing Quantum Gates by Laser Pulses

Quantization of the Collective Oscillations

Using the generalized coordinates

Q_{l}(t) = ℜ (λ_{l}^{+} e^{−i ω̌_{l} t} + λ_{l}^{−} e^{+i ω̌_{l} t})  (3.41)

we get

x_{j}(t) = x̌_{j} + ∑_{l=1}^{N} C_{lj} Q_{l}(t) ∀ j ∈ {1, …, N}  (3.42)

for the x-coordinates of the ions and

L = \frac{m}{2} ∑_{l=1}^{N} (Q̇_{l}^{2} − ω̌_{l}^{2} Q_{l}^{2})  (3.43)

for the Lagrangian corresponding to (3.36). With the canonically conjugate momenta

P_{j} =^{def} \frac{∂}{∂Q̇_{j}} L = m Q̇_{j}

we get the Hamiltonian

H(Q, P, t) = \frac{1}{2m} ∑_{l=1}^{N} P_{l}^{2} + \frac{m}{2} ∑_{l=1}^{N} ω̌_{j}^{2}Q_{l}^{2}

and standard quantization of the system amounts to selecting a (normalized) ground state (vector) |0⟩_{osc} and replacing the Q_{l}, P_{l} by operators

\begin{aligned}Q̂_{l} & = & \sqrt{\frac{ℏ}{2m ω̌_{l}}} (ˇâ_{l} + ˇ â_{l}^{†}) , \\ P̂_{l} & = & −i \sqrt{\frac{ℏ m ω̌_{l}}{2}} (ˇâ_{l} − ˇ â_{l}^{†}) , \end{aligned}  (3.44)

with annihilation operators ˇ â_{l} and creation operators ˇ â_{l}^{†} obeying the canonical commutation relations^{44}

[ˇâ_{l}, ˇ â_{k}]_{−} = 0 , [ˇâ_{l}, ˇ â_{k}^{†}]_{−} = δ_{lk}  (3.45)

DRAFT, June 26, 2009

44These commutation relations are easily seen to be equivalent to

[Q̂_{l}, Q̂_{k}]_{−} = 0 , [P̂_{l}, P̂_{k}]_{−} = 0 , [Q̂_{l}, P̂_{k}]_{−} = i ℏ δ_{lk} .

and the Fock condition

ˇâ_{l} |0⟩_{osc} = 0 ∀ l ∈ {1, …, N} .  (3.46)

The quantized Hamiltonian, then, is

Ĥ_{osc} = \frac{1}{2m} ∑_{l=1}^{N} P̂_{l}^{2} + \frac{m}{2} ∑_{l=1}^{N} ω̌_{j}^{2}Q̂_{l}^{2}

= ∑_{l=1}^{N} ℏ ω̌_{l} (ˇâ_{l}^{†} ˇ â_{l} + \frac{1}{2})  (3.47)

and a maximal orthonormal system of the state space (for the quantized oscillatory motion) is given by the Fock states

|n_{1}, …, n_{N}⟩_{osc} =^{def} \frac{1}{\sqrt{n_{1}! … n_{N}!}} ∏_{l=1}^{N} (ˇâ_{l}^{†})^{n_{l}} |0⟩_{osc} , n_{1}, …, n_{N} ∈ ℤ_{+} ,  (3.48)

fulfilling

ˇâ_{l}^{†} ˇ â_{l} |n_{1}, …, n_{N}⟩_{osc} = n_{l} |n_{1}, …, n_{N}⟩_{osc} .  (3.49)

The corresponding observables for the x-coordinates of the ions are

x̂_{j} =_{(3.42)} x̌_{j} ˆ1_{osc} + ∑_{l=1}^{N} C_{lj} Q̂_{l}(t)

=_{(3.44)} x̌_{j} ˆ1_{osc} + ∑_{l=1}^{N} \sqrt{\frac{ℏ}{2m ω̌_{l}}} C_{lj} (ˇâ_{l} + ˇ â_{l}^{†}) .  (3.50)

Of course, a more detailed description of the ions’ motion would require inclusion of the modes describing radial oscillations.

Laser-Ion Interaction

In the following we assume that all the ions are ^{40}Ca^{+} ions. Moreover we select a special mode ˇ â_{l_{0}} (typically l = 1 or 2) as data bus. All other modes are assumed cooled into their ground state and will not be included in the description. The Hamiltonian describing the collective motion of all ions the the internal state of the j-th ion not interacting with an external electromagnetic field then is

ℏ ω̌_{l_{0}} (ˇâ_{l_{0}}^{†} ˇ â_{l_{0}} + \frac{1}{2}) + \underbrace{E_{e_{j}} |e_{j}⟩⟨e_{j}| + E_{g_{j}} |g_{j}⟩⟨g_{j}|}_{= \frac{Ee_{j} −Egj}{2} (|e_{j}⟩⟨e_{j}|−|g_{j}⟩⟨g_{j}|)+ \frac{Ee_{j} +Egj}{2} \underbrace{(|e_{j}⟩⟨e_{j}| + |g_{j}⟩⟨g_{j}|)}_{=ˆ1_{int}}} ,

where E_{e_{j}} resp. E_{g_{j}} is the energy level of the qubit state |e_{j}⟩ resp. |g_{j}⟩. This Hamiltonian describes the same time evolution as

Ĥ_{0j} =^{def} −\frac{ℏ ω_{0j}}{2} σ̂_{3j} + ℏ ω̌_{l_{0}} ˇ â_{l_{0}}^{†} ˇ â_{l_{0}} ,  (3.51)

where

ω_{0j} =^{def} \frac{E_{e_{j}} − Egj}{2 ℏ} , σ̂_{3j} =^{def} |g_{j}⟩⟨g_{j}| − |e_{j}⟩⟨e_{j}| .  (3.52)

(3.52) will be used in the following. We assume the laser radiation to be strong enough for the exterior field formalism to be adequate (see, e.g., Sections 7.1.1 and 7.2.1 of (Lücke, nlqo) in this connection). Interaction with the classical laser radiation will be described in the dipole approximation^{45} by adding

V̂_{j} =^{def} −q_{el} r̂_{j} · E((x̂_{j}, 0, 0) t) ,

where

q_{el} =^{def} electric charge of the valence electron ,
r̂_{j} =^{def} \{ \begin{matrix}observable of the valence electron’s \\ position relative to the center of the ion , \end{matrix}
E(x, t) =^{def} external electric field at position x and time t .

We assume that the exterior field is of the form

E(x, t) = E_{0} (e^{−i(ω_{L} t−k_{L}·x+ϕ̌)} + e^{+i(ω_{L} t−k_{L}·x+ϕ̌)}) .  (3.53)

Replacing r̂_{j} by

ˆ1_{int} r̂_{j} ˆ1_{int} = (|g_{j}⟩⟨g_{j}| + |e_{j}⟩⟨e_{j}|) r̂_{j} (|g_{j}⟩⟨g_{j}| + |e_{j}⟩⟨e_{j}|)

= ⟨g_{j} | r̂_{j} e_{j}⟩ \overbrace{|g_{j}⟩⟨e_{j}|}^{σ̂_{j}^{−} =^{def}} + \overbrace{⟨g_{j} | r̂_{j} e_{j}⟩}^{r_{eg,j} =^{def}} \overbrace{|g_{j}⟩⟨e_{j}|}^{σ̂_{j}^{−} =^{def}}

we get

V̂_{j} = −q_{el} (r_{eg,j} σ̂_{j}^{+} + (r_{eg,j})^{†} σ̂_{j}^{−}) · E_{0}(e^{−i(ω_{L} t−η_{l_{0}j}(ˇâ_{l_{0}}+ˇâ_{l_{0}}^{†})+ϕ_{j})}
+ e^{+i(ω_{L} t−η_{l_{0}j}(ˇâ_{l_{0}}+ˇâ_{l_{0}}^{†})+ϕ_{j})}) ,

where

η_{l_{0}j} =^{def} k_{L}^{1} \sqrt{\frac{ℏ}{2m ω̌_{l_{0}}}} , ϕ_{j} =^{def} ϕ̌ − k_{L}^{1} x̌_{j} .

DRAFT, June 26, 2009

45In the quadrupole approximation we would have to add

V̂′_{j} =^{def} −q_{el} r̂_{j} · ((r̂_{j} · ∇_{r′}) E(x′, t))_{|_{r′=(x̂_{j} ,0,0)}} .

In the interaction picture (see, e.g., Section 7.1.1 of (Lücke, nlqo))the time evolution is determined by

V̂_{j}^{I} = e^{+ \frac{i}{ℏ} Ĥ_{0j} t}V̂_{j} e^{− \frac{i}{ℏ} Ĥ_{0j} t}

instead of Ĥ_{0j} + V̂_{j}. Using the Campbell-Hausdorff formula,^{46} here in the form

e^{+ \frac{i}{ℏ} Ĥ_{0j} t}V̂_{j} e^{− \frac{i}{ℏ} Ĥ_{0j} t} = exp (ad_{\frac{i}{ℏ} Ĥ_{0j} t}) V̂_{j} ,

and

[Ĥ_{0j}, ˇ â_{l_{0}}^{†}]_{−} = +ℏ ω̌_{l_{0}} ˇ â_{l_{0}}^{†} ,
[Ĥ_{0j}, ˇ â_{l_{0}}]_{−} = −ℏ ω̌_{l_{0}} ˇ â_{l_{0}} ,
[Ĥ_{0j}, σ̂_{j}^{±}]_{−} = \frac{ℏ ω_{0j}}{2} [σ̂_{3j}, σ̂_{j}^{±}]_{−}
= ±ℏ ω_{0j} σ̂_{j}^{±} ,

we get^{47}

V̂_{j}^{I} = −q_{el} (r_{eg,j} · E_{0} σ̂_{j}^{+} e^{+i ω_{0j} t} + H.c.) (e^{−i(Ω_{L} t−η_{l_{0}j}(ˇâ_{l_{0}} e^{−i ω̌_{l_{0}} t}+H.c.)+ϕ_{j})} + H.c.) .

Hence in the rotating wave approximation,^{48} i.e. if we neglect the contributions of the higher frequencies ± (ω_{L} + ω_{0j}) :

V̂_{j}^{I} ≈ λ_{j} \frac{ℏ}{2} σ̂_{j}^{+} e^{+i η_{l_{0}j}(ˇâ_{l_{0}} e^{−i ω̌_{l_{0}} t}+H.c.)} e^{−i(ω_{L}−ω_{0j})t} + H.c. ,

where^{49}

λ_{j} =^{def} −\frac{2}{ℏ} q_{el} r_{eg,j} · E_{0} e^{−i ϕ_{j}} .  (3.54)

Using the Baker-Hausdorff formula^{50}

[Â, [Â, B̂]_{−}]_{−} = [B̂, [Â, B̂]_{−}]_{−} = 0 ⟹ e^{Â+B̂} = e^{− \frac{1}{2} [Â,B̂]_{−}} eÂeB̂  (3.55)

DRAFT, June 26, 2009

46Recall Exercise 15.

47By ‘H.c.’ we denote the hermitian conjugate of the preceding term.

48See (Aniello et al., 2003) for some criticism of this approximation.

49In the quadrupole approximation (recall Footnote 45) we have to add

λ′_{j} =^{def} −\frac{2}{ℏ} q_{el} ⟨e_{j}|(r̂_{j} · E_{0})(r̂_{j} · k_{L}) g_{j}⟩ .

50For operators in finite dimensional vector spaces (3.55) may be proved as follows: Since e^{λÂ}B̂ e^{−λÂ} and exp(ad_{λÂ}) B̂ fulfill the same first order differential equation and initial condition (for λ = 0), the Campbell-Hausdorff formula (3.12) holds for arbitrary Â , B̂. Therefore, also

f_{1}(λ) =^{def} e^{λ Â} e^{λ B̂}

and

f_{2}(λ) =^{def} e^{λ(Â+B̂)+ \frac{λ^{2}}{2} [Â,B̂]_{−}}

fulfill the same first order differential equation and initial condition (for λ = 0), if the l.h.s. of (3.55) holds, and hence f_{1} = f_{2} .

3.3. COLD TRAPPED IONS 85 and (3.45) we get

e^{+i η_{l_{0}j}(ˇâ_{l_{0}} e^{−i ω̌_{l_{0}} t}+H.c.)} = e^{+i η_{l_{0}j} ˇ â_{l_{0}}^{†} e^{+i ω̌_{l_{0}} t}} e^{+i η_{l_{0}j} ˇ â_{l_{0}} e^{−i ω̌_{l_{0}} t}} e^{− \frac{1}{2} η_{l_{0}j}^{2}} .

Therefore, if

ω_{L} − ω_{0j} = k ω̌_{l_{0}} , k ∈ ℤ ,  (3.56)

then the rotating wave approximation becomes

V̂_{j}^{I} ≈ λ_{j} \frac{ℏ}{2} σ̂_{j}^{+} e^{− \frac{1}{2} η_{l_{0}j}^{2}} ∑_{μ,ν=0}^{∞} (i η_{l_{0}j})^{μ+ν} \frac{(ˇâ_{l_{0}}^{†})^{μ}}{μ!} \frac{(ˇâ_{l_{0}})^{ν}}{ν!} e^{+i (μ−ν−k) ω_{0j} t} + H.c.

For sufficiently small λ_{j} non-resonant transitions and hence terms with μ−ν −k ≠ 0 may be neglected. Then we may use

V̂_{j}^{I} = \{ \begin{matrix}λ_{j} \frac{ℏ}{2} σ̂_{j}^{+} (ˇâ_{l_{0}}^{†})^{|k|} ˇ F̂_{k}(ˇâ_{l_{0}}^{†} ˇ â_{l_{0}}) + H.c. & for k ≥ 0 , \\ λ_{j} \frac{ℏ}{2} σ̂_{j}^{+} ˇ F̂_{k}(ˇâ_{l_{0}}^{†} ˇ â_{l_{0}}) (ˇâ_{l_{0}})^{|k|} + H.c. & for k ≥ 0 , \end{matrix}  (3.57)

where

ˇF̂_{k}(ˇâ_{l_{0}}^{†} ˇ â_{l_{0}}) =^{def} e^{− \frac{1}{2} η_{l_{0}j}^{2}} (i η_{l_{0}j})^{|k|} ∑_{ν=0}^{∞} (−η_{l_{0}j}^{2})^{ν} \frac{(ˇâ_{l_{0}}^{†})^{ν} (ˇâ_{l_{0}})^{ν}}{ν! (ν + |k|)!} .  (3.58)

Multiplying V̂_{j}^{I} from the right with

ˆ1_{osc} = ∑_{n=0}^{∞} |n⟩_{osc} _{osc}⟨n|

and inserting

σ̂_{j}^{+} = |e_{j}⟩⟨g_{j}|

finally yields

V̂_{j}^{I} = \frac{ℏ}{2} ∑_{n=0}^{∞}(Ω_{j}^{n,k}Â_{j}^{n,k} + (Ω_{j}^{n,k}Â_{j}^{n,k})^{†}) ,  (3.59)

with

Â_{j}^{n,k} =^{def} \{ \begin{matrix}|e_{j}⟩⟨g_{j}| ⊗ |n + |k|⟩_{osc} _{osc}⟨n| & for k ≥ 0 , \\ |e_{j}⟩⟨g_{j}| ⊗ |n⟩_{osc} _{osc}⟨n + |k|| & for k ≤ 0 , \end{matrix}  (3.60)

and

Ω_{j}^{n,k} =^{def} λ_{j} e^{− \frac{1}{2} η_{l_{0}j}^{2}} (i η_{l_{0}j})^{|k|} \sqrt{\frac{n!}{(n + |k|)!}} L_{n}^{|k|}(η_{l_{0}j}^{2}) ,  (3.61)
L_{n}^{α}(x) =^{def} ∑_{ν=0}^{n} \frac{(−x)^{ν}}{ν!} \binom{n + α}{n − ν} (generalized Laguerre polynomials) .

86 CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES

Outline of proof for (3.61): We have to show

∑_{ν=0}^{∞} (−x)^{ν} \frac{(ˇâ_{l_{0}}^{†})^{ν+|k|} (ˇâ_{l_{0}})^{ν}}{ν! (ν + |k|)!} |n⟩_{osc} _{osc}⟨n| = \sqrt{\frac{n!}{(n + |k|)!}} L_{n}^{|k|}(x) |n + |k|⟩_{osc} _{osc}⟨n|

and

∑_{ν=0}^{∞} (−c)^{ν} \frac{(ˇâ_{l_{0}}^{†})^{ν} (ˇâ_{l_{0}})^{ν+|k|}}{ν! (ν + |k|)!} |n + |k|⟩_{osc} _{osc}⟨n + |k|| = \sqrt{\frac{n!}{(n + |k|)!}} L_{n}^{|k|}(x) |n⟩_{osc} _{osc}⟨n + |k|| .

Since

(ˇâ_{l_{0}}^{†})^{ν} (ˇâ_{l_{0}})^{ν} = (ˇâ_{l_{0}}^{†} ˇ â_{l_{0}})(ˇâ_{l_{0}}^{†} ˇ â_{l_{0}} − 1) … (ˇâ_{l_{0}}^{†} ˇ â_{l_{0}} − (n − 1))

(see Equation 1.63 of (Lücke, nlqo)), the first of these equations follows from

(ˇâ_{l_{0}}^{†})^{ν+|k|} (ˇâ_{l_{0}})^{ν} |n⟩_{osc} = (ˇâ_{l_{0}}^{†})^{|k|} n (n − 1) … (n − (ν − 1)) |n⟩_{osc}
=_{(3.48)} n (n − 1) … (n − (ν − 1)) \sqrt{\frac{(n+|k|)!}{n!}} |n + |k|⟩_{osc}
= \{ \begin{matrix}\sqrt{\frac{n!}{(n + |k|)!}} \frac{(n + |k|)!}{(n − ν)!} |n + |k|⟩_{osc} & for n ≥ ν \\ 0 & else , \end{matrix}

the second from

(ˇâ_{l_{0}}^{†})^{ν} (ˇâ_{l_{0}})^{ν+|k|} |n + |k|⟩_{osc} = |n⟩_{osc} _{osc}⟨n| (ˇâ_{l_{0}}^{†})^{ν} (ˇâ_{l_{0}})^{ν+|k|} |n + |k|⟩_{osc}
= (_{osc}⟨n + |k| | (ˇâ_{l_{0}}^{†})^{ν+|k|} (ˇâ_{l_{0}})^{ν} | n⟩_{osc})^{†} |n⟩_{osc}
= \{ \begin{matrix}\sqrt{\frac{n!}{(n + |k|)!}} \frac{(n + |k|)!}{(n − ν)!} |n⟩_{osc} & for n ≥ ν \\ 0 & else . \end{matrix}

The propagator in the interaction picture is the exponential^{51}

e^{− \frac{i}{ℏ} V̂_{j}^{I} t} = ∑_{n=0}^{∞}(cos (|Ω_{j}^{n,k}| \frac{t}{2}) (B̂_{j}^{n,k} + Ĉ_{j}^{n,k})

− i sin (|Ω_{j}^{n,k}| \frac{t}{2}) (Â_{j}^{n,k} e^{−i Φ̃_{k,j}} + H.c.)) + D̂_{j}^{n,k} ,

(3.62)

DRAFT, June 26, 2009

51Usually |Ω_{j}^{n,k}| is called the Rabi frequency for the transition

|g_{j}⟩ ⊗ |n + |k|⟩_{osc} ⇌ |e_{j}⟩ ⊗ |n⟩_{osc} if k ≤ 0 ,

resp.

|g_{j}⟩ ⊗ |n⟩_{osc} ⇌ |e_{j}⟩ ⊗ |n + |k|⟩_{osc} if k ≥ 0 .

3.3. COLD TRAPPED IONS 87 where

Φ̃_{k,j} =^{def} arg (Ω_{j}^{n,k}) =_{(3.61),(3.54)} ϕ_{j} − \frac{π}{2} |k|  (3.63)

and

B̂_{j}^{n,k} =^{def} \{ \begin{matrix}|e_{j}⟩⟨e_{j}| ⊗ |n + |k|⟩_{osc} _{osc}⟨n + |k|| & for k ≥ 0 , \\ |e_{j}⟩⟨e_{j}| ⊗ |n⟩_{osc} _{osc}⟨n| & for k ≤ 0 , \end{matrix}
Ĉ_{j}^{n,k} =^{def} \{ \begin{matrix}|g_{j}⟩⟨g_{j}| ⊗ |n⟩_{osc} _{osc}⟨n| & for k ≥ 0 , \\ |g_{j}⟩⟨g_{j}| ⊗ |n + |k|⟩_{osc} _{osc}⟨n + |k|| & for k ≤ 0 , \end{matrix}
D̂_{j}^{n,k} =^{def} \{ \begin{matrix}∑_{n=0}^{|k|−1} |e_{j}⟩⟨e_{j}| ⊗ |n⟩_{osc} _{osc}⟨n| & for k ≥ 0 , \\ ∑_{n=0}^{|k|−1} |g_{j}⟩⟨g_{j}| ⊗ |n⟩_{osc} _{osc}⟨n| & for k ≤ 0 . \end{matrix}

Outline of proof for (3.62): By (3.60) we have

\binom{}{Â_{j}^{n,k}}^{†} = \{ \begin{matrix}|g_{j}⟩⟨e_{j}| ⊗ |n⟩_{osc} _{osc}⟨n + |k|| & for k ≥ 0 \\ |g_{j}⟩⟨e_{j}| ⊗ |n + |k|⟩_{osc} _{osc}⟨n| & for k ≤ 0 \end{matrix}

and therefore

\begin{aligned}Â_{j}^{n_{1},k} Â_{j}^{n_{2},k} & = & 0 , \\ Â_{j}^{n_{1},k} \binom{}{Â_{j}^{n_{2},k}}^{†} & = & δ_{n_{1}n_{2}} B̂_{j}^{n_{1},k} , \\ \binom{}{Â_{j}^{n_{1},k}}^{†} Â_{j}^{n_{2},k} & = & δ_{n_{1}n_{2}} Ĉ_{j}^{n_{1},k} \end{aligned}  (3.64)

for k ≥ 0 as well as for k ≤ 0 . This implies, first of all,

e^{− \frac{i}{ℏ} V̂_{j}^{I} t} = ∏_{n=0}^{∞} exp (−i \frac{t}{2} (Ω_{j}^{n,k} Â_{j}^{n,k} + H.c.)) .  (3.65)

Moreover, (3.64) implies

(Ω_{j}^{n,k} Â_{j}^{n,k} + H.c.)^{2} = |Ω_{j}^{n,k}|^{2} (B̂_{j}^{n,k} + Ĉ_{j}^{n,k}) ,
(Ω_{j}^{n,k} Â_{j}^{n,k} + H.c.)^{2ν+1} = |Ω_{j}^{n,k}|^{2ν+1} (Â_{j}^{n,k} e^{−i Φ̃_{k,j}} + H.c.)

and hence

exp (−i \frac{t}{2} (Ω_{j}^{n,k} Â_{j}^{n,k} + H.c.)) = ˆ1 + (1 − cos(|Ω_{j}^{n,k}| \frac{t}{2}))(B̂_{j}^{n,k} + Ĉ_{j}^{n,k})

−i sin(|Ω_{j}^{n,k}| \frac{t}{2})(Â_{j}^{n,k} e^{−i ϕ̃_{j}} + H.c.) .

Inserting this into (3.65) yields (3.62).

88 CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES Especially, we have

e^{− \frac{i}{ℏ} V̂_{j}^{I} t}

= ∑_{n=0}^{∞}(cos (|Ω_{j}^{n,0}| \frac{t}{2}) (|e_{j}⟩⟨e_{j}| ⊗ |n⟩_{osc} _{osc}⟨n| + |g_{j}⟩⟨g_{j}| ⊗ |n⟩_{osc} _{osc}⟨n|)

− i sin (|Ω_{j}^{n,0}| \frac{t}{2}) (|e_{j}⟩⟨g_{j}| ⊗ |n⟩_{osc} _{osc}⟨n| e^{−i ϕ_{j}} + H.c.)) \underline{\underline{for k = 0}}

(3.66)

and e^{− \frac{i}{ℏ} V̂_{j}^{I} t}

= ∑_{n=0}^{∞}(cos (|Ω_{j}^{n,1}| \frac{t}{2}) (|e_{j}⟩⟨e_{j}| ⊗ |n⟩_{osc} _{osc}⟨n| + |g_{j}⟩⟨g_{j}| ⊗ |n + 1⟩_{osc} _{osc}⟨n + 1|)

− i sin (|Ω_{j}^{n,1}| \frac{t}{2}) (|e_{j}⟩⟨g_{j}| ⊗ |n⟩_{osc} _{osc}⟨n + 1| e^{−i (ϕ_{j}−π/2)} + H.c.))

+ |g_{j}⟩⟨g_{j}| ⊗ |0⟩_{osc} _{osc}⟨0| \underline{\underline{for k = −1}} .

(3.67)

Laser Pulses for Quantum Computation An exterior electromagnetic field of the form^{52} (plane-wave), (3.56) acting on the j-th ion during the time interval

Δt = 2 φ/ |Ω_{j}^{n,k}|

is called a φ-pulse for (n, k) . Examples for the action exp(−\frac{i}{ℏ} V̂_{j}^{I} Δt) of these pulses are:

1. φ-pulse for (0,0) with ϕ_{j} = ψ :

α |g_{j}⟩ ⊗ |0⟩_{osc} + β |e_{j}⟩ ⊗ |0⟩_{osc} ⟼ (α cos φ − i e^{+i ψ} β sin φ) |g_{j}⟩ ⊗ |0⟩_{osc}
+ (β cos φ − i e^{−i ψ}α sin φ) |e_{j}⟩ ⊗ |0⟩_{osc} .

2.

\frac{π}{2}-pulse for (0,-1) with ϕ_{j} = \frac{π}{2} :

|g_{j}⟩ ⊗ |0⟩_{osc} ⟼ + |g_{j}⟩ ⊗ |0⟩_{osc} ,
|e_{j}⟩ ⊗ |0⟩_{osc} ⟼ −i |g_{j}⟩ ⊗ |1⟩_{osc} .

DRAFT, June 26, 2009

52Note that for electromagnetic radiation the electric field uniquely fixes the magnetic field and vice versa.

3. π-pulse for (0,-1) with ϕ_{j} = 0 :

|g_{j}⟩ ⊗ |0⟩_{osc} ⟼ + |g_{j}⟩ ⊗ |0⟩_{osc} ,
|g_{j}⟩ ⊗ |1⟩_{osc} ⟼ − |g_{j}⟩ ⊗ |1⟩_{osc} ,
|e_{j}⟩ ⊗ |0⟩_{osc} ⟼ − |g_{j}⟩ ⊗ |0⟩_{osc} ,
|e_{j}⟩ ⊗ |1⟩_{osc} ⟼ + |g_{j}⟩ ⊗ |1⟩_{osc} .

Obviously, the action of a φ-pulse is that of a 1-qubit rotation, i.e. it is described by the matrix

( \begin{matrix}cos φ & −i e^{+iψ} sin φ \\ −i e^{−iψ} sin φ & cos φ \end{matrix} ) w.r.t. \{|g_{j}⟩ ˆ= |0⟩ , |e_{j}⟩ ˆ= |1⟩\} ,

as long as n_{l_{0}} = 0 . Moreover, suitable laser pulses for implementing the CNOT gate as described in 3.3.1 are the following:

(3.17) : a \frac{π}{2} -pulse for (0, −1) with ϕ_{j} = \frac{π}{2} on ion j_{1}
(3.18) : a π-pulse for (0, −1) with ϕ_{j_{2}} = 0 on ion j_{2} ,
tuned to a transition |g_{j_{2}}⟩ ⊗ |1⟩_{osc} ⇌ |e′_{j_{2}}⟩ |0⟩_{osc}
with |e′_{j_{2}}⟩ sufficiently different from |e_{j_{2}}⟩ ,
(3.19) : a \frac{π}{2} -pulse for (0, −1) with ϕ_{j} = \frac{π}{2} on ion j_{1} .

This shows that a sufficiently large class of quantum gates may be implemented, provided that the oscillatory modes can be cooled to their ground state.

Exercise 16 Show the following:

a) The matrices e^{i σ̂_{1} φ} and e^{i σ̂_{2} φ} correspond to special 1-qubit rotations, for all φ ∈ ℝ.

b) e^{−i σ̂_{1} \frac{π}{4}} σ̂_{2} e^{+i σ̂_{1} \frac{π}{4}} = σ̂_{3} .

c) Every Û ∈ SU(2) may be represented in the form^{53}

Û = e^{i σ̂_{2} \frac{ψ}{2}} e^{i σ̂_{1} \frac{θ}{2}} e^{i σ̂_{2} \frac{ϕ}{2}} , ψ , θ , ϕ ∈ ℝ .

3.3.4 Laser Cooling

See (Metcalf and van der Straten, 1999).

DRAFT, June 26, 2009

53Recall the beginning of the proof of Lemma 1.2.2.

90 CHAPTER 3. PHYSICAL REALIZATIONS OF QUANTUM GATES

Part II

Fault Tolerant Quantum Information Processing General Aspects of Quantum Information

An open system is nothing more than one which has interactions with some other environment system, whose dynamics we wish to neglect, or average over.

(Nielsen and Chuang, 2001, p. 353)

4.1 Introduction

Quantum information theory deals with transmission and quantification of quantum information. Roughly speaking, quantum information is the information carried by n-qubit systems (n ∈ ℕ). Of the utmost importance for quantum information — as opposed to classical information,^{1} — are:

1. The quantum mechanical superposition principle:

The pure states of a quantum system are in 1-1-correspondence with the 1-dimensional subspaces of a complex Hilbert space (unless there are superselection rules^{2}).

2. The no-cloning theorem (Dieks, 1982; Wootters and Zurek, 1982; Peres, 2002):

There cannot exist any recipe for preparing any two or more systems such that each of them carries the same quantum information as a given quantum system — unless the state of the latter is completely known, of course.

DRAFT, June 26, 2009

1See (Shannon, 1949) for the theory of classical information. 2See (Verstraete and Cirac, 2003; Bartlett and Wiseman, 2003) for the case with superselection

rules.

Remark: There are several justifications for the no-cloning theorem,^{3} e.g.:

• Given Ψ_{0} ∈ H, there is no linear extension of the mapping

C(Ψ ⊗ Ψ_{0}) =^{def} Ψ ⊗ Ψ for all normalized Ψ ∈ H

to all of H ⊗ H.

• Cloning would allow measurement of incommensurable quantum observables — impossible according to quantum mechanics.

• Cloning would, by proper use of one of the available Bell sources, allow for superluminal communication^{4} — impossible according to special relativity.

While the superposition principle opens up the fascinating possibilities of quantum computation, the impossibility of cloning unknown quantum states strongly limits the amount of information that can be read out from quantum states.^{5} Thus, e.g., it is impossible to distinguish nonorthogonal states^{6} by a single measurement. Nevertheless, quantum information seems to offer a wealth of useful applications without any classical equivalent.

We say that quantum information is transmitted — to whatever degree intact — through a quantum (rather than classical) channel if the information is sent using systems whose quantum character cannot be neglected in this respect. In other words:

We identify quantum channels with open^{7} quantum systems, used for quantum communication.

For practical communication it is important that information can be transferred in a reversible way from one physical system to another. The no-cloning theorem implies that unknown quantum information cannot be transferred from quantum to purely classical systems in a reversible way. This is why quantum channels and entanglement-assisted classical channels are the main topic of these lectures.

In the following, unless stated otherwise, we will always work in the interaction picture and make extensive use of Dirac’s bra-ket notation.^{8} For simplicity, we consider only finite-dimensional Hilbert spaces. This is sufficient for clarifying the main points.

DRAFT, June 26, 2009

3See (Werner, 2001, Section 2.3) for a detailed discussion.

4See (Herbert, 1982). Similarly, joint measurability (without cloning) of non-commuting observables would enable superluminal communication.

5On the other hand, the non-cloning theorem constitutes the basis for secure quantum cryptography (see http://www.idquantique.com) and quantum passwords (Gu and Weedbrook, 2005).

6We tacitly identify states with their density matrices or wave functions (if pure).

7We have to consider open quantum systems since quantum information is prone to quantum noise caused by interaction with the environment.

8See, e.g., (Lücke, eine).

4.2 Quantum Channels

4.2.1 Open Quantum Systems and Quantum Operations

Let us consider two quantum systems S_{1} resp. S_{2} with (finite dimensional) state spaces H_{1} resp. H_{2} and consider S_{2} as the environment of S_{1} ; i.e. let us assume the bipartite System S composed of S_{1} and S_{2} to be closed. Moreover, let us assume that S_{1} is prepared in a pure state^{9} ρ̂^{(1)} ∈ S(H_{1}) at time 0 . Then, at time 0 there are no correlations between S_{1} and S_{2} and the state of S is of the form

ρ̂_{0} = ρ̂^{(1)} ⊗ ρ̂^{(2)}  (4.1)

with ρ̂^{(2)} ∈ S(H_{2}) . At time t, then (see, e.g., Section 6.1.2 of (Lücke, nlqo)), the state of S is

ρ̂_{t} = Ûρ̂^{(1)} ⊗ ρ̂^{(2)} Û^{†} ,  (4.2)

where the unitary operator Û on the state space H_{1} ⊗ H_{2} of system S is given by

Û =^{def} e^{+ \frac{i}{ℏ} Ĥ_{0} t}e^{− \frac{i}{ℏ} Ĥ t} ,

Ĥ resp. Ĥ_{0} = Ĥ_{0}^{(1)} ⊗ Ĥ_{0}^{(2)} being the actual resp. free Hamiltonian of S. The partial state of S_{1} at time t, therefore, is

ρ̂_{t}^{(1)} = ℭ(ρ̂^{(1)}) ,  (4.3)

where^{10}

ℭ(ρ̂′) =^{def} trace_{2}(Û (ρ̂′ ⊗ ρ̂^{(2)}) Û^{†}) ∀ ρ̂′ ∈ S(H_{1}) .  (4.4)

Thanks to the spectral theorem, the initial state of the environment can be written in the form

ρ̂^{(2)} = ∑_{β=1}^{n_{2}} \underbrace{λ_{β}}_{≥0} |ϕ_{β}^{(2)}⟩⟨ϕ_{β}^{(2)}| , ∑_{β=1}^{n_{2}} λ_{β} = 1  (4.5)

with some orthonormal basis {ϕ_{1}^{(2)}, …, ϕ_{n_{2}}^{(2)}} of H_{2} . (4.4) and (4.5) imply^{11}

ℭ(ρ̂′) = ∑_{α,β=1}^{n_{2}} λ_{β} ⟨ψ_{α}^{(2)}| Û (ρ̂′ ⊗ |ϕ_{β}^{(2)}⟩⟨ϕ_{β}^{(2)}|) Û^{†} |ψ_{α}^{(2)}⟩

= ∑_{α,β=1}^{n_{2}} K̂_{α,β} ρ̂′ K̂_{α,β}^{†} ∀ ρ̂′ ∈ S(H_{1})  (4.6)

DRAFT, June 26, 2009

9By S(H) we denote the set of all states, i.e. of all positive operators ρ̂ on the Hilbert space H with trace (ρ̂) = 1 . L(H) , as usual, denotes the set of all bounded linear operators on H.

10Note that ℭ depends not only on Ĥ and Ĥ_{0} but also on the initial state of the environment! 11We use the notation

⟨ϕ^{(2)}|(∑_{k=1}^{N} Â_{k} ⊗ B̂_{k})|ψ^{(2)}⟩ =^{def} ∑_{k=1}^{N} Â_{k} ⟨ϕ^{(2)} | B̂_{k} ψ^{(2)}⟩

for ϕ^{(2)}, ψ^{(2)} ∈ H_{2} , Â_{1}, …, Â_{N} ∈ H_{1} , and B̂_{1}, …, B̂_{N} ∈ H_{2} .

for every orthonormal basis {ψ_{1}^{(2)}, …, ψ_{n_{2}}^{(2)}} of H_{2} , where

K̂_{α,β} =^{def} \sqrt{λ_{β}} ⟨ψ_{α}^{(2)}| Û |ϕ_{β}^{(2)}⟩ ∈ L(H_{1}) ∀ α, β ∈ \{1, …, n_{2}\}

and, therefore,

∑_{α,β=1}^{n_{2}} K̂_{α,β}^{†} K̂_{α,β} = ∑_{α,β=1}^{n_{2}} λ_{β} ⟨ϕ_{β}^{(2)}| Û^{†} |ϕ_{α}^{(2)}⟩ ⟨ϕ_{α}^{(2)}| Û |ϕ_{β}^{(2)}⟩

= ∑_{β=1}^{n_{2}} λ_{β} \underbrace{⟨ϕ_{β}^{(2)} | Û^{†} Û ϕ_{β}^{(2)}⟩}_{=1}

=_{(4.5)} 1 .

If, from an ensemble in the state (4.2), those individuals are selected (by projective measurement of the environment) for which S_{2} is in the partial state |ψ_{α}^{(2)}⟩⟨ψ_{α}^{(2)}| then (4.6) has to be replaced by^{12}

ℭ(ρ̂′) = ∑_{β=1}^{n_{2}} K̂_{α,β} ρ̂′ K̂_{α,β}^{†} ∀ ρ̂′ ∈ S(H_{1})

and

trace (ℭ(ρ̂)) ≤ 1

is the relative number of individuals selected. In either case E is just a special quantum operation:^{13}

Definition 4.2.1 Let H and H′ be Hilbert-spaces. Then by Q(H, H′) we denote the set of all mappings ℭ from S(H) into L(H′) of the form

ℭ(ρ̂) = ∑_{k=1}^{N} K̂_{k} ρ̂K̂_{k}^{†} ∀ ρ̂ ∈ S(H) ,  (4.7)

with suitable N ∈ ℕ and K̂_{k} ∈ L(H, H′) fulfilling^{14}

∑_{k=1}^{N} K̂_{k}^{†} K̂_{k} ≤ ˆ1 .  (4.8)

DRAFT, June 26, 2009

12For a corresponding representation of general linear maps see (Shabani and Lidar, 2006, Theorem1).

13 See (Buscemi et al., 2003) for efficient realizations and (Werner, 2001, Section 2.6.2) for the dual action of quantum operations on the observable algebras.

14Note that the adjoint K̂^{†} of a linear mapping K̂ ∈ L(H, H) is characterized by

⟨ψ′ | K̂ ψ⟩_{H} = ⟨K̂^{†} ψ′ | ψ⟩_{H_{′}} ∀ ψ ∈ H , ψ′ ∈ H′ .

Thus, e.g., (|ψ′⟩⟨ψ|)^{†} = |ψ⟩⟨ψ′|.

The elements of Q(H, H′) are called quantum operations and the K̂_{k} in (4.7) are called Kraus operators for ℭ.

Remarks:

1. Note that^{15}

0 ≤ trace (\underbrace{ℭ(ρ̂)}_{≥0}) ≤ 1 ∀ ρ̂ ∈ S(H_{1})

and that

∑_{k=1}^{N} K̂_{k}^{†} K̂_{k} = ˆ1 ⟺ trace (ℭ(ρ̂)) = 1 ∀ ρ̂ ∈ S(H_{1}) .

2. Obviously, we also have

ℭ_{1} ∈ Q(H_{1}, H_{2}) , ℭ_{2} ∈ Q(Ĥ_{2}, H_{3}) ⟹ ℭ_{2} ∘ ℭ_{1} ∈ Q(H_{1}, H_{3}) .

3. But, given ℭ_{1} ∈ Q(H_{1}, H_{2}) and ℭ_{3} ∈ Q(H_{1}, H_{3}) , this does not guarantee existence of an ℭ_{2} ∈ Q(H_{2}, H_{3}) with ℭ_{3} = ℭ_{2} ∘ ℭ_{1} .

4. Of course, nonexistence of such ℭ_{2} is only possible if ℭ_{1} is not invertible. The latter is obviously the case if, e.g., H_{1} = H_{2} and

\begin{aligned}ℭ_{1}(ρ̂) & = & \frac{1}{4} ∑_{ν=0}^{3} τ̂^{ν} ρ̂τ̂^{ν} \\ & = & \frac{1}{2} ˆ1 ∀ ρ̂ ∈ S(H) . \end{aligned}

5. The linear extension

ℭ̄(Â) =^{def} ∑_{k=1}^{N} K̂_{k} Â K̂_{k}^{†} ∀ Â ∈ L(H_{1}) ,

of ℭ to all of L(H_{1}) is \underline{completely} positive,^{16} i.e.:

Â′ ≥ 0 ⟹ (1 ⊗ ℭ̄) (Â′) ≥ 0 ∀ Â′ ∈ \underline{ℂ^{n}⊗}H_{1} , n ∈ ℕ .

DRAFT, June 26, 2009

15The possibility trace (\underbrace{ℭ(ρ̂)}_{≥0}) < 1 is included to allow for absorption.

16Every trace preserving affine mapping of S(H) into S(H) can be represented as (the restriction of) a difference of two completely positive mappings (Kuah and Sudarshan, 2005).

98 CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

Lemma 4.2.2 Let H_{1} and H_{2} be finite-dimensional Hilbert-spaces^{17} and let ℭ be a mapping from S(H_{1}) into L(H_{2}) . Then ℭ is a quantum operation, i.e. ℭ ∈ Q(H_{1}, H_{2}) , iff the following three conditions are fulfilled:

1.

trace (ℭ(ρ̂)) ≤ 1 ∀ ρ̂ ∈ S(H_{1}) .

2.

ℭ(λ ρ̂_{1} + (1 − λ) ρ̂_{2}) = λ ℭ(ρ̂_{1}) + (1 − λ) ℭ(ρ̂_{2}) ∀ λ ∈ [0, 1] , ρ̂_{1}, ρ̂_{2} ∈ S(H_{1}) .

3.

(1 ⊗ ℭ̄)(|Ψ⟩⟨Ψ|) ≥ 0 ∀ Ψ ∈ H_{1} ⊗ H_{1} ,

where ℭ̄ denotes the unique linear extension^{18} of ℭ to all of L(H_{1}) .

Outline of proof: Assume that ℭ fulfills the requirements 1–3. Choose an orthonormal basis {ϕ_{1}^{(1)}, …, ϕ_{n_{1}}^{(1)}} of H_{1} and defining

ψ^{∗} =^{def} ∑_{ν=1}^{n_{1}} ⟨ψ | ϕ_{ν}^{(1)}⟩ ϕ_{ν}^{(1)} ∀ ψ ∈ H_{1}  (4.9)

we get

⟨ψ^{∗} | ϕ_{ν}^{(1)}⟩ = ⟨ϕ_{ν}^{(1)} | ψ⟩ ∀ ψ ∈ H_{1} , ν ∈ {1, …, n_{1}}

and hence^{19}

ℭ(|ψ⟩⟨ψ|) _{=_{2. req.}} ∑_{ν,μ=1}^{n_{1}} ⟨ϕ_{ν}^{(1)} | ψ⟩⟨ψ | ϕ_{μ}^{(1)}⟩ ℭ^{}¯ (|ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}|)

= ∑_{ν,μ=1}^{n_{1}} ⟨ψ^{∗}| (|ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}| ⊗ ℭ^{}¯ (|ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}|)) |ψ^{∗}⟩
= ⟨ψ^{∗}| Â |ψ^{∗}⟩ ∀ ψ ∈ H_{1} , ∥ψ∥ = 1 ,  (4.10)

DRAFT, June 26, 2009

17Since we agreed to consider only finite-dimensional Hilbert spaces, all operators on H_{1} resp. H_{2} are of trace class. For infinite-dimensional Hilbert spaces H_{1} = H_{2} see (Davies, 1976, Theorem 2.3 and notes on page 147).

18Existence of this extension, is guaranteed by the second condition. For self-adjoint Â it is given by

ℭ^{}¯ (Â) =^{def} trace (Â_{+}) ℭ(\frac{Â_{+}}{(trace Â_{+})}) + trace (Â_{−}) ℭ(\frac{Â_{−}}{(trace Â_{−})}) ,

where Â_{+} resp. Â_{−} denotes the positive resp. negative part of Â :

Â = Â_{+} + Â_{−} ± Â_{±} ≥ 0 , Â_{+}Â_{−} = ˆ0 .

Linear mappings from L(H_{1}) into L(H_{2}) are also called superoperators.

19We use the notation explained in Footnote 11, with the roles of the tensor factors H_{1}, H_{2} interchanged.

4.2. QUANTUM CHANNELS 99

where

= (1 ⊗ ℭ^{}¯ ) ( \underbrace{∑_{ν,μ=1}^{n_{1}} |ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}| ⊗ |ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}|}_{=|∑_{ν=1}^{n_{1}} ϕ_{ν}^{(1)}⊗ϕ_{ν}^{(1)}⟩⟨∑_{ν=1}^{n_{1}} ϕ_{ν}^{(1)}⊗ϕ_{ν}^{(1)}|≥0})^{Â =^{def} ∑_{ν,μ=1}^{n_{1}} |ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}| ⊗ ℭ^{}¯ (|ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}|)}

and hence

0 ≤_{3. req.} Â ∈ L(H_{1} ⊗ H_{2}) .

This, together with the spectral theorem implies

Â = ∑_{k=1}^{N} |Ψ_{k}⟩⟨Ψ_{k}| for suitable Ψ_{1}, …, Ψ_{N} ∈ H_{1} ⊗ H_{2} .

The latter, together with (4.10) gives

ℭ^{}¯ (|ψ⟩⟨ψ|) = ∑_{k=1}^{N} ⟨ψ^{∗}| |Ψ_{k}⟩⟨Ψ_{k}| |ψ^{∗}⟩ ∀ ψ ∈ H_{1} .

Thus, defining the linear mappings^{20}

K̂_{k} ψ =^{def} ⟨ψ^{∗}| |Ψ_{k}⟩ ∀ ψ ∈ H_{1} , k ∈ {1, …, N}

from H_{1} into H_{2} we get

ℭ(|ψ⟩⟨ψ|) = ∑_{k=1}^{N} K̂_{k} |ψ⟩⟨ψ| K̂_{k}^{†} ∀ ψ ∈ H_{1} , ∥ψ∥ = 1 .

By condition 2 this gives (4.7). The latter together with condition 1, finally, gives^{21} (4.8). Conversely, conditions 1–3 are easily seen to be fulfilled for ℭ ∈ Q(H_{1}, H_{2}) .

Remark: At first glance one might consider conditions 1–3 of Lemma 4.2.2 as natural requirements on the action of quantum channels. Note, however, that the output state of a quantum channel need not even be determined by the input state in case of initial correlations with the environment.^{22}

DRAFT, June 26, 2009

20Here, we use the notation

⟨ψ| |∑_{j=1}^{N′} ψ_{j}^{(1)} ⊗ ψ_{j}^{(2)}⟩ =^{def} ∑_{j=1}^{N′} ⟨ψ | ψ_{j}^{(1)}⟩ ψ_{j}^{(2)}

for ψ, ψ_{1}^{(1)}, …, ψ_{N′}^{(1)} ∈ H_{1} and ψ_{1}^{(2)}, …, ψ_{N′}^{(2)} ∈ H_{2} .

21Violation of (4.8) would imply existence of a normalized eigenvector ψ_{+} of ∑_{k=1}^{N} K̂_{k}^{†} K̂_{k} with eigenvalue greater than 1 and thus trace (ℭ(ρ̂)) > 1 for ρ̂ = |ψ_{+}⟩⟨ψ_{+}|.

22Recall Footnote 10. See also (Kuah and Sudarshan, 2005), in this connection.

Definition 4.2.1 and Lemma 4.2.1 describe the most general quantum operations. Simple versions are, among others:

1. Unitary transformations

ρ̂′ ⟼ ℭ(ρ̂′) = V̂ ρ̂V̂^{†}

given, e.g., by (4.4) for

Û = V̂ × V̂′ , V̂ unitary .

2. Complete projective measurement operations

ρ̂′ ⟼ ℭ(ρ̂′) = ∑_{j=1}^{n_{1}} P̂_{ϕ_{j}^{(1)}} ρ̂P̂_{ϕ_{j}^{(1)}} , {ϕ_{1}^{(1)}, …, ϕ_{n_{1}}^{(1)}} orthonormal basis of H_{1} ,

given, e.g., by (4.4) if there are ρ̂_{1}^{(2)}, …, ρ̂_{n_{1}}^{(2)} ∈ S(H_{2}) with pairwise orthogonal supports and such that^{23}

ÛP̂_{ϕ_{j}^{(1)}} ⊗ ρ̂^{(2)} Û^{†} = P̂_{ϕ_{j}^{(1)}} ⊗ ρ̂_{j}^{(2)} ∀ j ∈ {1, …, n_{1}} .
Example:^{24}
Û = action of CNOT ,
S_{1} = control qubit ,
S_{2} = target qubit ,
ρ̂^{(2)} = |0⟩⟨0| (or |1⟩⟨1|) .

3. Cascaded complete projective measurement operations

ρ̂′ ⟼ ℭ(ρ̂′) = ∑_{j_{1},...,j_{r}=1}^{n_{1}} K̂_{(j_{1},...,j_{r})} ρ̂′ K̂_{(j_{1},...,j_{r})}^{†} ,

where

K̂_{(j_{1},...,j_{r})} = P̂_{r,j_{r}} ⋯ P̂_{1,j_{1}} ∀ (j_{1}, …, j_{r}) ∈ {1, …, n_{1}}^{r}

with P̂_{k,1}, …, P̂_{k,n_{1}} being the projectors of the k-th projective measurement.

Remark: Note that

∑_{j_{1},...,j_{r}=1}^{n_{1}} K̂_{(j_{1},...,j_{r})}^{†} K̂_{(j_{1},...,j_{r})} = 1

but that, in general, the K̂_{(j_{1},...,j_{r})} are no longer projection operators.

DRAFT, June 26, 2009

23In this case nondemolition measurements on S_{1} can be performed by corresponding (usually destructive) measurements on S_{2} . Note, however, that for the resulting partial state of S_{1} it does not matter whether the state of the environment, including the measurement apparatus, is checked or not (the possibility of postselection is sufficient).

24Another example would be the Stern-Gerlach measurement if it really worked as usually described.

4.2. QUANTUM CHANNELS 101 Theorem 4.2.3 Let H_{1} and H_{2} be Hilbert-spaces and let K̂_{1}, …, K̂_{N}, K̂′_{1}, …, K̂_{N} ∈ L(H_{1}, H_{2}) . Then

∑_{k=1}^{N} K̂_{k} ρ̂(K̂_{k})^{†} = ∑_{k=1}^{N} K̂′_{k} ρ̂(K̂′_{k})^{†} ∀ ρ̂ ∈ S(H_{1})  (4.11)

iff there is a unitary N × N-matrix (U_{k}^{j}) with

K̂′_{j} = ∑_{k=1}^{N} U_{k}^{j} K̂_{k} ∀ j ∈ {1, …, N} .  (4.12)

Outline of proof: Assume that (4.11) holds. Then, if we choose an orthonormal basis {ϕ_{1}, …, ϕ_{n}} of H_{1} and define

Ψ_{k} =^{def} ∑_{ν=1}^{n} ϕ_{ν} ⊗ (K̂_{k} ϕ_{ν}) ,
Ψ′_{k} =^{def} ∑_{ν=1}^{n} ϕ_{ν} ⊗ (K̂′_{k} ϕ_{ν})

for k ∈ {1, …, N}, we get

∑_{k=1}^{N} |Ψ_{k}⟩⟨Ψ_{k}| = ∑_{k=1}^{N} ∑_{ν,μ=1}^{n}(|ϕ_{ν}⟩⟨ϕ_{μ}|) ⊗ (K̂_{k} |ϕ_{ν}⟩⟨ϕ_{μ}| (K̂_{k})^{†})
=_{(4.11)} ∑_{k=1}^{N} ∑_{ν,μ=1}^{n}(|ϕ_{ν}⟩⟨ϕ_{μ}|) ⊗ (K̂′_{k} |ϕ_{ν}⟩⟨ϕ_{μ}| (K̂′_{k})^{†})
= ∑_{k=1}^{N} |Ψ′_{k}⟩⟨Ψ′_{k}| .

Therefore, by Corollary A.4.3, there is a unitary N × N-matrix (U_{j}^{k}) with

Ψ_{k} = ∑_{j=1}^{N} U_{k}^{j} Ψ′_{j} ∀ k ∈ {1, …, N} ,

i.e. with

∑_{ν=1}^{n} ϕ_{ν} ⊗ (K̂_{k} ϕ_{ν}) = ∑_{ν=1}^{n} ϕ_{ν} ⊗ (∑_{j=1}^{N} U_{k}^{j} K̂′_{j} ϕ_{ν} ) ∀ k ∈ {1, …, N} .

This implies

K̂_{k} ϕ_{ν} = ∑_{j=1}^{N} U_{k}^{j} K̂′_{j} ϕ_{ν} ∀ ν ∈ {1, …, n} , k ∈ {1, …, N}

and hence (4.12). Conversely, it is obvious that (4.12) implies (4.11).

The standard example, in case n_{1} > 1 , for a mapping ℭ of S(H_{1}) into S(H_{1}) fulfilling conditions 1 and 2, but not 3, of Lemma 4.2.2 is the transposition^{25}

ρ̂′ = ∑_{j,k=1}^{n_{1}} ρ_{k}^{j} |ϕ_{j}^{(1)}⟩⟨ϕ_{k}^{(1)}| ⟼ ℭ(ρ̂′) = 𝔗(ρ̂′) =^{def} ∑_{j,k=1}^{n_{1}} ρ_{k}^{j} |ϕ_{k}^{(1)}⟩⟨ϕ_{j}^{(1)}| . (4.13)

depending on the basis {ϕ_{1}^{(1)}, …, ϕ_{n_{1}}^{(1)}} of H_{1}.

Proof of positivity: Thanks to the spectral theorem it is sufficient to show that S_{pure}(H_{1}) is left invariant under transposition. This, however follows from^{26}

𝔗( |ϕ⟩⟨ψ|) = |ψ^{∗}⟩⟨ϕ^{∗}| ∀ ϕ, ψ ∈ H_{1} .  (4.14)

Disproof of complete positivity: Obviously, it is sufficient to check the case n_{1} = 2 . Then, with

|jk⟩ =^{def} ϕ_{j}^{(1)} ⊗ ϕ_{k}^{(1)} ∀ j, k ∈ {1, 2} ,

we have

\begin{aligned}(1 ⊗ 𝔗^{}¯ )(Ŵ_{λ}) & = & \frac{1 − λ}{4}(|0, 0⟩⟨0, 0| + |1, 1⟩⟨1, 1|) \\ & & + \frac{1 + λ}{4}(|0, 1⟩⟨0, 1| + |1, 0⟩⟨1, 0|) \\ & & − \frac{λ}{2}(|1, 1⟩⟨0, 0| + |0, 0⟩⟨1, 1|) \end{aligned}  (4.15)

for the Werner states^{27}

Ŵ_{λ} =^{def} \frac{1 − λ}{4} ˆ1 ⊗ ˆ1 + λ |Ψ^{−}⟩⟨Ψ^{−}| , λ ∈ [0, 1] .

But for λ > 1/3 (4.15) cannot be positive, since, e.g.

((1 ⊗ 𝔗^{}¯ )(Ŵ_{λ}))(|0, 0⟩ + |1, 1⟩) = \frac{1 − 3λ}{4}(|0, 0⟩ + |1, 1⟩) .

DRAFT, June 26, 2009

25Note that, for ρ̂′ ∈ S(H_{1}) , transposition is equivalent to complex conjugation

ρ̂′ = ∑_{j,k=1}^{n_{1}} ρ_{k}^{j} |ϕ_{j}^{(1)}⟩⟨ϕ_{k}^{(1)}| ⟼ ∑_{j,k=1}^{n_{1}} (ρ_{k}^{j})^{∗} |ϕ_{j}^{(1)}⟩⟨ϕ_{k}^{(1)}| .

26Recall (4.9). 27These states are distinguished by their invariance property

(Û ⊗ Û) Ŵ_{λ} (Û ⊗ Û)^{†} = Ŵ_{λ} for all unitary Û ∈ L(H)

(Werner, 1989, Section II). Note that the flip operator

|0, 0⟩⟨0, 0| + |1, 0⟩⟨0, 1| + |0, 1⟩⟨1, 0| + |1, 1⟩⟨1, 1|

coincides with ˆ1 ⊗ ˆ1 − 2 |Ψ^{−}⟩⟨Ψ^{−}|.

4.2.2 Quantum Noise and Error Correction

The action of noisy quantum channels corresponds to non-invertible quantum operations ℭ. Nevertheless such an operation may be become invertible by restriction to states with support on a suitable subspace C of H — and thus allow for error correction on the code space C.

Theorem 4.2.4 Let H be Hilbert space, ℭ ∈ Q(H, H) and let C be a linear subspace of H. Then the following three statements are equivalent:

1. There are Kraus operators K̂_{1}, …, K̂_{N} for ℭ and a_{1}, …, a_{N} ≥ 0 with

P̂_{C} K̂_{j}^{†} K̂_{k} P̂_{C} = δ_{jk} a_{j} P̂_{C} ∀ j, k ∈ {1, …, N} .  (4.16)

2. There is a trace preserving quantum operation ℜ with

(ℜ ∘ ℭ) (|ψ⟩⟨ψ|) ∝ |ψ⟩⟨ψ| ∀ ψ ∈ C ∥ψ∥ = 1 .  (4.17)

3. There are Kraus operators K̂′_{1}, …, K̂′_{N} for ℭ with

P̂_{C} K̂′_{j}^{†} K̂′_{k} P̂_{C} = a_{jk} P̂_{C} ∀ j, k ∈ {1, …, N}  (4.18)

for some self-adjoint matrix (a_{jk}) .

Outline of proof: Assume the first statement to be true. Then, using the polar decomposition

and hence

For

ℜ(ρ̂) =^{def} ∑_{j=1}^{N} (Û_{j}^{†} P̂_{Û_{j} C}) ρ̂(Û_{j}^{†} P̂_{Û_{j} C})^{†} + P̂_{0} ρ̂P̂_{0} ∀ ρ̂ ∈ S(H) ,  (4.22)

where

P̂_{0} =^{def} ˆ1 − ∑_{j=1}^{N} P̂_{Û_{j} C}  (4.23)

and thus

P̂_{0} ℭ(P̂_{C}ρ̂P̂_{C}) P̂_{0} _{=_{(4.19)}} ∑_{j=1}^{N} a_{j} P̂_{0} Û_{j} P̂_{C} ρ̂P̂_{C} Û_{j}^{†} P̂_{0}_{=_{(4.21)} 0 ∀ ρ̂ ∈ S(H) ,}

K̂_{j} P̂_{C} = Û_{j} ^{+} \sqrt{P̂_{C} K̂_{j}^{†} K̂_{j} P̂_{C}}

= ^{+} \sqrt{a_{j}} Û_{j} P̂_{C}  (4.19)

(see, e.g., Lemma 7.3.20 of (Lücke, eine)) we have

δ_{jk} P̂_{C} = P̂_{C} Û_{j}^{†} Û_{k} P̂_{C}  (4.20)

P̂_{Û_{j} C} P̂_{Û_{k} C} = δ_{jk} P̂_{Û_{j} C} .  (4.21)

this gives^{28}

(ℜ ∘ ℭ)(|ψ⟩⟨ψ|) = ∑_{j,k=1}^{N} P̂_{C} Û_{j}^{†} \underbrace{P̂_{Û_{j} C} \underbrace{K̂_{k} |ψ⟩}_{=_{(4.19)} ^{+} \sqrt{a_{k}} Û_{k}|ψ⟩}}_{=_{(4.21)} δ_{jk} ^{+} \sqrt{a_{k}} Û_{k}|ψ⟩}⟨ψ| K̂_{k}^{†} P̂_{Û_{j} C} Û_{j} P̂_{C} (4.24)

=_{(4.20)} ∑_{j=1}^{N} a_{j} |ψ⟩⟨ψ| ∀ ψ ∈ C .

Since

∑_{j=1}^{N} (Û_{j}^{†} P̂_{Û_{j}C})^{†} (Û_{j}^{†} P̂_{Û_{j}C}) + P̂_{0}^{†} P̂_{0} =_{(4.21)} ˆ1 ,

this implies the second statement.^{29} Now assume the second statement to be

true. Then (4.17) holds and, of course,^{30} trace (ℭ(|ψ⟩⟨ψ|)) must be constant for

normalized ψ ∈ C. Therefore,

(ℜ ∘ ℭ) (P̂_{C} ρ̂P̂_{C}) = γ P̂_{C} ρ̂P̂_{C} ∀ ρ̂ ∈ S(H)  (4.25)

holds for some γ ≥ 0 . If K̂′_{1}, …, K̂′_{N} are Kraus operators for ℭ and R̂_{1}, …, R̂_{N} are Kraus operators for ℜ then (4.25) is equivalent to

∑_{j,k=1}^{N} R̂_{j} K̂′_{k} P̂_{C} ρ̂P̂_{C} K̂′_{k}^{†} R̂_{j}^{†} = γ P̂_{C} ρ̂P̂_{C} ∀ ρ̂ ∈ S(H) .

Then,^{31} by Theorem 4.2.3, there are complex numbers λ_{jk} with

R̂_{j} K̂′_{k} P̂_{C} = λ_{jk} P̂_{C} ∀ j, k ∈ {1, …, N}

and hence

P̂_{C} K̂′_{l}^{†} R̂_{j}^{†} R̂_{j} K̂′_{k} P̂_{C} = λ_{jl}^{∗} λ_{jk} P̂_{C} ∀ j, k, l ∈ {1, …, N} .

Since ℜ is trace preserving, this implies the third statement with

a_{lk} = ∑_{j=1}^{N} λ_{jl}^{∗} λ_{jk} ∀ k, l ∈ {1, …, N} .

Finally, assume the third statement to be true. Then, by the spectral theorem, there are a unitary matrix (u_{jk}) and real numbers a_{1}, …, a_{N} with

∑_{j′,k′=1}^{N} u_{j′j}^{∗} a_{j′k′} u_{k′k} = a_{j} δ_{j,k} ∀ j, k ∈ {1, …, N} .

DRAFT, June 26, 2009

28Note that P̂_{C} Û_{j}^{†} P̂_{Û_{j} C} = Û_{j}^{†} P̂_{Û_{j} C} . 29Recall Remark 1 to Definition 4.2.1. 30Note that for ρ̂_{1}, ρ̂_{2} ∈ S(H) and λ_{1}, λ_{2} ∈ ℂ we have

\begin{matrix}λ_{1} ρ̂_{1} + λ_{2} ρ̂_{2} ∝ ρ̂_{1} + ρ̂_{2} \\ ρ̂_{1} ≠ ρ̂_{2} \end{matrix} \} ⟹ λ_{1} = λ_{2} .

31We may add N − 1 zeros as Kraus operators to \sqrt{γ} P̂_{C} .

This, together with (4.18), implies the first statement with

K̂_{j} = ∑_{k=1}^{N} u_{kj} K̂′_{k} ∀ j ∈ {1, …, N} .

Remark: (4.19), (4.21), and (4.22) show how errors produced by ℭ can be corrected for the code C :

Perform a projective measurement w.r.t. the orthogonal subspaces Û_{j} C and apply Û_{j}^{†} according to the result of this ‘measurement’.

Corollary 4.2.5 Let C be a linear subspace of the Hilbert space H and let K̂_{1}, …, K̂_{N} resp. K̂′_{1}, …, K̂′_{N} be Kraus operators for ℭ ∈ Q(H, H) resp. ℭ′ ∈ Q(H, H) . If (4.16) holds and if the K̂′_{j} are complex linear combinations of the K̂_{j} then, with ℜ as constructed in the proof of Theorem 4.2.4, (4.17) holds also for ℭ replaced by ℭ′.

Outline of proof: Assume that

K̂′_{j} = ∑_{k=1}^{N} λ_{jk} K̂_{k}

and let ℜ be defined as in the proof of Theorem 4.2.4. Then

(ℜ ∘ ℭ′)(|ψ⟩⟨ψ|) ; \begin{matrix}=_{(4.24)} & ∑_{j,k,l,r=1}^{N} (P̂_{C} Û_{j}^{†} P̂_{Û_{j}C}) λ_{kl} ^{+} \sqrt{a_{k}} Û_{l} P̂_{C} |ψ⟩⟨ψ| λ_{kr}^{∗} ^{+} \sqrt{a_{r}} P̂_{C} Û_{r}^{†} (P̂_{C} Û_{j}^{†} P̂_{Û_{j}C})^{†} \\ =_{(4.21)} & ∑_{j,k=1}^{N} a_{j} |λ_{kl}|^{2} |ψ⟩⟨ψ| ∀ ψ ∈ C . \end{matrix}

4.3 Error Correcting Codes

4.3.1 General Apects

According to standard formulations (Lücke, 1996) the time evolution of closed^{32} quantum mechanical systems is unitary.

Let us consider the closed system of one qubit together with its environment. A unitary transformation of the corresponding state space maps separated pure states to entangled pure states:

\begin{matrix}|χ⟩ = (α |0⟩ + β |1⟩) ⊗ |E⟩ \\ ⟼ |χ′⟩ = α(|0⟩ ⊗ |E_{0,0}⟩ + |1⟩ ⊗ |E_{0,1}⟩) + β(|0⟩ ⊗ |E_{1,0}⟩ + |1⟩ ⊗ |E_{1,1}⟩) \end{matrix} (4.26)

DRAFT, June 26, 2009

32For open quantum systems see (Alicki, 2003) and references given there.

106 CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

(∀ α, β ∈ ℂ). This way the originally pure partial state^{33}

⟨χ| Â ⊗ ˆ1 |χ⟩ = |α|^{2} ⟨0| Â |0⟩ + |β|^{2} ⟨1| Â |1⟩ + 2 ℜ(\overline{α} β ⟨0| Â |1⟩) ∀ Â ∈ L_{sa}(ℂ^{2}) may become a mixture:^{34}

⟨χ′| Â ⊗ ˆ1 |χ′⟩ = |α|^{2} ⟨0| Â |0⟩ + |β|^{2} ⟨1| Â |1⟩ if^{35} ⟨E_{j,k} | E_{lm}⟩ = δ_{jl}δ_{k0}δ_{m0} . In other words:

The environment may cause decoherence.

Therefore, we have to be able to undo unwanted changes caused by the environment. Now, for arbitrary vectors |E_{r}⟩ from the state space of the environment we have

∑_{r=0}^{3}(σ̂_{r} |0⟩) ⊗ |E_{r}⟩ = |0⟩ ⊗ (|E_{0}⟩ + |E_{3}⟩) + |1⟩ ⊗ (|E_{1}⟩ + i |E_{2}⟩)  (4.27)

and

∑_{r=0}^{3}(σ̂_{r} |1⟩) ⊗ |E_{r}⟩ = |0⟩ ⊗ (|E_{1}⟩ − i |E_{2}⟩) + |1⟩ ⊗ (|E_{0}⟩ − |E_{3}⟩) ,  (4.28)

where the σ̂_{r} correspond to the Pauli matrices:^{36}

σ_{0} =^{def} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , σ_{1} =^{def} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , σ_{2} =^{def} ( \begin{matrix}0 & −i \\ +i & 0 \end{matrix} ) , σ_{3} =^{def} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

(4.29)

Equations (4.26)–(4.28) imply

|χ′⟩ = ∑_{r=0}^{3}(σ_{r}(α |0⟩ + β |1⟩ )) ⊗ |E_{r}⟩  (4.30)

if

\begin{matrix}|E_{0,0}⟩ = |E_{0}⟩ + |E_{3}⟩ , & |E_{0,1}⟩ = |E_{1}⟩ + i |E_{2}⟩ , \\ |E_{1,0}⟩ = |E_{1}⟩ − i |E_{2}⟩ , & |E_{1,1}⟩ = |E_{0}⟩ − |E_{3}⟩ , \end{matrix}

i.e. if

\begin{matrix}|E_{0}⟩ = \frac{|E_{0,0}⟩ + |E_{1,1}⟩}{2} , & |E_{1}⟩ = \frac{|E_{0,1}⟩ + |E_{1,0}⟩}{2} , \\ |E_{2}⟩ = \frac{|E_{0,1}⟩ − |E_{1,0}⟩}{2 i} , & |E_{3}⟩ = \frac{|E_{0,0}⟩ − |E_{1,1}⟩}{2} . \end{matrix}

DRAFT, June 26, 2009

33We assume that |α|^{2} + |β|^{2} = 1 and ⟨E | E⟩ = 1 .

34This is why open quantum quantum mechanical systems (Davies, 1976) do not evolve unitarily.

35Consider, e.g., the simple example |E⟩ = |0⟩ , |χ′⟩ = CNOT |χ⟩.

36Hence σ̂_{0} = ˆ1 , σ̂_{1} = ˆ¬ , σ̂_{3} = Ŝ_{π} , σ̂_{2} = i σ̂_{1} σ̂_{3} .

(4.30) tells us that there are only three types of errors to be corrected, corresponding to^{37} σ̂_{1}, σ̂_{2}, σ̂_{3} . In this sense the set of possible errors for single-qubit systems is discrete.

More generally, a unitary operation of the state space of an n-qubit system and its environment acts according to^{38}

∑_{b∈{0,1}^{n}} λ_{b} |b⟩ ⊗ |E⟩ ⟼ ∑_{b∈{0,1}^{n}} λ_{b} ∑_{b′∈{0,1}^{n}} |b′⟩ ⊗ |E_{b,b′}⟩ ,  (4.31)

where the |E_{b,b′}⟩ are suitable state vectors of the environment depending on |E⟩ (and b, b′, of course), but not on the λ_{b} . Now, from (4.27)/(4.28) we see that for arbitrary states |E_{b,b′}⟩ of the environment there are vectors |E_{r}⟩ with

∑_{r=0}^{3}(σ̂_{r} |b⟩) ⊗ |E_{r}⟩ = ∑_{b′∈{0,1}} |b′⟩ ⊗ |E_{b,b′}⟩ ∀ b ∈ {0, 1} .

Straightforward induction shows that for arbitrary state vectors |E_{b,b′}⟩ there are corresponding state vectors |E_{r}(b)⟩ ; r ∈ {0, …, 3}^{n}; with

∑_{r∈{0,1,2,3}^{n}}(σ̂_{r} |b⟩) ⊗ |E_{r}(b)⟩ = ∑_{b′∈{0,1}^{n}} |b′⟩ ⊗ |E_{b,b′}⟩ ∀ b ∈ {0, 1}^{n} ,

where

σ̂_{r} =^{def} σ̂_{r_{1}} ⊗ … ⊗ σ̂_{r_{n}} ∀ r ∈ {0, …, 3}^{n} .

Together with (4.31) this shows that every unitary action on an n-qubit system^{39} and its environment is of the form

∑_{b∈{0,1}^{n}} λ_{b} |b⟩ ⊗ |E⟩ ⟼ ∑_{b∈{0,1}^{n}} λ_{b} ∑_{r∈{0,1,2,3}^{n}}(σ̂_{r} |b⟩) ⊗ |E_{r}(b)⟩ , ∀ b ∈ {0, 1}^{n} .

Usually, in the theory of quantum error correction, only the case

|E_{r}(b)⟩ = |E_{r}⟩ ∀ b ∈ {0, 1}^{n} , r ∈ {0, 1, 2, 3}^{n}

DRAFT, June 26, 2009

37Of course, the σ̂_{ν} are not the only set of operators serving this purpose:

\begin{matrix}σ̂′_{r} =^{def} ∑_{s=0}^{3} u_{rs} σ̂_{s} , |E′_{r}⟩ =^{def} ∑_{s=0}^{3} u_{rs}^{∗} |E_{s}⟩ , ∑_{q=0}^{3} u_{rq} u_{qs}^{∗} = δ_{rs} \\ ⟹ ∑_{r=0}^{3}(σ̂_{r} |b⟩) ⊗ |E_{r}⟩ = ∑_{r=0}^{3}(σ̂′_{r} |b⟩) ⊗ |E′_{r}⟩ ∀ b ∈ \{0, 1\} . \end{matrix}

38This is a simple consequence of linearity and the fact that the |b⟩ form a basis of the n-qubit state space.

39We assume that the qubits are not destroyed. For instance, if a qubit is identified with an atom in a superposition of its ground state and its first excited state, then exciting a higher level destroys this qubit.

is considered.^{40} Then the error action is of the form

Ψ ⊗ |E⟩ ⟼ ∑_{r∈{0,1,2,3}^{n}}(σ̂_{r} Ψ) ⊗ |E_{r}⟩ , Ψ = ∑_{b∈{0,1}^{n}} λ_{b} |b⟩ , (4.32)

and error correction for such quantum noise should be possible along the lines indicated below.

Remark: Alternatively, (4.32) may be written in the form

Ψ ⊗ |E⟩ ⟼ ∑_{a,b∈{0,1}^{n}}(X̂_{a} Ẑ_{b} Ψ) ⊗ |E_{a,b}⟩ ,  (4.33)

where:

X̂_{b} =^{def} (δ_{0b_{1}} + δ_{1b_{1}} σ̂_{1}) ⊗ … ⊗ (δ_{0b_{n}} + δ_{1b_{n}} σ̂_{1}) ,
Ẑ_{b} =^{def} (δ_{0b_{1}} + δ_{1b_{1}} σ̂_{3}) ⊗ … ⊗ (δ_{0b_{n}} + δ_{1b_{n}} σ̂_{3}) .

(4.34)

To explain the essential idea of quantum error correction, let us assume that also for multi-qubits systems only one-qubit errors corresponding to σ̂_{3} (phase errors) occur. In order to conserve an unknown one-qubit state (disentangled from the environment) we first of all encode

Ψ = α |0⟩ + β |1⟩

for arbitrary α, β ∈ ℂ into the (pure) three-qubit state

Ψ̂ = α \left|{ \atop ŵ_{0}}\right⟩ + β \left|{ \atop ŵ_{1}}\right⟩ ,

where

\left|{ \atop ŵ_{0}}\right⟩ =^{def} \frac{1}{\sqrt{4}}(|0, 0, 0⟩ + |0, 1, 1⟩ + |1, 0, 1⟩ + |1, 1, 0⟩) (even parity)
\left|{ \atop ŵ_{1}}\right⟩ =^{def} \frac{1}{\sqrt{4}}(|1, 1, 1⟩ + |1, 0, 0⟩ + |0, 1, 0⟩ + |0, 0, 1⟩) (odd parity) .

(4.35)

This may be done in the following way:

[figure: α |0⟩ + β |1⟩; |0⟩; H; |0⟩; H] \} α \left|{ \atop ŵ_{0}}\right⟩ + β \left|{ \atop ŵ_{1}}\right⟩ .

DRAFT, June 26, 2009

40 For the general case see (Knill et al., 1999). The b-independence of the |E_{r}(Ψ)⟩ is easily derived if every qubit interacts only with its own environment. Note, however, that the |E_{r}⟩ need neither be orthogonal nor normalized nor unique!

Decoding is not more difficult:

α \left|{ \atop ŵ_{0}}\right⟩ + β \left|{ \atop ŵ_{1}}\right⟩ \{ [figure: α |0⟩ + β |1⟩; H; |0⟩; H; |0⟩.]

If the state vector of the total system (three-qubit system plus environment) is Ψ̂⊗|Ê⟩ then, according to our assumption, the interaction between both subsystems can cause only transitions of the form

Ψ̂ ⊗ |Ê⟩ ⟼ ∑_{ν=0}^{3} (ˆσ̂_{3}^{(ν)}Ψ̂) ⊗ |Ê_{3}^{(ν)}⟩

with suitable state vectors |Ê_{3}^{(ν)}⟩ of the environment, where

ˆσ̂_{3}^{(ν)} |b_{1}, b_{2}, b_{3}⟩ =^{def} \{ \begin{matrix}|b_{1}, b_{2}, b_{3}⟩ & if ν = 0 \\ (−1)^{b_{ν}} |b_{1}, b_{2}, b_{3}⟩ & else \end{matrix} ∀ b ∈ \{0, 1\}^{3} .

The essential point is that the subspaces

H_{ν} =^{def} ˆ σ̂_{3}^{(ν)} \{α̂ \left|{ \atop ŵ_{0}}\right⟩ + β̂ \left|{ \atop ŵ_{1}}\right⟩ : α̂, β̂ ∈ ℂ\}

are pairwise orthogonal. Therefore, to restore the original encoded state vector α |0⟩ + β |1⟩, it suffices to perform an optimal test (measurement of first kind ) to which of the four subspaces H_{ν} this state vector belongs and apply σ̂_{3}^{(ν)} according to the outcome.

Exercise 17 Show that the (n + 1)-qubit network

[figure: ⋯]
⋯
⋯
⋮  ⋮

transforms (λ_{0} |0⟩ + λ_{1} |1⟩) ⊗ |0, …, 0⟩ into λ_{0} |0, …, 0⟩ + λ_{1} |1, …, 1⟩ and discuss its possible use for error correction.

4.3.2 Classical Codes

The general idea of classical error correction (Hamming, 1950; Pless, 1989) is the following:

• Consider a channel transmitting n-bit words without changing more than m bits of any word.

• Then the original words can be uniquely reconstructed from the received ones if only special code words w = (w_{1}, …, w_{n}) ∈ {0, 1}^{n} are sent which are chosen such that the Hamming distance

d(w, w′) =^{def} ∑_{ν=1}^{n} |w_{ν} − w′_{ν}| (= ∥w − w′∥^{2})

between any two code words w, w′ is > 2 m.

Obviously, 2^{n} must be larger than the number of code words (the more the larger m is) for error correction to work this way.^{41} Actually:

“Error-correcting coding is the art of adding redundancy efficiently so that most messages, if distorted, can be correctly decoded.”

(Pless, 1989, p. 2)

Especially convenient are the [n, k] linear classical codes, for which a set C ⊂ {0, 1}^{n} of 2^{k} code words — the code — is selected by means of an (n − k) × n-matrix Ĥ as^{42}

C = ker(Ĥ) =^{def} {b ∈ {0, 1}^{n} : Ĥ b = 0} .

Of course, the n − k rows of the so-called parity check^{43} matrix Ĥ have to be independent in order to have

dim(ker(Ĥ)) = k .

Warning: The code C may contain transposed row vectors of the parity check matrix Ĥ.

Without restriction of generality the parity check matrix can be assumed to be of the form^{44}

Ĥ = (Â 1l_{n−k}) ,

DRAFT, June 26, 2009

41Of course this reduces the capacity of the communication channel.

42Here, we identify matrices with the corresponding linear maps. Note that the components of Ĥ are in {0, 1} and that all arithmetic is to be understood modulo 2. Hence, e.g., Ĥ ≡ −Ĥ.

43Every row (h_{1}, …, h_{n}) gives rise to a parity check ∑_{ν=1}^{n} h_{ν} b_{ν} =^{?} 0 on the substring of those bits

of b in places where the row has 1’s .

44This is easily seen using Gaußian elimination. Eventually the bits have to be relabeled.

where Â is some (n − k) × k-matrix. In this form we easily see that

ĤĜ = 0

holds with the n × k-matrix

Ĝ = ( 1l_{k} ; Â ) ,

hence^{45}

C = {Ĝ a : a ∈ {0, 1}^{k}} .

Remark: A possible coding would be

\{0, 1\}^{k} ∋ \underbrace{a}_{word of message} ⟼ \underbrace{Ĝ a}_{corresp. code word} ∈ \{0, 1\}^{n} .

While Ĝ may directly be used as a generator of the code, Ĥ is more convenient for error detection:

Let E ⊂ {0, 1}^{n} be the set of possible ‘errors’ and let Ĥ/^{\}E be an injection. Then the distortion

w ↦ w′ = w + e

of a code word w ∈ C by an error e ∈ E can be identified by checking the error syndrome

Ĥ w′ = Ĥ e

and corrected by adding (=subtracting) e .

Exercise 18 The rows of the r × (2^{r} − 1) parity check matrix characterizing the so-called binary Hamming code Ham[r, 2] are the nonzero elements of {0, 1}^{r}, ordered^{46} according to the value of the corresponding binary numbers.^{47}

a) Show for every r ≥ 2 that Ham[r, 2] is suitable for correcting errors on single bits.

b) Discuss Ham[2, 2] in detail.

Let C be a a [n, k] linear classical code. Then

C^{⊥} =^{def} {b ∈ {0, 1}^{n} : b · b′ = 0 mod 2 ∀ b′ ∈ C}

is called its dual code.

DRAFT, June 26, 2009

45Note that, thanks to 1l_{k}, the rows of Ĝ are all independent.

46Actually, different orderings give rise to equivalent codes.

47See (4.53) for r = 3 .

Exercise 19 Let C be a [n, k] linear classical code with parity check matrix Ĥ and generator Ĝ. Show the following:^{48}

a) C^{⊥} is a [n − k, n] linear classical code with parity check matrix Ĥ^{⊥} = Ĝ^{T} and generator Ĝ^{⊥} = Ĥ^{T} .

b)

(C^{⊥})^{⊥} = C .

c)

∑_{b′∈C} (−1)^{b·b′} = \{ \begin{matrix}|C| & if b ∈ C^{⊥}, \\ 0 & if b ∈ \{0, 1\}^{n} \ C^{⊥}. \end{matrix}

4.3.3 Quantum Codes

In classical communication the received message may be inspected and corrected according to the error syndrome. In quantum communication, however, we should carefully avoid too detailed ‘measurement’ (associated with uncontrollable ‘collapse’) of the state before reconstruction. Therefore, in order to be able to correct all errors corresponding to error operations σ̂ ∈ 𝔈 we have to look for quantum codes^{49} of the following form:

• The n-qubit state space H containing the quantum code words is a direct sum of specified subspaces H_{d}.

• Every σ̂ ∈ 𝔈 is of a definite type d, i.e. σ̂ \left|{ \atop ŵ}\right⟩ ∈ H_{d} holds for all quantum code words^{50} \left|{ \atop ŵ}\right⟩.

• If σ̂, σ̂′ ∈ 𝔈 are of the same type d then σ̂ |w⟩ ∼ σ̂′ |w⟩ holds for all quantum code words |w⟩ (but not necessarily for other state vectors).

Under these conditions — if only errors corresponding to operations σ̂ ∈ 𝔈 are superimposed^{51} — quantum error correction is possible as indicated in 4.3.1:

• The ‘received’ state is forced — via corresponding ‘measurement’ — to ‘collapse’ into a state described by an element of one of the subspaces H_{d}.

• Since the ‘collapsed’ state is just the sent code word distorted by an error of type d we only have to apply the inverse of some unitary error operation of type d to reconstruct the correct code word.

DRAFT, June 26, 2009

48As usual, we denote the number of elements of a finite set C by |C|.

49By n-qubit quantum code we always mean the set of pairwise orthogonal n-qubit state vectors used as quantum code words. Note, however, that many authors mean by quantum code the complex linear span of quantum code words.

50In 4.3.1 we already used linear superpositions \left|{ \atop ŵ_{0}}\right⟩ , \left|{ \atop ŵ_{1}}\right⟩ of the computational base states as quantum code words, in order to indicate additional possibilities in quantum coding.

51Of course, 𝔈 should include the trivial ‘error operation’ σ̂ = ˆ1 .

4.3. ERROR CORRECTING CODES 113 Exercise 20

a) Show that for Shor’s 9-qubit code words

\begin{matrix}|ˆŵ_{0}⟩ =^{def} 2^{−3/2}(|0, 0, 0⟩ + |1, 1, 1⟩) ⊗ (|0, 0, 0⟩ + |1, 1, 1⟩) ⊗ (|0, 0, 0⟩ + |1, 1, 1⟩) , \\ |ˆŵ_{1}⟩ =^{def} 2^{−3/2}(|0, 0, 0⟩ − |1, 1, 1⟩) ⊗ (|0, 0, 0⟩ − |1, 1, 1⟩) ⊗ (|0, 0, 0⟩ − |1, 1, 1⟩) \end{matrix}

every superposition of single-qubit errors, i.e. every distortion of the form

(α |ˆŵ_{0}⟩ + β |ˆŵ_{1}⟩) ⊗ |E⟩ ⟼ ∑_{r∈R_{9}}(α σ̂_{r} |ˆŵ_{0}⟩ + β σ̂_{r} |ˆŵ_{1}⟩) ⊗ |E_{r}⟩ ,

R_{9} =^{def} ⋃_{r=0}^{3}{permutations of (r, 0, 0, 0, 0, 0, 0, 0, 0)} ,

may be corrected as described above.

b) Show that encoding Ψ = α |0⟩ + β |1⟩ into α |ˆŵ_{0}⟩ + β |ˆŵ_{1}⟩ may be achieved as follows:

[figure: Ψ; H; |0⟩; |0⟩; |0⟩; H; |0⟩; |0⟩; |0⟩; H; |0⟩; |0⟩] } α |ˆŵ_{0}⟩ + β |ˆŵ_{1}⟩

Recall that, according to (4.33), for every n ∈ ℕ the possible error n-qubit error operations are elements of the Pauli group^{52}

𝔖_{n} =^{def} {i^{ν} X̂_{b_{1}} Ẑ_{b_{3}} : ν ∈ {0, …, 3} , b_{1}, b_{3} ∈ {0, 1}^{n}}  (4.36)

DRAFT, June 26, 2009

52The X̂_{b} and Ẑ_{b} were defined by (4.34) and (4.29).

That the latter is a group w.r.t. operator multiplication follows immediately from the algebra of Pauli matrices:

\begin{aligned}σ̂_{ν} σ̂_{ν} & = & ˆ1  ∀ ν ∈ \{0, 1, 2, 3\} , \\ σ̂_{j} σ̂_{k} & = & − σ̂_{k} σ̂_{j} ∀ j, k ∈ \{1, 2, 3\} , j ≠ k , \\ σ̂_{1} σ̂_{2} & = & i σ̂_{3} , \\ σ̂_{2} σ̂_{3} & = & i σ̂_{1} , \\ σ̂_{3} σ̂_{1} & = & i σ̂_{2} . \end{aligned}  (4.37)

The following statements also follow directly from these relations:

σ̂_{ν} = σ̂_{ν}^{∗} = σ̂_{ν}^{−1} ∀ ν ∈ {0, 1, 2, 3} ,  (4.38)
𝔖_{n} ⊃ 𝔖_{n}^{0} =^{def} {σ̂_{r_{1}} ⊗ ⋯ ⊗ σ̂_{r_{n}} : r_{1}, …, r_{n} ∈ {0, …, 3}} is not a group , (4.39)
i^{ν} X̂_{b_{1}} Ẑ_{b_{3}} ∈ 𝔖_{n}^{0} ⟺ i^{ν} = i^{b_{1}·b_{3}}  ∀ ν ∈ {0, 1, 2, 3} , b_{1}, b_{3} ∈ {0, 1}^{n} , (4.40)
σ̂σ̂′ ∈ {+σ̂′σ̂ , −σ̂′σ̂} ∀ σ̂, σ̂′ ∈ 𝔖_{n} .  (4.41)

Theorem 4.3.1 Let W ⊂ H be an n-qubit quantum code and let 𝔈 ⊂ 𝔖_{n} be a set of error operations including the trivial operation ˆ1 . Assume that the linear span H_{W} of W is stabilized by the subset 𝔖_{W} of 𝔖_{n}, i.e. that

H_{W} = {Ψ̂ ∈ H : ĝ Ψ̂ = Ψ̂ ∀ ĝ ∈ 𝔖_{W}} .  (4.42)

Moreover, assume

σ̂^{∗} σ̂′ ∉ 𝔑(𝔖_{W}) \ 𝔖_{W} ∀ σ̂, σ̂′ ∈ 𝔈 ,  (4.43)

where 𝔑(𝔖_{W}) denotes the normalizer of 𝔖_{W} :

𝔑(𝔖_{W}) = {σ̂ ∈ 𝔖_{n} : σ̂ ĝ σ̂^{∗} = ĝ ∀ ĝ ∈ 𝔖_{W}} .

Then there is a unique mapping dσ̂ from 𝔖_{W} into {+1, −1} such that:

ĝ (σ̂ Ψ̂) = dσ̂(ĝ) (σ̂ Ψ̂) ∀ ĝ ∈ 𝔖_{W} , σ̂ ∈ 𝔈 , Ψ̂ ∈ H_{W} , (4.44)
σ̂ H_{W} ⊂ H_{dσ̂}  =^{def} {Φ̂ ∈ H : ĝ Φ̂ = dσ̂(ĝ) Φ̂} ∀σ̂ ∈ 𝔈 ,  (4.45)
dσ̂ ≠ d_{}_{σ̂′} ⟹ H_{dσ̂} ⊥ H_{d_{σ̂′}} ∀σ̂, σ̂′ ∈ 𝔈 ,  (4.46)
dσ̂ = d_{}_{σ̂′} ⟹ σ̂ Ψ̂ = σ̂′ Ψ̂ ∀σ̂, σ̂′ ∈ 𝔈 , Ψ̂ ∈ H_{W} .  (4.47)

Outline of proof: (4.44) is a direct consequence of (4.41) and (4.42). (4.44) directly implies (4.45). Since^{53}

ĝ = ĝ^{∗} ∀ ĝ ∈ 𝔖_{W}  (4.48)

DRAFT, June 26, 2009

53This is because for all σ̂ ∈ 𝔖_{n} we have σ̂^{2} = ˆ1 ⟺ σ̂ = σ̂^{∗}.

— and since eigenvectors corresponding to different eigenvalues of a self-adjoint operator are always orthogonal — (4.44) also implies (4.46). Finally, (4.44) and (4.41) imply

ĝ σ̂^{(∗)} = dσ̂(ĝ) σ̂^{(∗)} ĝ ∀ ĝ ∈ 𝔖_{W} , σ̂ ∈ 𝔈

and hence

\begin{aligned}(σ̂^{∗} σ̂′) ĝ (σ̂^{∗} σ̂′)^{∗} & = & dσ̂(ĝ) d_{}_{σ̂′}(ĝ) ĝ (σ̂^{∗} σ̂′) (σ̂^{∗} σ̂′)^{∗} \\ & =_{(4.38)} & dσ̂(ĝ) d_{}_{σ̂′}(ĝ) ĝ ∀ ĝ ∈ 𝔖_{W} , σ̂, σ̂′ ∈ 𝔈 . \end{aligned}

Therefore (4.47) follows according to

dσ̂ = d_{}_{σ̂′} ⟹ (σ̂^{∗} σ̂′) ĝ (σ̂^{∗} σ̂′)^{∗} = ĝ ∀ ĝ ∈ 𝔖_{W}
⟹ σ̂^{∗} σ̂′ ∈ 𝔑(𝔖_{W})
⟹_{(4.43)} σ̂^{∗} σ̂′ ∈ 𝔖_{W}
⟹_{(4.42)} σ̂^{∗} σ̂′ Ψ̂ = Ψ̂ ∀Ψ̂ ∈ H_{W}
⟹_{(4.38)} σ̂′ Ψ̂ = σ̂ Ψ̂ ∀Ψ̂ ∈ H_{W} .

Remarks:

1. In view of (4.43), 𝔖_{W} should be chosen as large as possible.

2. The maximal 𝔖_{W} fulfilling the requirements of Theorem 4.3.1 for given H_{W} is an abelian group called the stabilizer of H_{W} .

3. Quantum codes W fulfilling the requirements of Theorem 4.3.1 are called stabilizer codes.

4. If there are σ̂ , σ̂′ ∈ 𝔈 and \left|{ \atop ŵ}\right⟩ ∈ W with

σ̂ \left|{ \atop ŵ}\right⟩ = σ̂′ \left|{ \atop ŵ}\right⟩ but σ̂ ≠ σ̂′

then the code is called degenerate w.r.t. 𝔈.

5. Shor’s 9-qubit code, described in Exercise 20, is degenerate w.r.t. the set of single qubit error operations. This can be easily seen by considering phase flip errors on different qubits.

6. A stabilizer code is nondegenerate w.r.t 𝔈 iff σ̂ ≠ σ̂′ ⟹ σ̂^{∗}σ̂′ ∉ 𝔖_{W} ∀ σ̂, σ̂′ ∈ 𝔈 .

7. In classical coding there is no analog for degeneracy.

Lemma 4.3.2 For j ∈ {1, 2}, let the C_{j} be a [n, k_{j}] linear classical codes with C_{2} ⊂ C_{1} ≠ C_{2} , and define

W = CSS (C_{1}, C_{2}) =^{def} \{\left|{ \atop ŵ_{b}}\right⟩ =^{def} \frac{1}{\sqrt{|C_{2}|}} ∑_{b′∈C_{2}} |b + b′⟩ : b ∈ C_{1}\} .  (4.49)

where H denotes the n-qubit state space. Then (4.42) holds for

𝔖_{W} = {X̂_{a} Ẑ_{b} : a ∈ C_{2} , b ∈ C_{1}^{⊥}} .  (4.50)

Outline of proof: Let

Ψ̂ = ∑_{b∈{0,1}^{n}} λ_{b} |b⟩ ∈ {Ψ̂ ∈ H : ĝ Ψ̂ = Ψ̂ ∀ ĝ ∈ 𝔖_{W}} .

Then

Ψ̂ =_{(4.50)} \frac{1}{|C_{1}^{⊥}|} ∑_{b′∈C_{1}^{⊥}} Ẑ_{b′} ∑_{b∈\{0,1\}^{n}} λ_{b} |b⟩
= ∑_{b∈\{0,1\}^{n}} λ_{b} \frac{1}{|C_{1}^{⊥}|} ∑_{b′∈C_{1}^{⊥}}(−1)^{b·b′} |b⟩
=_{Ex. 19} ∑_{b∈C_{1}} λ_{b} |b⟩
=_{(4.50)} \frac{1}{|C_{2}|} ∑_{b′∈C_{2}} X̂_{b′} ∑_{b∈C_{1}} λ_{b} |b⟩
= ∑_{b∈C_{1}} λ_{b} \left|{ \atop ŵ_{b}}\right⟩ .

Since, obviously,

X̂_{a} Ẑ_{b′} \left|{ \atop ŵ_{b}}\right⟩ = \left|{ \atop ŵ_{b}}\right⟩ ∀ a ∈ C_{2} , b′ ∈ C_{1}^{⊥} , b ∈ C_{1}

this proves the lemma.

Remarks:

1. The quantum codes CSS(C_{1}, C_{2}) described by Lemma 4.3.2 are called Calderbank-Shor-Steane codes .

2. The number of code words for these codes is

|CSS (C_{1}, C_{2})| = \frac{|C_{1}|}{|C_{2}|} = 2^{k_{1}−k_{2}} .

Lemma 4.3.3 Let C_{j}, W and 𝔖_{W} be given as in Lemma 4.3.2. If C_{1} as well as C_{2}^{⊥} is suitable for correcting errors on up to t bits then

σ̂ ≠ σ̂′ ⟹ σ̂^{∗} σ̂′ ∉ 𝔑(𝔖_{W})  ∀ σ̂, σ̂′ ∈ 𝔈  (4.51)

holds for

𝔈 = {X̂_{e_{1}} Ẑ_{e_{3}} : e_{1}, e_{3} ∈ {b ∈ {0, 1}^{n} : ∑_{ν=1}^{n} |b^{ν}| ≤ t}} .  (4.52)

4.3. ERROR CORRECTING CODES 117

Outline of proof: Consider σ̂, σ̂′ ∈ 𝔈 with σ̂ ≠ σ̂′. Then there are

e_{j}, e′_{j} ∈ {b ∈ {0, 1}^{n} : ∑_{ν=1}^{n} |b^{ν}| ≤ t}

with^{54}

{e_{1} + e′_{1} , e_{3} + e′_{3}} ≠ {0}

and

σ̂^{∗} σ̂′ = (X̂_{e_{1}} Ẑ_{e_{3}})^{∗} X̂_{e′}_{1} Ẑ_{e′}_{3}
= Ẑ_{e_{3}} X̂_{e_{1}+e′_{1}} Ẑ_{e′}_{3}
= (−1)^{e_{3}·(e_{1}+e′_{1})} X̂_{e_{1}+e′_{1}} Ẑ_{e_{3}+e′_{3}}

Hence

(σ̂^{∗} σ̂′) X̂_{a} Ẑ_{b} (σ̂^{∗} σ̂′)^{∗} = X̂_{e_{1}+e′_{1}} Ẑ_{e_{3}+e′_{3}} X̂_{a} Ẑ_{b} ˆ Ẑ_{e_{3}+e′_{3}}X_{e_{1}+e′_{1}}
= i^{(e_{3}+e′_{3})·a} i^{b·(e_{1}+e′_{1})} X̂_{a} Ẑ_{b} ∀ a b ∈ {0, 1}^{n} .

If e_{3} + e′_{3} ≠ 0 then (e_{3} + e′_{3}) · a ≠ 0 mod 2 and, consequently,

(σ̂^{∗} σ̂′) X̂_{a} (σ̂^{∗} σ̂′)^{∗} = −X̂_{a}

for some a ∈ C_{2} since the generator of C_{2} is the parity check matrix of C_{2}^{⊥}. On the other hand, if e_{1} + e′_{1} ≠ 0 then b · (e_{1} + e′_{1}) ≠ 0 mod 2 and, consequently,

(σ̂^{∗} σ̂′) Ẑ_{b} (σ̂^{∗} σ̂′)^{∗} = −Ẑ_{b}

for some b ∈ C_{1}^{⊥}. Thus σ̂^{∗} σ̂′ ∉ 𝔑 (𝔖_{W}).

Remark: Obviously, CSS(C_{1}, C_{2}) is nondegenerate w.r.t. 𝔈 specified by Lemma 4.3.3.

In general, the number of operations^{55} σ̂_{r} affecting at most t ∈ {0, …, n} qubits

of an n-qubit system is^{56} ∑_{j=0}^{t} \binom{n}{j}3^{j}. Therefore, in order to correct all corresponding

errors for a nondegenerate n-qubit code according to the scheme described above, that many subspaces H_{d} are needed. Moreover, the dimension of each of these subspaces must not be smaller than the number of code words. Therefore:

Correction of all errors on at most t qubits of a nondegenerate nqubit code spanned by 2^{k} orthogonal code words is not possible if the quantum Hamming bound

∑_{j=0}^{t} \binom{n}{j}3^{j} 2^{k} ≤ 2^{n}

is violated.

DRAFT, June 26, 2009

54Recall Footnote 42. 55Recall (4.32). 56The index j = 0 corresponds to the trivial error operation (unit operator).

Note that for k = t = 1 the quantum Hamming bound becomes 2 + 6n ≤ 2^{n}, hence n ≥ 5 .

For further details on quantum codes see (Preskill, 01, Chapter 7), and (Schlingemann and Werner, 2000; Schlingemann, 2001; Keyl and Werner, 2002).

4.3.4 Reliable Quantum Computation

Let us discuss the implementation of error correction in more detail. For simplicity, we consider only the quantum code CSS(Ham[3, 2], Ham[3, 2]^{⊥}), called the Steane code. According to Exercise 18 a parity check matrix for Ham[3, 2] is

Ĥ_{3} =^{def} ( \begin{matrix}0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} ) .  (4.53)

Exercise 21

a) Show that the code words corresponding to the parity check matrix Ĥ_{3} are the same as those corresponding to the parity check matrix

Ĥ′_{3} =^{def} ( \begin{matrix}0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 \end{matrix} ) .

b) Show that^{57}

Ĥ_{4} =^{def} ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{matrix} )

is a parity matrix for Ham[3, 2]^{⊥}.

c) Show that

Ham[3, 2]^{⊥} = {b ∈ {0, 1}^{7} : Ĥ_{3} b = 0 , (−1)^{b_{1}+...+b_{7}} = 1} .

According to Exercise 21, the quantum code words of the Steane code are

\begin{aligned}\left|{ \atop ŵ_{0}}\right⟩ & = & \frac{1}{\sqrt{8}}(|0000000⟩ + |1101001⟩ + |1011010⟩ + |0111100⟩ \\ & & + |0110011⟩ + |1100110⟩ + |1010101⟩ + |0001111⟩) , \\ \left|{ \atop ŵ_{1}}\right⟩ & = & \frac{1}{\sqrt{8}}(+ |1111111⟩ + |0010110⟩ + |0100101⟩ + |1000011⟩ \\ & & + |1001100⟩ + |0011001⟩ + |0101010⟩ + |1110000⟩) , \end{aligned}  (4.54)

Exercise 22 Show for (4.54) that the following network acts as indicated:

[figure: α |0⟩ + β |1⟩; |0⟩; H; |0⟩; H; |0⟩; H; |0⟩; |0⟩; |0⟩] \} α \left|{ \atop ŵ_{0}}\right⟩ + β \left|{ \atop ŵ_{1}}\right⟩ .

Exercise 23 Show that the following 10-qubit network acts as indicated:

|b_{1}⟩  |b_{1}⟩
|b_{2}⟩  |b_{2}⟩
|b_{3}⟩  |b_{3}⟩
|b_{4}⟩  |b_{4}⟩
|b_{5}⟩  |b_{5}⟩
|b_{6}⟩  |b_{6}⟩
|b_{7}⟩  |b_{7}⟩
[figure: |0⟩]
[figure: |0⟩; } |Ĥ_{3} b⟩ ,]
[figure: |0⟩]

where, e.g.,

[figure: ⋮; ≡^{def}; ^{⋮  ⋮}; .]

The network of Exercise 23 may be used to reduce single-qubit errors of type σ̂_{1} or/and σ̂_{2} to those of type σ̂_{0} or/and σ̂_{3} :

An ideal test for the computational basis of the last 3 (ancillary) qubits causes the 10-qubit state to collapse into some state being the direct product of

• a (possibly only partially coherent) superposition of distortions of the original code word by single-qubit errors, appearing in one and the same position if of type σ̂_{1} or σ̂_{2} , and

• a base state of the ancillary 3-qubit system which corresponds either to (0, 0, 0) , if the collapsed 7-qubit state is a distortion of the original code word by single-qubit errors of only type σ̂_{0} or/and σ̂_{3} , or else corresponds to the classical error syndrome of a bit-flip in the position, where the code word is distorted by an error of type σ̂_{1} or/and σ̂_{2} .

If the collapsed state of the ancillary system does not correspond to (0, 0, 0) then σ̂_{1} should be applied to the qubit in the position where the code word is distorted. In any case, then, the resulting 7-qubit state will be a (possibly only partially coherent) superposition of distortions of the original code word by single-qubit errors of type σ̂_{0} or/and σ̂_{3} .

Exercise 24 Show that the following 10-qubit network flips the (∑_{j=1}^{3} e_{j} 2^{3−j})-th

qubit for input of the specified type with |e_{1}, e_{2}, e_{3}⟩ ≠ |0, 0, 0⟩ and, therefore, may be used to avoid testing the error syndrome for single-qubit errors of type σ̂_{1} or/and σ̂_{2} :

[figure: |e_{1}⟩; |e_{1}⟩]
[figure: |e_{2}⟩; |e_{2}⟩]
[figure: |e_{3}⟩; |e_{3}⟩]

where

≡^{def}  ^{ˆ¬}  ^{}^{ˆ¬} ,  ^{}≡^{def}  ^{ˆ¬}  ^{}^{ˆ¬} .

The eventually remaining single-qubit errors of only type σ̂_{0} or/and σ̂_{3}. may be converted into errors of type σ̂_{0} or/and σ̂_{1} by applying Û_{H}^{⊗n}. Correction these errors as just described and applying Û_{H}^{⊗n} once more restores the original message.

Up to now we tacitly assumed that all devices used for error correction work perfectly error free. Of course this is unrealistic and, actually, special care has to be taken to prevent these devices from making things worse.

For instance, if a phase error appears for the first ancillary qubit of the error syndrome network presented in Exercise 23 then according to Exercise this error may propagate into all of the last four data qubits. To prevent this one could use

[figure: |0⟩]
[figure: |0⟩]
[figure: |0⟩]
[figure: |0⟩]

instead of

[figure: |0⟩]

and implement in a suitable way.^{58}

While such precautions prohibit propagation of errors of the ancillary part of the network into the data part they do not guarantee a correct error syndrome. Therefore the ‘measurement’ of the error syndrome should be repeated and only used for error correction if confirmed.

In order to protect calculations against quantum noise they should be performed directly on the encoded data.

Of course, the encoded data should be error checked sufficiently often. Especially, the actual computation should not be started before the initial encoded state has been checked to be free of errors.

DRAFT, June 26, 2009

58See (Möttönen and Vartiainen, 2005, Fig. 8), in this connection.

Altogether it seems possible to implement reliable quantum computation, if sufficient care is taken. For further details see (Preskill, 1998b; Preskill, 1998a; Leung, 2000).

4.4 Entanglement Assisted Channels^{59}

“Entanglement is monogamous — the more entangled Bob is with Alice, the less entangled he can be with anyone else.”

Charles Bennett^{60}

4.4.1 Quantum Dense Coding^{61}

Let S = S_{1} ⊕ S_{2} be a bipartite^{62} 2-qubit system with state space H ⊗ H and computational basis {|ν, μ⟩ =^{def} ϕ_{ν} ⊗ ϕ_{μ}}_{ν,μ∈{0,1}}. Then the so-called Bell states

Φ^{+} =^{def} \frac{1}{\sqrt{2}} (|0, 0⟩ + |1, 1⟩) ,

Φ^{−} =^{def} \frac{1}{\sqrt{2}} (|0, 0⟩ − |1, 1⟩) ,

Ψ^{+} =^{def} \frac{1}{\sqrt{2}} (|0, 1⟩ + |1, 0⟩) ,

Ψ^{−} =^{def} \frac{1}{\sqrt{2}} (|0, 1⟩ − |1, 0⟩)

(4.55)

form an orthonormal basis of H ⊗ H and may be locally transformed into each other:^{63}

\begin{aligned}Φ^{∓} & = & (σ̂_{3} ⊗ ˆ1) Φ^{±} , \\ Ψ^{∓} & = & (σ̂_{3} ⊗ ˆ1) Ψ^{±} , \\ Ψ^{+} & = & (σ̂_{1} ⊗ ˆ1) Φ^{+} . \end{aligned}  (4.56)

Obviously, 2 classical bits of information may be encoded via the Bell states, e.g.:

(0, 0) ˆ= Φ^{+} , (0, 1) ˆ= Φ^{−} , (1, 0) ˆ= Ψ^{+} , (1, 1) ˆ= Φ^{−} .

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59See Section 1.2.2 for the network models of dense coding, teleportation, and entanglement swapping. See also (Devetak and Winter, 2003; Devetak et al., 2004) for related protocols.

  60http://qpip-server.tcs.tifr.res.in/ qpip/HTML/Courses/Bennett/TIFR2.pdf

61See also (Mermin, 2002).

62See Appendix A.4.3.

63As usual, we denote by σ̂_{1}, …, σ̂_{3} the Pauli operators, i.e. w.r.t. (|0⟩ , |1⟩) :

σ̂_{1} ˆ= ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , σ̂_{2} ˆ= ( \begin{matrix}0 & −i \\ +i & 0 \end{matrix} ) , σ̂_{3} ˆ= ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

Thus, if Alice and Bob (situated arbitrarily far apart) initially share a pair of qubits forming S_{1} ⊕ S_{2} in a Bell state,^{64} say Ψ_{−}, then Alice may communicate 2 bits of information by sending Bob her single qubit (system S_{1}) after acting on it in an appropriate way:

2-bit information operation by Alice new Bell state
(0,0)  σ̂_{1} σ̂_{3}  Φ^{+}
(0,1)  σ̂_{3} σ̂_{1} σ̂_{3}  Φ^{−}
(1,0)  σ̂_{3}  Ψ^{+}
(1,1)  none  Ψ^{−}

After receiving Alice’s qubit Bob just has to perform a projective measurement w.r.t. the Bell basis^{65} {Φ^{+}, Φ^{−}, Ψ^{+}, Ψ^{−}} in order to decode the 2-bit information.

Needless to say, without entanglement Alice would not have any chance to transmit more than a single bit by sending just a single qubit. Therefore, the described procedure to communicate 2 bits by sending just 1 qubit is called quantum dense coding (of classical information). Of course, the crucial point is that Alice has to be given one partner of an entangled pair of qubits first. Note, however, that Alice and Bob may store their qubits for some time in suitable quantum memory^{66} before starting to communicate. Then the information carried by the sent qubit is of no use for any potential eavesdropper.

4.4.2 Quantum Teleportation

Consider, e.g., a 3-qubit system S = S_{1} ⊕ S_{2} ⊕ S_{3} with state space H ⊗ H ⊗ H and computational basis {|α, β, γ⟩ =^{def} ϕ_{α} ⊗ ϕ_{β} ⊗ ϕ_{γ}}_{α,β,γ∈{0,1}} in the initial state^{67}

Ψ_{0} = ψ ⊗ \frac{1}{\sqrt{2}} (ϕ_{0} ⊗ ϕ_{1} − ϕ_{1} ⊗ ϕ_{0}) .  (4.57)

In order to determine the effect of a projective measurement on the subsystem S_{1}⊕S_{2} w.r.t. its Bell basis we rewrite this state in the form

Ψ_{0} = Φ^{+} ⊗ χ_{0} + Φ^{−} ⊗ χ_{1} + Ψ^{+} ⊗ χ_{2} + Ψ^{−} ⊗ χ_{3} .  (4.58)

Writing

ψ = α |0⟩ + β |1⟩

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64For the creation of photon pairs in Bell states via parametric down conversion see, e.g., (Gatti et al., 2003) and references given there.

65For the implementation of such measurements see, e.g., (Paris et al., 2000; Tomita, 2000; Kim et al., 2001).

66For the possibility of storing optical qubits see (Gingrich et al., 2003).

67Obviously, then, the subsystem S_{2} ⊕ S_{3} is in the Bell state corresponding to Ψ^{−}.

and comparing

\begin{matrix}\sqrt{2} ψ ⊗ \frac{1}{\sqrt{2}} (ϕ_{0} ⊗ ϕ_{1} − ϕ_{1} ⊗ ϕ_{0}) \\ = α |0, 0, 1⟩ − α |0, 1, 0⟩ − β |1, 1, 0⟩ + β |1, 0, 1⟩ \end{matrix}

with

\sqrt{2} Φ^{+} ⊗ χ_{0} + Φ^{−} ⊗ χ_{1} + Ψ^{+} ⊗ χ_{2} + Ψ^{−} ⊗ χ_{3} ; \begin{aligned}= & \frac{1}{2}(|0, 0⟩ ⊗ χ_{0} + |1, 1⟩ ⊗ χ_{0} + |0, 0⟩ ⊗ χ_{1} − |1, 1⟩ ⊗ χ_{1} \\ & + |0, 1⟩ ⊗ χ_{2} + |1, 0⟩ ⊗ χ_{2} + |0, 1⟩ ⊗ χ_{3} − |1, 0⟩ ⊗ χ_{3}) \\ = & |0, 0⟩ ⊗ \frac{χ_{0} + χ_{1}}{2} + |0, 1⟩ ⊗ \frac{χ_{2} + χ_{3}}{2} + |1, 0⟩ ⊗ \frac{χ_{2} − χ_{3}}{2} + |1, 1⟩ ⊗ \frac{χ_{0} − χ_{1}}{2} \end{aligned}

we get

\begin{aligned}χ_{0} + χ_{1} & = & +α |1⟩ , \\ χ_{2} + χ_{3} & = & −α |0⟩ , \\ χ_{2} − χ_{3} & = & +β |1⟩ , \\ χ_{0} − χ_{1} & = & −β |0⟩ \end{aligned}

and hence

\begin{aligned}χ_{0} & = & α |1⟩ − β |0⟩ , \\ & = & σ̂_{1} σ̂_{3} ψ \\ χ_{1} & = & α |1⟩ + β |0⟩ , \\ & = & σ̂_{1} ψ \\ χ_{2} & = & β |1⟩ − α |0⟩ , \\ & = & −σ̂_{3} ψ \\ χ_{3} & = & −α |0⟩ − β |1⟩ , \\ & = & −ψ . \end{aligned}  (4.59)

(4.57)–(4.59) show the possibility of quantum teleportation (of quantum information):^{68}

If Alice and Bob (situated arbitrarily far apart) initially share a pair of qubits forming S_{2} ⊕ S_{3} in the Bell state \frac{1}{\sqrt{2}} (ϕ_{0} ⊗ ϕ_{1} − ϕ_{1} ⊗ ϕ_{0}) then Alice may communicate to Bob the quantum information contained in the unknown state ψ in the following way:

Alice performs a projective measurement on S_{1} ⊕ S_{2} w.r.t. the Bell basis {Φ^{+}, Φ^{−}, Ψ^{+}, Ψ^{−}} and tells Bob her result via classical communication. Bob just has to perform one of the operations σ̂_{1}σ̂_{3} , σ̂_{1} , σ̂_{3} or none — according to the outcome of Alice’s measurement — on his

DRAFT, June 26, 2009

68See (Sanctuary et al., 2003) for critical remarks on corresponding experiments.

qubit (system S_{3}) in order to have the latter in the state ±ψ :

result of Alice’s measurement Bob’s operation
Φ^{+}  σ̂_{3}σ̂_{1} = (σ̂_{1}σ̂_{3})^{−1}
Φ^{−}  σ̂_{1} = \binom{}{σ̂_{1}}^{−1}
Ψ^{+}  σ̂_{3} = \binom{}{σ̂_{3}}^{−1}
Ψ^{−}  none

Note that the classical information sent by Alice would be of no use to an eavesdropper and that sending classical information avoids the decoherence problems connected with sending qubits.

4.4.3 Entanglement Swapping

Consider a 4-qubit system S = S_{0} ⊕ S_{1} ⊕ S_{2} ⊕ S_{3} in the initial state

Ψ̂_{0} = Ψ^{−} ⊗ Ψ^{−} .  (4.60)

Then the calculations of Section 4.4.2 show that

Ψ̂_{0} = (ˆ1 ⊗ ˆ1 ⊗ ˆ1 ⊗ σ̂_{1}σ̂_{3}) \frac{1}{2}(|0⟩ ⊗ Φ^{+} ⊗ |1⟩ − |1⟩ ⊗ Φ^{+} ⊗ |0⟩)

(ˆ1 ⊗ ˆ1 ⊗ ˆ1 ⊗ σ̂_{1}) \frac{1}{2}(|0⟩ ⊗ Φ^{−} ⊗ |1⟩ − |1⟩ ⊗ Φ^{−} ⊗ |0⟩)

− (ˆ1 ⊗ ˆ1 ⊗ ˆ1 ⊗ σ̂_{3}) \frac{1}{2}(|0⟩ ⊗ Ψ^{+} ⊗ |1⟩ − |1⟩ ⊗ Ψ^{+} ⊗ |0⟩)

− \frac{1}{2}(|0⟩ ⊗ Ψ^{−} ⊗ |1⟩ − |1⟩ ⊗ Ψ^{−} ⊗ |0⟩) .

(4.61)

Now assume that

Victor has access to S_{0} ,
Alice has access to S_{1} ⊕ S_{2} ,
Bob has access to S_{3} .

Then — even though Victor, Alice, and Bob may be arbitrarily far apart, the entanglement of the subsystem S_{0} ⊕ S_{1} may be swapped to the subsystem S_{0} ⊕ S_{3} in the following way:

Alice performs a projective measurement on S_{1}⊕S_{2} w.r.t. the Bell basis {Φ^{+}, Φ^{−}, Ψ^{+}, Ψ^{−}} and tells Bob her result via classical communication. Bob just has to perform one of the operations σ̂_{1}σ̂_{3} , σ̂_{1} , σ̂_{3} or none — according to the outcome of Alice’s measurement — on his qubit (system S_{3}) in order to have the partial state of the subsystem S_{0} ⊕ S_{3} in the state Ψ^{−}.

Thus Alice may act as an entanglement provider:

Alice prepares pairs of entangled qubits and distributes one partner of each pair to various customers including Victor and Bob. If Victor and Bob need to share an entangled pair they instruct Alice to perform a projective measurement on S_{1} ⊕ S_{2} w.r.t. the Bell basis {Φ^{+}, Φ^{−}, Ψ^{+}, Ψ^{−}} and communicate the result to either Victor and Bob who then knows the type of entanglement of the pair shared with Bob.

4.4.4 Quantum Cryptography^{69}

As explained in 2.2.1 the security of the RSA encryption scheme relies on the extreme difficulty to factorize large numbers n by classical means (and the possibility of authentication of submitted messages). Otherwise d could be determined from n and e. However, in view of Shor’s quantum factoring algorithm (Shor, 1994) and the possible implementation of quantum computers such classical cryptosystems as RSA may become insecure. Fortunately, quantum mechanics itself offers means for secure communication exploiting the Vernam cipher^{70} (also called one time pad):

Exploiting quantum mechanisms, Victor and Bob agree on a purely random secret key c = (c_{1}, …, c_{n}) ∈ {0, 1}^{n}. Then, instead of sending Bob the pure the plaintext message c = (c_{1}, …, c_{n}) ∈ {0, 1}^{n} Victor sends him the encoded message^{71}

e = (b_{1} ⊕ c_{1}, …, b_{n} ⊕ c_{n})

(through some public channel) which Bob may decrypt as

b = (e_{1} ⊕ c_{1}, …, e_{n} ⊕ c_{n})

but appears purely random to all eventual eavesdroppers.

As shown by Claude Shannon (Shannon, 1949a), this cryptosystem is absolutely secure if e is kept secret and used only once.

If Victor and Bob share enough (nearly) maximally entangled pairs of qubits^{72} they may establish a secret key in the following way:

DRAFT, June 26, 2009

69See (Bowmeester et al., 2000, Chapter 2) for a nice introduction and (Elliott et al., 2005) for actual implementation. A commercial quantum cryptosystem is offered at: www.idquantique.com

70Developed by Gilbert Vernam at AT&T in 1917 (first published in 1926). 71Note that

b ⊕ b′ =^{def} b + b′ mod 2 ∀ b, b′ ∈ {0, 1} .

72If the entanglement is not good enough even if it is fairly bad they may perform entanglement distillation resulting in a smaller number of nearly perfectly entangled pairs; see Section 6.2.3.

Via public communication they agree on an orthonormal basis (e_{1}, e_{2}, e_{3}) of ℝ^{3} and on a series of joint measurements of the following type on definite pairs:

For every tested pair the momenta of the partners are directed parallel or antiparallel e_{2} and a check for linear polarization is performed, but Victor and Bob independently and randomly between two possibilities: Either they test whether the linear polarization of their photon is parallel or orthogonal to e_{1} or check whether the linear polarization is parallel or orthogonal

to e′_{1} = \frac{1}{\sqrt{2}} (e_{1} + e_{3}).

As long as their choices are different their results for the corresponding pairs are completely uncorrelated. Whenever they choose the same type of measurement their results are (nearly) perfectly correlated. Thus they may agree via public communication on a random secret key in the following way:

• Alice and Bob identify those pairs for which, by chance, they had chosen the same type of measurement and discard those pairs of which at least one barter got lost and thus did not provide a definite result.

• The remaining pairs are split into two groups.

• Bob and Alice (publicly) compare their results for the first group in order to check perfect correlation.

• If the correlations turn out to be (nearly) perfect for the first group then Alice and Bob agree to use their results on the second group for a common key. Thank to the correlations they need not communicate the results concerning the second group. Therefore a possible eavesdropper has no access to the chosen key.

• Weak deviations from perfect correlation may be error corrected after communicating results of various parity checks^{73} — of almost no use for any eavesdropper.

This cryptosystem can only be attacked by manipulating the entangled pairs before Victor’s and Bob’s measurements. But such attack will be detected by Victor and Bob, who can eventually discard the current key and create another one.

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73Alternatively, if the correlations are only marginally spoiled, one could apply standard classical error correction. to the appropriately encoded (and slightly disturbed) plaintext after use of the one time pad.

128 CHAPTER 4. GENERAL ASPECTS OF QUANTUM INFORMATION

Quantifying Quantum Information

In fact, the mathematical machinery we need to develop quantum information theory is very similar to Shannon’s mathematics (typical sequences, random coding, …); so similar as to sometimes obscure that the conceptual context is really quite different.

(Preskill, 01, Section 5.2)

5.1 Shannon Theory for Pedestrians

For simplicity, let us consider an information source (Z, p) of the following type: 1. Letters x are randomly drawn from a finite alphabet Z = {z_{1}, …, z_{N}}.

2. The probability for drawing the n-letter word w = (z_{j_{1}}, …, z_{j_{n}}) is^{1}

p(z_{j_{1}}, …, z_{j_{n}}) = ∏_{ν=1}^{n} p(z_{j_{ν}}) ∀ z_{j_{1}}, …, z_{j_{n}} ∈ Z ,

where p(z) = probability for drawing z ∀ z ∈ Z .

Consistency, of course requires

p(z_{1}) + … + p(z_{N}) = 1 .  (5.1)

Then, for the corresponding Shannon entropy

H(Z, p) =^{def} − ∑_{j=1}^{N} p(z_{j}) log_{2}(p(z_{j}))  (5.2)

one may prove^{2}

DRAFT, June 26, 2009

1Thus we assume that the probabilities for the successively drawn letters are independent. 2See (Shannon, 1949, Appendix 3).

Shannon’s noiseless coding theorem:

For arbitrarly given δ > 0 , we may associate with every n ∈ ℕ a set W_{δ,n} of typical n-letter words^{3} such that:

1.

|W_{δ,n}| ≤ 2^{n(H(Z,p)+δ)} ∀ n ∈ ℕ .

2. The probability for a drawing a n-letter word w ∉ W_{δ,n} tends to 0 for n → ∞.

In other words:

Asymptotically, the relevant words can be indexed by ⌈n (H(Z, p) + δ)⌉ bits,^{4} for every fixed δ > 0 .

In this sense, the information gained by drawing the letter z is − log_{2}(p(z)) bits. The average information gained by drawing a letter, correspondingly, is H(Z, p) bits.

Remarks:

1. As to be expected, we have:

H(Z, p) = 0 ⟺ ∃ z_{0} ∈ Z : p(z_{0}) = 1 .  (5.3)

2. Straightforward calculation shows that^{5}

\begin{matrix}Z = Z_{1} ∪ Z_{2} , Z_{1} ≠ ∅ = Z_{1} ∩ Z_{2} ≠ Z_{2} \\ ⟹ H(Z, p) = H(\{Z_{1}, Z_{2}\} , \overline{p}) + \overline{p}(Z_{1}) H(Z_{1}, p_{1}) + \overline{p}(Z_{2}) H(Z_{2}, p_{2}) , \end{matrix}

(5.4)

where

\begin{matrix}\overline{p}(Z_{j}) & =^{def} & ∑_{z∈Z_{j}} p(z) \\ p_{j}(z) & =^{def} & p(z)/\overline{p}(Z_{j}) ∀ z ∈ Z_{j} \end{matrix} \} ∀ j ∈ \{1, 2\} .

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3The typical words have to include essentially all those containing the letter x approximately ⌈n p(z)⌉-times for every z ∈ Z. For large n the number of such words is of the order

\frac{n!}{∏_{z}(n p(z))!} ≈ 2^{n H(Z,p)} (since N! ≈_{Stirling} e^{−N} N^{N} for large N) .

4Obviously, such coding can be used for data compression, if H(Z, p) < log_{2}(|Z|).

5(5.4) together with (5.5) and continuity in the p(z) fixes H uniquely (Shannon, 1949, Theorem

5.1. SHANNON THEORY FOR PEDESTRIANS 131

3. H(Z, p) as a functional of p is maximal^{6} for constant p, i.e. for p(z_{1}) = … = p(z_{N}) = 1/N .

4.

p(z) = \frac{1}{|Z|} ∀ z ∈ Z  ⟹  H(Z, p) = log_{2}(|Z|) . (5.5)

5. Since

log_{2}(|Z|) = N if |Z| = 2^{N} ,

compression is not possible for constant p(z) (essentially all words are typical).

Now assume

Z = X × Y

and define

p_{1}(x) =^{def} ∑_{y∈Y} p(x, y) ∀ x ∈ X ,  (5.6)

p_{2}(y) =^{def} ∑_{x∈X} p(x, y) ∀ y ∈ Y ,  (5.7)

p_{1}(x|y) =^{def} \{ \begin{matrix}p(x, y)/p_{2}(y) & if p_{2}(y) > 0 \\ 1/ |X| & else \end{matrix} ∀ (x, y) ∈ Z ,  (5.8)

p_{2}(y|x) =^{def} \{ \begin{matrix}p(x, y)/p_{1}(x) & if p_{1}(x) > 0 \\ 1/ |Y | & else \end{matrix} ∀ (x, y) ∈ Z .  (5.9)

Remark: A possible application is the following:

X  = {elements drawn and sent via some channel} ,
Y  = {elements received through that channel} ,
p(x, y) = joint probability for sending x and receiving y ,
p_{1}(x) = probability for sending x ,
p_{2}(y) = probability for receiving y ,
p_{2}(y|x) = probability for receiving y when x is sent ,
p_{1}(x|y) = probability for x having been sent when y is received .

Then, thanks to^{7}

a ≠ b ⟹ a (ln a − ln b) > a − b ∀ a, b > 0  (5.10)

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6The simplest way to check this this is by means of a Lagrange multiplier λ : Determine λ ∈ ℝ and p(z_{1}), …, p(z_{N}) ≥ 0 (not a priori postulating (5.1)) for which H(Z, p) + λ (1 − ∑_{z∈Z} p(z)) is maximal.

7Setting x = b/a, (5.10) follows from: 0 < x ≠ 1 ⟹ ln x < x − 1 .

132 CHAPTER 5. QUANTIFYING QUANTUM INFORMATION

we have^{8}

H(X × Y, p) ≤ H(X, p_{1}) + H(Y, p_{2}) (subadditivity)  (5.11)

and

\begin{matrix}H(X × Y, p) = H(X, p_{1}) + H(Y, p_{2}) \\ ⟹ p(x, y) = p_{1}(x) p_{2}(y) ∀ (x, y) ∈ X × Y . \end{matrix}  (5.12)

Outline of proof: Replacing b by b c in (5.10) and using

log_{2}(x) = \frac{ln(x)}{ln(2)} ∀ x > 0 ,

we get

a ≠ b c ⟹ a (log_{2}(a) − log_{2}(b) − log_{2}(c)) > \frac{a − b c}{ln(2)} ∀ a, b, c > 0

and hence

H(X, p_{1}) + H(Y, p_{2}) − H(X × Y, p)
= − ∑_{x∈X} (∑_{y∈Y} p(x, y)) log_{2}(p_{1}(x)) − ∑_{y∈Y} (∑_{x∈X} p(x, y)) log_{2}(p_{2}(y))

+ ∑_{(x,y)∈X×Y} p(x, y) log_{2}(p(x, y))

= ∑_{(x,y)∈X×Y} p(x, y) (log_{2}(p(x, y)) − log_{2}(p_{1}(x)) − log_{2}(p_{2}(y)))
≥ ∑_{(x,y)∈X×Y} (p(x, y) − p_{1}(x) p_{2}(y))/ ln(2)
≥ 0

with equality iff

p(x, y) − p_{1}(x) p_{2}(y) ∀ (x, y) ∈ X × Y .

Moreover, according to Shannon’s noiseless coding theorem, the conditional entropy^{9}

H_{1}(X|Y) =^{def} ∑_{y∈Y} p_{2}(y) H(X, p_{1}(.|y))  (5.13)

is the average asymptotic amount of bits of information needed in addition per Y - part of the drawn z ∈ Z, if only these parts are known, in order to determine also the X-parts. Accordingly, we have

H(X × Y, p) = H(Y, p_{2}) + H(X|Y) .  (5.14)

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8This corresponds to the fact that correlations between X and Y contain additional information. 9Its quantum analogue can be negative (Horodecki et al., 2005).

5.2. ADAPTION TO QUANTUM COMMUNICATION 133

Outline of proof:

H(X|Y)

=_{(5.13)} − ∑_{y∈Y} p_{2}(y) ∑_{x∈X} p_{1}(x|y) log_{2}( \underbrace{p_{1}(x|y)}_{=p(x,y)/p_{2}(y)})

= − ∑_{(x,y)∈Z} \underbrace{p_{2}(y) p_{1}(x|y)}_{=p(x,y)} log_{2}(p(x, y)) + ∑_{y∈Y} p_{2}(y) \underbrace{∑_{x∈X} p_{1}(x|y)}_{=1} log_{2}(p_{2}(y)) .

Similarly we have

H(X × Y, p) = H(X, p_{1}) + H(Y |X)

for

H_{2}(Y |X) =^{def} ∑_{x∈X} p_{1}(x) H(Y, p_{2}(.|x)) .

Thanks to subadditivity (5.11), the mutual information

I(X : Y) =^{def} H(X, p_{1}) + H(Y, p_{2}) − H(X × Y, p)  (5.15)

is non-negative, as required by its interpretation as amount of information contained in the correlations between X and Y . Its relation to the conditional entropy is given by

\begin{aligned}I(X : Y) & = & H(X, p_{1}) − H_{1}(X|Y) \\ & = & H(Y, p_{2}) − H_{2}(Y |X) \\ & ≥ & 0 . \end{aligned}  (5.16)

5.2 Adaption to Quantum Communication

5.2.1 Von Neumann Entropy^{10}

The von Neumann entropy^{11}

S_{1}(ρ̂) =^{def} −trace (ρ̂ log_{2} ρ̂) ∀ ρ̂ ∈ S(H)  (5.17)

(von Neumann, 1927) can be considered as a generalization of the Shannon entropy in the following sense:

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10See also (Wehrl, 1978; Ohya and Petz, 1993; Petz, 2001; Ruskai, 2002).

11One can easily prove that, at least for the (normalized) statistical operator ρ̂ of the microcanonical or canonical ensemble, k ln(2) S_{1}(ρ̂) is the usual thermodynamic entropy; see, e.g., (Gardiner and Zoller, 2000, Section 2.4.1). Note, however, that the von Neumann entropy — contrary to the thermodynamic entropy — is non-extensive (not additive) for homogeneous non-equilibrium systems. For generalizations of the von Neumann entropy as a measure of ‘mixedness’ see (Berry and Sanders, 2003) and references given there.

If

ρ̂ = ∑_{ν=1}^{n} \underbrace{λ_{ν}}_{≥0} |ϕ_{ν}⟩⟨ϕ_{ν}| , ⟨ϕ_{α} | ϕ_{β}⟩ = \{ \begin{matrix}1 & for α = β, \\ 0 & else , \end{matrix}

then

S_{1}(ρ̂) = H(X, p_{1})  (5.18)

holds for

X =^{def} {|ϕ_{1}⟩⟨ϕ_{1}| , …, |ϕ_{n}⟩⟨ϕ_{n}|}  (5.19)
p_{1}(|ϕ_{ν}⟩⟨ϕ_{ν}|) =^{def} λ_{ν} ∀ ν ∈ {1, …, n} .  (5.20)

Warning: If {ϕ_{1}, …, ϕ_{n}} is not an orthonormal system then (5.19) and (5.20) do not imply (5.18) in general!

Theorem 5.2.1 (Klein’s inequality^{12}) For all Â, B̂ ∈ L(H) we have

0 ≤ Â ≠ B̂ ≥ 0 , ker(B̂) ⊂ ker(Â)
⟹ trace (Â (ln Â − ln B̂)) > trace (Â − B̂) .

Outline of proof: Thanks to the spectral theorem there are orthonormal systems {ϕ_{1}, …, ϕ_{n}} and {ϕ′_{1}, …, ϕ′_{n}} of H with

Â = ∑_{ν=1}^{n} \underbrace{a_{ν}}_{≥0} |ϕ_{ν}⟩⟨ϕ_{ν}| , B̂ = ∑_{ν=1}^{n} \underbrace{b_{ν}}_{≥0} |ϕ′_{ν}⟩⟨ϕ′_{ν}|

for suitable a_{1}, …, b_{n}. Then

trace (Â ln Â) = ∑_{ν=1}^{n} a_{ν} ln a_{ν}

and

trace (Â ln B̂) = ∑_{α=1}^{n} ⟨ϕ_{α} | ∑_{β=1}^{n} a_{β} |ϕ_{β}⟩⟨ϕ_{β}| ∑_{γ=1}^{n} ln b_{γ} |ϕ_{γ}⟩⟨ϕ_{γ}| ϕ_{α}⟩

= ∑_{α,γ=1}^{n} a_{α} ln(b_{γ}) |⟨ϕ_{α} | ϕ′_{γ}⟩|^{2} ,

hence

trace (Â ln Â) − trace (Â ln B̂) = ∑_{α=1}^{n} a_{α} (ln a_{ν} − ∑_{γ=1}^{n} ln(b_{γ}) |⟨ϕ_{α} | ϕ′_{γ}⟩|^{2}) .

DRAFT, June 26, 2009

12See (Klein, 1931).

5.2. ADAPTION TO QUANTUM COMMUNICATION 135

Since, for x > 0 , ln(x) is a strictly concave function,^{13} the latter implies

trace (Â ln Â) − trace (Â ln B̂)  (5.21)
≥ ∑_{α=1}^{n} a_{α} (ln a_{α} − ln(∑_{γ=1}^{n} b_{γ} |⟨ϕ_{α} | ϕ′_{γ}⟩|^{2}))  (5.22)
≥_{(5.10)} ∑_{α=1}^{n} (a_{α} − ∑_{γ=1}^{n} b_{γ} |⟨ϕ_{α} | ϕ′_{γ}⟩|^{2})  (5.23)

= ∑_{α=1}^{n} a_{α} − ∑_{γ=1}^{n} b_{γ}

= trace (Â − B̂) .

In (5.22) equality holds only if, for every α ∈ {1, …, n}, at most one of products a_{α}b_{γ} |⟨ϕ_{α} | ϕ′_{γ}⟩| is different from zero. Then, with suitable relabelling of the ϕ′_{γ} , we have ∑_{γ=1}^{n} b_{γ} |⟨ϕ_{α} | ϕ′_{γ}⟩|^{2} = b_{α} and equality in (5.23) holds only if a_{α} = b_{α} for all α ∈ {1, …, n}, i.e. if Â = B̂.

Remark: As an immediate consequence of Theorem 5.2.1 we have strict positivity of the quantum relative entropy

S(ρ̂∥ρ̂′) =^{def} trace (ρ̂(ln ρ̂ − ln ρ̂′))

for all states ρ̂, ρ̂′ ∈ S(H) with ker(ρ̂′) ⊂ ker(ρ̂) and ρ̂ ≠ ρ̂′. Corollary 5.2.2 Let H_{1}, H_{2} be Hilbert spaces and ρ̂ ∈ S(H_{1} ⊗ H_{2}) . Then

S_{1}(ρ̂) ≤ S_{1}(trace_{2}(ρ̂)) + S_{1}(trace_{1}(ρ̂)) (subadditivity)

and S_{1}(ρ̂) = S_{1}(trace_{2}(ρ̂)) + S_{1}(trace_{1}(ρ̂)) ⟺ ρ̂ = trace_{2}(ρ̂) ⊗ trace_{1}(ρ̂) .

Outline of proof: Application of Klein’s inequality to

Â = ρ̂ , B̂ = trace_{2}(ρ̂) ⊗ trace_{1}(ρ̂)

gives

0 = trace (Â − B̂)
≤ −S_{1}(ρ̂) − trace (Â log_{2} B̂)

DRAFT, June 26, 2009

13Strictly concave functions f of x > 0 are those fulfilling λ f(x_{1}) + (1 − λ) f(x_{2}) < f(λ x_{1} + (1 − λ) x_{2}) ∀ λ ∈ (0, 1) , x_{1}, x_{2} > 0 . with equality only for ρ̂ = trace_{2}(ρ̂) ⊗ trace_{1}(ρ̂) . Since

trace (Â log B̂)

= trace (Â(log_{2}(trace_{2}(ρ̂)) ⊗ ˆ1 + ˆ1 ⊗ log_{2}(trace_{1}(ρ̂))))
= trace (trace_{2}(ρ̂) log_{2}(trace_{2}(ρ̂))) + trace (trace_{1}(ρ̂) log_{2}(trace_{1}(ρ̂)))
= −S_{1}(trace_{2}(ρ̂)) − S_{1}(trace_{1}(ρ̂)) ,

this proves the corollary.

The von Neumann entropy of a state ρ increases if the latter is changed by a complete projective measurement operation (ignoring the results):

Corollary 5.2.3 Let H be a Hilbert space, ρ̂_{0} ∈ S(H) , {P̂_{1}, …, P̂_{l}} a set of pairwise orthogonal projection operators on H with

P̂_{1} + … + P̂_{l} = ˆ1

and define

ρ̂′_{0} =^{def} ∑_{k=1}^{l} P̂_{k} ρ̂_{0} P̂_{k} .

Then

ρ̂_{0} ≠ ρ̂′_{0} ⟹ S_{1}(ρ̂_{0}) < S_{1}(ρ̂′_{0}) .

Proof: Since

trace (ρ̂ log_{2} ρ̂′)  =  trace (\underbrace{∑_{k=1}^{l} P̂_{k} P̂_{k}}_{=ˆ1} ρ̂ log_{2} ρ̂′)

=  trace (∑_{k=1}^{l} P̂_{k} ρ̂ log_{2}(ρ̂′) P̂_{k})
=_{[ρ̂′,P̂_{k}]_{−}=0} trace (∑_{k=1}^{l} P̂_{k} ρ̂P̂_{k} log_{2}(ρ̂′))
=  −S_{1}(ρ̂′) ,

The statement follows from Klein’s inequality (Theorem 5.2.1) applied to Â = ρ̂, B̂ = ρ̂′.

Warning: In general, trace preserving quantum operations may decrease the von Neumann entropy.^{14}

DRAFT, June 26, 2009

14E.g., if

K̂_{1} = |0⟩⟨0| K̂_{2} = |0⟩⟨1|

5.2.2 Accessible Information

Assume that the ‘alphabet’ X = {ρ̂_{1}, …, ρ̂_{N_{1}}} is a set of (pairwise different) states on H. Then, by “drawing the ‘letter’ ρ̂ from X” we mean the random choice of an individual from an ensemble in the state ρ̂. Again, for simplicity, we assume that for every letter ρ̂ its probability p_{1}(ρ̂) for being drawn is given, positive, and does not depend on which letters have been drawn before.

The best one can do, in order to acquire information about a drawn letter, is perform a POV measurement^{15} corresponding to some set Y = {Ê_{1}, …, Ê_{N_{2}}} of

events Ê represented by positive bounded operators on H with^{16}

Ê_{1} + … + Ê_{N_{2}} = ˆ1 .

According to quantum mechanical rules, the probability for Ê is trace (ρ̂Ê) , if ρ̂ was drawn. Hence, the probability for ρ̂ being drawn and Ê being detected is^{17}

p(ρ̂, Ê) =^{def} p_{1}(ρ̂) trace (ρ̂Ê) .  (5.24)

If the letter drawn is unknown, the probability for Ê is

\begin{aligned}p_{2}(Ê) & =_{(5.7)} & ∑_{j=1}^{N_{1}} p(ρ̂_{j}, Ê) \\ & = & trace (ρ̂_{0} Ê) , \end{aligned}

where

ρ̂_{0} =^{def} ∑_{j=1}^{N_{1}} p_{1}(ρ̂_{j})ρ̂_{j}

is the state of the source providing the letters. Of course,^{18} ρ̂_{0} does not uniquely determine (X, p_{1}) . Nevertheless, the von Neumann entropy fulfills the Holevo bound

I(X : Y) ≤ S_{1}(ρ̂_{0}) − ∑_{j=1}^{N_{1}} p_{1}(ρ̂_{j}) S_{1}(ρ̂_{j})  (5.25)

|0⟩⟨0| and hence

DRAFT, June 26, 2009 are the Kraus operator of the quantum operation ℭ acting on a qubit system we have ℭ(ˆ1/2) =

S_{1}(ˆ1/2) > S_{1}(ℭ(ˆ1/2)) = 0 .

15If the elements of X are linearly independent, then the best result can be achieved by projective measurement, i.e. with Ê_{1}, …, Ê_{N_{2}} being projection operators (Eldar, 2003). For the importance of considering also linearly dependent Ê_{ν}’s see, e.g., (Kaszlikowski et al., 2003).

16Recall Corollary A.4.3, in this connection.

17Obviously, (5.24) is consistent with (5.6).

18Recall Corollary A.4.3.

(Nielsen and Chuang, 2001, Theorem 12.1) and the condition

\begin{matrix}∃ j_{1}, j_{2} ∈ \{1, …, N_{1}\} : trace (ρ̂_{j_{1}} ρ̂_{j_{2}}) ≠ 0 , j_{1} ≠ j_{2} \\ ⟹ S_{1}(ρ̂_{0}) < H(X, p_{1}) + ∑_{j=1}^{N_{1}} p_{1}(ρ̂_{j}) S_{1}(ρ̂_{j}) \end{matrix}  (5.26)

(Nielsen and Chuang, 2001, Theorem 11.10). A direct consequence of (5.25) and (5.26) is the upper bound

\begin{matrix}∃ j_{1}, j_{2} ∈ \{1, …, N_{1}\} : trace (ρ̂_{j_{1}} ρ̂_{j_{2}}) ≠ 0 , j_{1} ≠ j_{2} \\ ⟹ A(X, p) < H(X, p_{1}) \end{matrix}  (5.27)

on the accessible information

A(X, p) =^{def} max_{Y ˆ=POV} I(X : Y) .  (5.28)

Hence:

It is impossible to get full information on the letters actually drawn unless^{19}

ρ̂ ≠ ρ̂′ ⟹ ρ̂ρ̂′ = ˆ0 ∀ ρ̂, ρ̂′ ∈ X .

Remark: Note that, for arbitrary ρ̂, ρ̂′ ∈ S(H) we have^{20}

ρ̂ρ̂′ = ˆ0 ⟺ trace (ρ̂ρ̂′) = 0
⟺ ρ̂H ⊥ ρ̂′H .

On the other hand, (5.27) (together with continuity of the entropies) implies the bound

A(X, p) ≤ S_{1}(ρ̂_{0}) .  (5.29)

DRAFT, June 26, 2009

19Otherwise, distinguishability of arbitrary states could be used for superluminal communication. 20Here, positivity of the operators ρ̂, ρ̂′ is essential! For Â = Â^{†} ∈ L(H) the range Â H of Â is

also called the support of Â, since

Â Ψ ≠ 0 ⟺ 0 ≠ Ψ ∈ ÂH

for such Â.

5.2.3 Distance Measures for Quantum States^{21}

A natural distance measure for quantum states ρ̂, ρ̂′ of a finite-dimensional^{22} quantum system is the trace distance^{23}

D(ρ̂, ρ̂′) =^{def} \frac{1}{2} ∥ρ̂ − ρ̂′∥_{1} ,  (5.30)

where ∥.∥_{1} denotes the trace norm

∥A∥_{1} =^{def} trace ( ^{+} \sqrt{Â^{†}Â})  (5.31)

for trace class operators Â on a Hilbert space H. Especially for qubits we have

ρ̂ = \frac{1}{2} (ˆ1 + ρ · τ̂) , ρ̂′ = \frac{1}{2} (ˆ1 + ρ′ · τ̂)

and hence^{24}

D(ρ̂, ρ̂′) = \frac{1}{4} trace ^{+} \sqrt{(ρ · τ̂ − ρ′ · τ̂)^{2}}

= \frac{1}{4} trace (|ρ − ρ′| ^{+} \sqrt{ˆ1})

= \frac{1}{2} |ρ − ρ′| ,

i.e.:

For qubit states ρ̂, ρ̂′ the trace distance is half the Euclidean distance of the corresponding Bloch vectors ρ , ρ′.

For general mixed states we have the following:

Lemma 5.2.4 Let H be a finite-dimensional Hilbert space and ρ̂, ρ̂′ ∈ S(H) . Then^{25}

D(ρ̂, ρ̂′) = max_{P̂∈P}(trace (P̂ ρ̂) − trace (P̂ ρ̂′)) ,

(5.32)

where P denotes the set of all projection operators on H.

(Streater, 2003).

23Note that

\begin{matrix}ρ̂ = ∑_{ν=1}^{∞} p_{ν} |Φ_{ν}⟩⟨Φ_{ν}| , \\ ρ̂′ = ∑_{ν=1}^{∞} p′_{ν} |Φ_{ν}⟩⟨Φ_{ν}| \end{matrix} \} ⟹ D(ρ̂, ρ̂′) = \frac{1}{2} ∑_{ν=1}^{∞} |p_{ν} − p′_{ν}|

24Recall that with trace ≤ 1 .

DRAFT, June 26, 2009 21See also (Gilchrist et al., 2005).

22For infinite-dimensional systems, however, the trace distance is not physically adequate if {Φ_{ν}}_{ν∈ℕ} is a MONS of H.

|e| = 1 ⟹ (e · τ̂)^{2} = ˆ1 ∀ e ∈ ℝ^{3} .

25This formula holds also with P replaced by the set of all events; i.e. of all positive operators

Outline of proof: The spectral theorem tells us that there are an orthonormal basis {ϕ_{ν}}_{ν∈ℕ} of H and real numbers λ_{1}, …, λ_{n} with

ρ̂ − ρ̂′ = ∑_{ν=1}^{n} λ_{ν} ϕ_{ν}  (5.33)

and

∑_{ν=1}^{n} λ_{ν} = 0 .  (5.34)

Then

+ \sqrt{(ρ̂ − ρ̂′)^{2}} = ∑_{ν=1}^{n} |λ_{ν}| ϕ_{ν}

and, therefore,

D(ρ̂, ρ̂′) = \frac{1}{2} ∑_{ν=1}^{n} |λ_{ν}|
=_{(5.34)} ∑_{ν∈{1,...,n}_{λν >0}} λ_{ν}

= trace( ∑_{ν∈{1,...,n}_{λν >0}} λ_{ν} ϕ_{ν})

≥ trace(P̂ ∑_{ν∈{1,...,n}_{λν >0}} λ_{ν} ϕ_{ν}) + trace(P̂ ∑_{ν∈{1,...,n}_{λν ≤0}} λ_{ν} ϕ_{ν}) ∀ P̂ ∈ P

with equality for

P̂ = ∑_{ν∈{1,...,n}_{λν >0}} |ϕ_{ν}⟩⟨ϕ_{ν}| .

By (5.33) and linearity of the trace, this implies (5.32).

Now it is obvious that the trace distance is a metric on the set of states:^{26}

\begin{aligned}D(ρ̂_{1}, ρ̂_{2}) & ≥ & 0 , \\ D(ρ̂_{1}, ρ̂_{2}) & = & 0 ⟺ ρ̂_{1} = ρ̂_{2} , \\ D(ρ̂_{1}, ρ̂_{2}) & = & D(ρ̂_{2}, ρ̂_{1}) , \\ D(ρ̂_{1}, ρ̂_{3}) & ≤ & D(ρ̂_{1}, ρ̂_{2}) + D(ρ̂_{2}, ρ̂_{3}) . \end{aligned}

Corollary 5.2.5 Let H_{1} and H_{2} be finite-dimensional Hilbert spaces and let ℭ ∈ Q(H_{1}, H_{1}) be trace-preserving. Then Q(H_{1}, H_{1}) is contractive w.r.t. the trace distance, i.e.:

D(ℭ(ρ̂), ℭ(ρ̂′)) ≤ D(ρ̂, ρ̂′) ∀ ρ̂, ρ̂′ ∈ S(H_{1}) .

DRAFT, June 26, 2009

26The inequality follows from (5.32) and

trace (ρ̂_{1}) − trace (ρ̂_{3}) = (trace (ρ̂_{1}) − trace (ρ̂_{2})) + (trace (ρ̂_{2}) − trace (ρ̂_{3})) .

5.2. ADAPTION TO QUANTUM COMMUNICATION 141

Outline of proof: With ϕ_{1}, …, ϕ_{n} and λ_{1}, …, λ_{n} chosen as in the proof for (5.32), we have^{27}

D(ρ̂, ρ̂′) = trace( ∑_{ν∈{1,...,n}_{λν >0}} λ_{ν} ϕ_{ν})
= trace(ℭ^{}¯ ( ∑_{ν∈{1,...,n}_{λν >0}} λ_{ν} ϕ_{ν}))
≥ trace(P̂ ℭ^{}¯ ( ∑_{ν∈{1,...,n}_{λν >0}} λ_{ν} ϕ_{ν})) + trace(P̂ ℭ^{}¯ ( ∑_{ν∈{1,...,n}_{λν ≤0}} λ_{ν} ϕ_{ν}))
= trace (P̂(ℭ(ρ̂) − ĉ(ρ̂′))) ∀ P̂ ∈ P .

This, together with (5.32), proves the theorem.

Corollary 5.2.6 Let H be a Hilbert space and N ∈ ℕ. Moreover, consider ρ̂_{1}, …, ρ̂′_{N} ∈ S(H) and p_{1}, …, p′_{N} ≥ 0 with

∑_{ν=1}^{N} p_{ν} = 1 = ∑_{ν=1}^{N} p′_{ν} .  (5.35)

Then

D (∑_{ν=1}^{N} p_{ν}ρ̂_{ν} , ∑_{ν=1}^{N} p′_{ν}ρ̂′_{ν}) ≤ \frac{1}{2} ∑_{ν=1}^{N} |p_{ν} − p′_{ν}| + ∑_{ν=1}^{N} p_{ν} D(ρ̂_{ν}, ρ̂′_{ν}) .  (5.36)

Outline of proof: By Lemma 5.2.4 there is a P̂ ∈ P with

D (∑_{ν=1}^{N} p_{ν} ρ̂_{ν} , ∑_{ν=1}^{N} p′_{ν}ρ̂′_{ν}) = trace (P̂ ∑_{ν=1}^{N} p_{ν} ρ̂_{ν} − P̂ ∑_{ν=1}^{N} p′_{ν} ρ̂′_{ν})
= ∑_{ν=1}^{N}(p_{ν} \underbrace{trace (P̂ ρ̂_{ν} − P̂ ρ̂′_{ν})}_{≤_{L.5.2.4} D(ρ̂_{ν},ρ̂′_{ν})} + (p_{ν} − p′_{ν}) \underbrace{trace (P̂ ρ̂′_{ν})}_{∈[0,1]})
≤ ∑_{ν=1}^{N} p_{ν} D(ρ̂_{ν}, ρ̂′_{ν}) + ∑_{ν∈{1,...,N}_{pν −p′_{ν} >0}}(p_{ν} − p′_{ν}) .

Together with

∑_{ν∈\{1,...,N\}_{pν −p′_{ν} >0}}(p_{ν} − p′_{ν}) =_{(5.35)} \frac{1}{2} ∑_{ν=1}^{N} |p_{ν} − p′_{ν}|

this implies (5.36).

DRAFT, June 26, 2009

27Recall Footnote 18 of Chapter 4.

142 CHAPTER 5. QUANTIFYING QUANTUM INFORMATION A direct consequence of (5.36) is

D (∑_{ν=1}^{N} p_{ν} ρ̂_{ν} , ∑_{ν=1}^{N} p_{ν} ρ̂′_{ν}) ≤ ∑_{ν=1}^{N} p_{ν} D(ρ̂_{ν} , ρ̂′_{ν}) .  (5.37)

Moreover, setting ρ̂′_{ν} ≡ ρ̂′ in (5.37) and recalling (5.35), we get

D (∑_{ν=1}^{N} p_{ν} ρ̂_{ν} , ρ̂′) ≤ ∑_{ν=1}^{N} p_{ν} D(ρ̂_{ν} , ρ̂′) .  (5.38)

Another important measure for the distance of states is the Bures fidelity:^{28}

F (ρ̂, ρ̂′) =^{def} (trace ^{+} \sqrt{( ^{+} \sqrt{ρ̂′} ^{+} \sqrt{ρ̂})^{†} ( ^{+} \sqrt{ρ̂′} ^{+} \sqrt{ρ̂})})^{2} .

Since ^{+} \sqrt{|ψ⟩⟨ψ|} = |ψ⟩⟨ψ| and hence

\begin{aligned}\sqrt[+]{^{+} \sqrt{|ψ⟩⟨ψ|} ρ̂′ ^{+} \sqrt{|ψ⟩⟨ψ|}} & = & \sqrt[+]{|ψ⟩⟨ψ| ρ̂′ |ψ⟩⟨ψ|} \\ & = & ^{+} \sqrt{⟨ψ | ρ̂′ ψ⟩} |ψ⟩⟨ψ| \end{aligned}

holds for all normalized ψ ∈ H, we have:

ρ̂ = |ψ⟩⟨ψ| ⟹ F (ρ̂, ρ̂′) = ⟨ψ | ρ̂′ ψ⟩ ∀ ρ̂, ρ̂′ ∈ S(H) . (5.39)

Symmetry of the fidelity in general is not evident from its definition but follows directly from Uhlmann’s theorem (Uhlman, 1976): Theorem 5.2.7 Let H be a Hilbert-space and ρ̂_{1} , ρ̂_{2} ∈ S(H) . Then

F (ρ̂_{1}, ρ̂_{2}) = max_{Ψ_{1}∈T_{ρ̂_{1}},Ψ_{2}∈T_{ρ̂_{2}}} |⟨Ψ_{1} | Ψ_{2}⟩|^{2} ,

where^{29}

Tρ̂ =^{def} {Ψ ∈ H ⊗ H : ρ̂ = trace_{2}(|Ψ⟩⟨Ψ|)} ∀ ρ̂ ∈ S(H) .

DRAFT, June 26, 2009

28For commuting ρ̂, ρ̂′ the Bures fidelity has a simple geometrical interpretation, since

\begin{matrix}ρ̂ = ∑_{ν} p_{ν} |ϕ_{ν}⟩⟨ϕ_{ν}| , \\ ρ̂′ = ∑_{ν} p′_{ν} |ϕ_{ν}⟩⟨ϕ_{ν}| \end{matrix} \} ⟹ F (ρ̂, ρ̂′) = (∑_{ν} \sqrt{p_{ν}p′_{ν}})^{2} .

See (Chen et al., 2002) for a geometrical interpretation of the Bures fidelity of general qubit states. 29Note that Tρ̂ is the set of all purifications of ρ̂ as introduced in Lemma A.4.8.

5.2. ADAPTION TO QUANTUM COMMUNICATION 143

Outline of proof: Let j ∈ {1, 2} and Ψ_{j} ∈ T_{ρ̂_{j}} . By the spectral theorem, there is are orthonormal basis {ϕ_{1}^{(j)}, …, ϕ_{n}^{(j)}} of H, some n′ ∈ {1, …, n}, and s_{1}^{(j)}, …, s_{n′}^{(j)} > 0 with

ρ̂_{j} = ∑_{ν=1}^{n′} (s_{ν}^{(j)})^{2} ϕ_{ν}^{(j)} .

Therefore, the Schmidt-decomposition of Ψ_{j} has to be of the form

\begin{aligned}Ψ_{j} & = & ∑_{ν=1}^{n′} s_{ν}^{(j)} ϕ_{ν}^{(j)} ⊗ ψ_{ν}^{(j)} \\ & = & ∑_{ν=1}^{n} ( ^{+} \sqrt{ρ̂_{j}} ϕ_{ν}^{(j)}) ⊗ ψ_{ν}^{(j)} \end{aligned}

with some orthonormal basis {ψ_{1}^{(j)}, …, ψ_{n}^{(j)}} of H. Considering j = 1 and j = 2 together we thus get

⟨Ψ_{2} | Ψ_{1}⟩ = ∑_{ν,μ=1}^{n′} ⟨ ^{+} \sqrt{ρ̂_{2}} ϕ_{ν}^{(2)} | ^{+} \sqrt{ρ̂_{1}} ϕ_{μ}^{(1)}⟩⟨ψ_{ν}^{(2)} | ψ_{μ}^{(1)}⟩ .  (5.40)

In order to rewrite the r.h.s. of (5.40) as a trace we use the unitary operators V̂ and V̂′ characterized by

V̂ ϕ_{μ}^{(2)} = ϕ_{μ}^{(1)} ∀ μ ∈ {1, …, n}

and

⟨ϕ_{μ}^{(2)} | V̂′ ϕ_{ν}^{(2)}⟩ = ⟨ψ_{ν}^{(2)} | ψ_{μ}^{(1)}⟩ ∀ ν, μ ∈ {1, …, n} .

Then

⟨Ψ_{2} | Ψ_{1}⟩ =_{(5.40)} ∑_{ν,μ=1}^{n′} ⟨ ^{+} \sqrt{ρ̂_{2}} ϕ_{ν}^{(2)} | ^{+} \sqrt{ρ̂_{1}} V̂ ϕ_{μ}^{(2)}⟩⟨ϕ_{μ}^{(2)} | V̂′ ϕ_{ν}^{(2)}⟩
= ∑_{ν=1}^{n′} ⟨ϕ_{ν}^{(2)} | ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}} V̂ V̂′ ϕ_{ν}^{(2)}⟩
= trace ( ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}} V̂ V̂′)
= trace (V̂ V̂′ ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}})

Applying Lemma 5.2.8, below, to the polar decomposition

V̂ V̂′ ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}} = Û ^{+} \sqrt{( ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}})^{†} ( ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}})} , Û unitary ,

(see, e.g., Lemma 7.3.20 of (Lücke, eine)) we always get

\begin{aligned}|⟨Ψ_{1} | Ψ_{2}⟩|^{2} & ≤ & (trace ^{+} \sqrt{( ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}})^{†} ( ^{+} \sqrt{ρ̂_{2}} ^{+} \sqrt{ρ̂_{1}})})^{2} \\ & = & F (ρ̂_{1} , ρ̂_{1}) . \end{aligned}

Obviously, by appropriate choice of the ψ_{ν}^{(j)} we get equality.

Lemma 5.2.8 Let H be a finite-dimensional^{30} Hilbert space. Then

|trace (B̂ ρ̂)| ≤ ∥B̂∥ ∀ ρ̂ ∈ S(H) , B̂ ∈ L(H) .

Outline of proof: By the spectral theorem, there are a an orthonormal basis {ϕ_{1}, …, ϕ_{n}} of H and p_{1}, …, p_{n} ≥ 0 with

ρ̂ = ∑_{ν=1}^{n} p_{ν} |ϕ_{ν}⟩⟨ϕ_{ν}| , ∑_{ν=1}^{n} p_{ν} = 1 .

Then

|trace (B̂ ρ̂)| = |∑_{ν=1}^{n} p_{ν} trace (B̂ |ϕ_{ν}⟩⟨ϕ_{ν}|)|

= |∑_{ν=1}^{n} p_{ν} ⟨ϕ_{ν} | B̂ ϕ_{ν}⟩|

≤ ∑_{ν=1}^{n} p_{ν} \underbrace{|⟨ϕ_{ν} | B̂ ϕ_{ν}⟩|}_{≤\left∥{ \atop B̂}\right∥}

≤ ∥B̂∥ .

Uhlmanns theorem and the definition of the Bures fidelity show:

F (ρ̂_{1}, ρ̂_{2}) = F (ρ̂_{2}, ρ̂_{1}) ,  (5.41)
F (ρ̂_{1}, ρ̂_{2}) ∈ [0, 1] ,  (5.42)
F (ρ̂_{1}, ρ̂_{2}) = \{ \begin{matrix}1 & iff ρ̂_{1} = ρ̂_{2} \\ 0 & iff ρ̂_{1}ρ̂_{2} = 0 . \end{matrix}  (5.43)

Using Uhlmanns theorem one may also show:^{31}

A(ρ̂_{1}, ρ̂_{2}) =^{def} arccos \sqrt{F (ρ̂_{1}, ρ̂_{2})} ∈ [0, π/2] is a metric , (5.44)
F (ℭ(ρ̂_{1}), ℭ(ρ̂_{2})) ≥ F (ρ̂_{1}, ρ̂_{2}) ,  (5.45)

F (∑_{ν} p_{ν}ρ̂_{ν}, ∑_{ν} p′_{ν}ρ̂′_{ν}) ≥ ∑_{ν} \sqrt{p_{ν}p′_{ν}} F (ρ̂_{ν}, ρ̂′_{ν}) .  (5.46)

For pure states ϕ, ψ :

\begin{aligned}F (P̂_{ϕ}, P̂_{ψ}) & = & |⟨ϕ | ψ⟩| \\ & = & \sqrt{1 − D(P̂_{ϕ}, P̂_{ψ})^{2}} . \end{aligned}

DRAFT, June 26, 2009

et al., 2003, Theorem 4).

30The given may be directly extended to the infinite-dimensional case.

31Necessary and sufficient conditions for a given set of pure states to be transformable via a quantum operation into another given set of (not necessarily pure) states are given in (Chefles

For general states:

1 − F (ρ̂_{1}, ρ̂_{2}) ≤ D(ρ̂_{1}, ρ̂_{2}) ≤ \sqrt{1 − F (ρ̂_{1}, ρ̂_{2})^{2}} .

Remark: In principle, fidelity and trace distance are of equal use to characterize the difference of states. However, usually, calculations are easier with fidelity. Therefore only the latter will be used in the following.

Relevant for transmission of (unknown) states:

F_{min}(ℭ) =^{def} min_{ρ̂} F (ρ̂, ℭ(ρ̂))
=_{(5.46)} min_{ψ} F (P̂_{ψ}, ℭ(P̂_{ψ})) .

Relevant for the (approximate) realization of a gate Û as the quantum operation ℭ is the gate fidelity

F (Û, ℭ) =^{def} min_{ρ̂} F (Ûρ̂Û^{−1}, ℭ(ρ̂))
= min_{ψ} F (P̂_{Ûψ}, ℭ(P̂_{Ûψ}))

for which we have, e.g.,^{32}

arccos \sqrt{F (Û_{1}Û_{2}, ℭ_{1} ∘ ℭ_{2})} ≤ arccos \sqrt{F (Û_{1}, ℭ_{1})} + arccos \sqrt{F (Û_{1}, ℭ_{1})} .

Relevant for quantum sources producing ρ̂_{j} with probability p_{j} and disturbed by ℭ is the ensemble average fidelity

\overline{F} =^{def} ∑_{j} p_{j} F (ρ̂_{j}, ℭ(ρ̂_{j})) .

5.2.4 Schumacher Encoding

Lemma 5.2.9 Let H be a finite-dimensional Hilbert space, ρ̂_{0} ∈ S(H) , and δ > 0 . Then

lim_{n→∞} trace (ρ̂_{0}^{⊗n}Λ̂_{ρ̂_{0},δ}^{(n)}) = 1

and^{33}

2^{n(S_{1}(ρ̂_{0})+δ)} ≥ dim(Λ̂_{ρ̂_{0},δ}^{(n)} H^{⊗n}) ≥ 2^{n(S_{1}(ρ̂_{0})−δ)} ∀ n ∈ ℕ ,

where, for n ∈ ℕ, Λ̂_{ρ̂_{0},δ}^{(n)} denotes the projector onto the subspace of all eigenvectors of ρ̂_{0}^{⊗n} with eigenvalues in [2^{n(S_{1}(ρ̂_{0})−δ)}, 2^{n(S_{1}(ρ̂_{0})+δ)}].

DRAFT, June 26, 2009

32Recall (5.44).

33See (Mitchison and Jozsa, 2003) in this connection.

146 CHAPTER 5. QUANTIFYING QUANTUM INFORMATION Lemma 5.2.10 Let H be a Hilbert space and Â a self-adjoint operator on H. The for arbitrary ψ_{1}, …, ψ_{N} ∈ H and p_{1}, …, p_{N} > 0 we have

∑_{ν=1}^{N} p_{ν} |⟨ψ_{ν} | Â ψ_{ν}⟩|^{2} ≥ 2 trace (Â ∑_{ν=1}^{N} p_{ν} |ψ_{ν}⟩⟨ψ_{ν}|) − ∑_{ν=1}^{N} p_{ν} .

Outline of proof: Apply the inequality

x^{2} ≥ 2 x − 1 ∀ x ∈ ℝ

to x = ⟨ψ_{ν} | Â ψ_{ν}⟩ .

...

make a guess, Fano inequality

5.2.5 A la Nielsen/Chuang

Klein inequality (Nielsen and Chuang, 2001, Theorem 11.7):^{34}

trace (ρ̂ log(ρ̂)) ≥ trace (ρ̂ log(ρ̂′)) ∀ ρ̂, ρ̂′ ∈ S(H) .  (5.47)

H_{bin}(p) =^{def} H({p, 1 − p}) ∀ p ∈ [0, 1] .

...

PPT criterion

Cat states implemented for Josephson junctions or coherent states (entanglement laser)

5.2.6 Entropy^{35}

Classically, every message can be encoded in a string of bits. But can quantum information always be encoded in a string of qubits?!

DRAFT, June 26, 2009

34The r.h.s of (5.47) may become −∞. Equality holds iff ρ̂ = ρ̂′.

35See also (Ohya and Petz, 1993) and (Petz, 2001).

148 CHAPTER 5. QUANTIFYING QUANTUM INFORMATION

Handling Entanglement^{1}

We have seen in 4.4 that perfect entanglement may be used for implementing noiseless quantum communication. Therefore, quantification and handling (e.g., distillation) of entanglement is important. Here, for simplicity, we consider only bipartite systems. One might expect, then, that separability is equivalent to the existence of corresponding (local) hidden variable models, hence to the validity of all (generalized) Bell inequalities.^{2} However, as shown in (Werner, 1989), that this is not the case.

6.1 Detecting Entanglement

Detecting entanglement of pure states is very easy:

ρ̂ ∈ S_{pure}(H_{1} ⊗ H_{2}) ∪ S_{sep}(H_{1} ⊗ H_{2}) ⟺_{Lemma A.4.6 TheoremA.4.7} \{ \begin{matrix}ρ̂ ∈ S_{pure}(H_{1} ⊗ H_{2}) \\ (trace_{2}(ρ̂))^{2} = trace_{2}(ρ̂) . \end{matrix}

It is mixedness which can make detection of entanglement a very hard problem (Gurvits, 2002).

6.1.1 Entanglement Witnesses and Non-Completely Positive Mappings

Lemma 6.1.1 A state ρ̂ of the bipartite system S with state space H = H_{1} ⊗ H_{2} is non-separable^{3} iff it possesses an entanglement witness, i.e. an operator Ŵ ∈ L(H_{1} ⊗ H_{2}) fulfilling the following two properties:^{4}

DRAFT, June 26, 2009

1See also (Horodecki et al., 2001) and references given there. 2See (Werner and Wolf, 201; Collins and Gisin, 2003) in this connection. 3Recall Definition A.4.4. 4The second property guarantees that Ŵ is Hermitian — thanks to the polarization identity

h(x_{1}, x_{1}) = \frac{1}{4} ∑_{α=0}^{3} (−i)^{α} |x_{1} + i^{α} x_{2}⟩⟨x_{1} + i^{α} x_{2}| ,

valid for every mapping h that is linear in the second and conjugate linear in the first argument; especially for h(ϕ, ψ) = |ϕ⟩⟨ψ|.

1.

trace (ρ̂Ŵ) < 0 .

2.

ρ̂′ separable ⟹ trace (ρ̂Ŵ) ≥ 0 ∀ ρ̂′ ∈ S(H_{1} ⊗ H_{2}) .

Outline of proof: The statement follows from the known fact^{5} that for every point X outside a convex set K there is a hyperplane separating X from K.

Lemma 6.1.2 (Jamiołkowski) For j ∈ {1, 2}, let {ϕ_{1}^{(j)}, …, ϕ_{n_{j}}^{(j)}} be a MONS of

the Hilbert space H_{j}. Then for every Ŵ ∈ L(H_{1} ⊗ H_{2}) there is a unique linear mapping 𝔏Ŵ : L(H_{1}) ⟶ L(H_{2}) with

Ŵ = n_{1} (1 ⊗ 𝔏Ŵ) (P̂_{H_{1}}^{+}) ,  (6.1)

where

P̂_{H_{1}}^{+} =^{def} \frac{1}{n_{1}} |∑_{ν=1}^{n_{1}} ϕ_{ν}^{(1)} ⊗ ϕ_{ν}^{(1)}⟩⟨∑_{μ=1}^{n_{1}} ϕ_{μ}^{(1)} ⊗ ϕ_{μ}^{(1)}| .  (6.2)

𝔏Ŵ fulfills

𝔏Ŵ (|ϕ_{ν_{1}}^{(1)}⟩⟨ϕ_{μ_{1}}^{(1)}|) = ∑_{ν_{2},μ_{2}=1}^{n_{2}} ⟨ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{ν_{2}}^{(2)} | Ŵ (ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{μ_{2}}^{(2)})⟩ |ϕ_{ν_{2}}^{(2)}⟩⟨ϕ_{μ_{2}}^{(2)}|  (6.3)

for all ν_{1}, ν_{2} ∈ {1, …, n_{1}} and:

𝔏Ŵ positive  ⟺  trace (Ŵ ρ̂) ≥ 0 ∀ ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) ,  (6.4)

where^{6}

S_{sep}(H_{1} ⊗ H_{2}) =^{def} {ρ̂ ∈ S(H_{1} ⊗ H_{2}) : ρ̂ separable} .  (6.5)

Outline of proof: Defining 𝔏Ŵ by (6.3) plus linear continuation gives

Ŵ  =  ∑_{ν_{1},μ_{1}=1}^{n_{1}} ∑_{ν_{2},μ_{2}=1}^{n_{2}} ⟨ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{ν_{2}}^{(2)} | Ŵ (ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{μ_{2}}^{(2)})⟩ |ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{ν_{2}}^{(2)}⟩⟨ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{μ_{2}}^{(2)}|

=_{(6.3)}  ∑_{ν_{1},μ_{1}=1}^{n_{1}}(|ϕ_{ν_{1}}^{(1)}⟩⟨ϕ_{μ_{1}}^{(1)}|) ⊗ 𝔏Ŵ (|ϕ_{ν_{1}}^{(1)}⟩⟨ϕ_{μ_{1}}^{(1)}|)

=_{lin. cont.} (1 ⊗ 𝔏Ŵ ) ( ∑_{ν_{1},μ_{1}=1}^{n_{1}} |ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{ν_{1}}^{(1)}⟩⟨ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{μ_{1}}^{(1)}|)

DRAFT, June 26, 2009

5See, e.g., (Neumark, 1959, $ 1, No. 9) and (Robertson and Robertson, 1967, Kap. 1, Satz 8). 6Note that

S_{sep}(H_{1} ⊗ H_{2}) =_{Theorem A.4.5} S(H_{1}) ⊗_{alg} S(H_{2}) .

6.1. DETECTING ENTANGLEMENT 151

and hence (6.1). Conversely, (6.1) gives

𝔏Ŵ (|ϕ_{ν_{1}}^{(1)}⟩⟨ϕ_{μ_{1}}^{(1)}|) = ⟨ϕ_{ν_{1}}^{(1)}| ∑_{ν_{1},μ_{1}=1}^{n_{1}}(|ϕ_{ν′_{1}}^{(1)}⟩⟨ϕ_{μ′_{1}}^{(1)}|) ⊗ 𝔏Ŵ (|ϕ_{ν′_{1}}^{(1)}⟩⟨ϕ_{μ′_{1}}^{(1)}|) |ϕ_{μ_{1}}^{(1)}⟩
=_{lin.} ⟨ϕ_{ν_{1}}^{(1)}| dim(H_{1}) (1 ⊗ 𝔏Ŵ ) (P̂H_{1}^{+}) |ϕ_{μ_{1}}^{(1)}⟩
=_{(6.1)} ⟨ϕ_{ν_{1}}^{(1)}| Ŵ |ϕ_{μ_{1}}^{(1)}⟩ ,

i.e. (6.3). Since

trace (Ŵ ∑_{k=1}^{N} ρ̂_{k}^{(1)} ⊗ ρ̂_{k}^{(2)}) =_{Jami} dim(H_{1}) ∑_{k=1}^{N} trace (ρ̂_{k}^{(1)} ⊗ 𝔏Ŵ (ρ̂_{k}^{(2)})) ,

positivity of 𝔏Ŵ implies nonnegativity of trace (Ŵ ρ̂) for separable^{7} ρ̂. Conversely the latter implies positivity of 𝔏Ŵ , since

⟨ϕ^{(2)} | 𝔏Ŵ (|ϕ^{(1)}⟩⟨ϕ^{(1)}|) ϕ^{(2)}⟩ =_{(6.3)} ⟨Ψ_{ϕ^{(1)},ϕ^{(2)}} | Ŵ Ψ_{ϕ^{(1)},ϕ^{(2)}}⟩ ∀ ϕ^{(1)} ∈ H_{1} , ϕ^{(2)} ∈ H_{2} ,

where

Ψ_{ϕ^{(1)},ϕ^{(2)}} =^{def} ( ∑_{ν_{1}=1}^{dim(H_{1})} ⟨ϕ^{(1)} | ϕ_{ν_{1}}^{(1)}⟩ ϕ_{ν_{1}}^{(1)}) ⊗ ϕ^{(2)} .

Corollary 6.1.3 For j ∈ {1, 2}, let {ϕ_{1}^{(j)}, …, ϕ_{n_{j}}^{(j)}} be a MONS of the Hilbert space H_{j}. Then for every Ŵ ∈ L(H_{1} ⊗ H_{2}) we have^{8}

trace (ρ̂Ŵ) < 0 ⟹ (1 ⊗ 𝔏† ; Ŵ )(ρ̂) ≱ 0 ∀ ρ̂ ∈ S(H_{1} ⊗ H_{2}) (6.6)

and

trace (ρ̂Ŵ) ≥ 0 ∀ ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) ⟺ 𝔏† ; Ŵ (ρ̂_{2}) ≥ 0 ∀ ρ̂_{2} ∈ S(H_{2}) , (6.7)

where 𝔏† ; Ŵ denotes the Hilbert-Schmidt adjoint of the linear mapping 𝔏Ŵ that is characterized by (6.3), i.e.:

trace (𝔏† ; Ŵ (Â_{2}) Â_{1}) = trace (Â_{2} 𝔏Ŵ (Â_{1})) ∀ Â_{1} ∈ L(H_{1}) , Â_{2} ∈ L(H_{2}) .

DRAFT, June 26, 2009

7Recall Footnote 6. 8Positivity of Â ∈ L(H) is easily to checked:

Â ≥ 0 ⟺ det(Â − x ˆ1) = \underbrace{c}_{∈ℝ} ∑_{ν=1}^{dim(H)}(−1)^{ν} \underbrace{c_{ν}}_{≥0} x^{ν} .

Outline of proof: (6.6) follows from

\frac{1}{n_{1}} trace (ρ̂Ŵ) =_{(6.1)} trace (ρ̂(1 ⊗ 𝔏Ŵ )(P̂H_{1}^{+}))
= trace ((1 ⊗ 𝔏† ; Ŵ )(ρ̂) P̂H_{1}^{+})

and (6.7) from (6.4).

We may conclude:

1. If Ŵ is an entanglement witness for ρ̂ then (1 ⊗ 𝔏† ; Ŵ )(ρ̂) ≱ 0 and, there-

fore, the positive map 𝔏† ; Ŵ cannot be completely positive.

2. For every positive mapping 𝔏′ : H_{2} ⟶ H_{1} we have:^{9}

(1 ⊗ 𝔏′)(ρ̂) ≱ 0 ⟹ ρ̂ ∉ S_{sep}(H_{1} ⊗ H_{2}) .

3. A state ρ̂ ∈ S(H_{1} ⊗ H_{2}) is separable if and only if (1 ⊗ 𝔏′)(ρ̂) is positive for all positive mappings 𝔏′ : H_{2} ⟶ H_{1} .

Remark: In general, for detecting entanglement, 𝔏† ; Ŵ is more useful than Ŵ since

𝔏† ; Ŵ (ρ̂) ≱ 0 ≠⇒^{i.g.} trace (Ŵ ρ̂) < 0 .

Ŵ , on the other hand, allows to detect entanglement by local measurements (Gühne et al., 2002).

6.1.2 Examples

Lemma 6.1.4 The flip operator of the bipartite system S with state space H⊗H, i.e. the linear Operator F̂ on H ⊗ H characterized by

F̂ (ψ_{1} ⊗ ψ_{2}) =^{def} ψ_{2} ⊗ ψ_{1} ∀ ψ_{1}, ψ_{2} ∈ H ,  (6.8)

has the following properties:^{10}

1.

F̂ = F̂^{†} , F̂^{2} = ˆ1 .

DRAFT, June 26, 2009

10Recall Definition A.4.4.

9Of course this statement is relevant for non-completely positive 𝔏′, only.

6.1. DETECTING ENTANGLEMENT 153

2.

F̂ = P̂_{+} − P̂_{−} ,

where the

P̂_{±} =^{def} \frac{1}{2} (ˆ1 ± F̂)

are the projectors onto the symmetric resp. anti-symmetric pure states of S :

Ψ ∈ P̂_{±}(H ⊗ H) ⟺ F̂ Ψ = ±Ψ̂ ∀ Ψ ∈ H ⊗ H .

3. For all β ∈ ℝ :

ˆ1 + β F̂ ≥ 0 ⟺ β ∈ [−1, +1] .

4.

trace (F̂) = − dim(H) .

5. For all ρ̂ ∈ S(H ⊗ H) :

ρ̂ separable ⟹ trace (F̂ ρ̂) ≥ 0 .

Outline of proof: The first four statements are more or less obvious and the last one follows from

trace(F̂(|ψ_{1}⟩⟨ψ_{1}| ⊗ |ψ_{2}⟩⟨ψ_{2}|)) = trace (F̂ |ψ_{1} ⊗ ψ_{2}⟩⟨ψ_{1} ⊗ ψ_{2}|)
= trace(|ψ_{2} ⊗ ψ_{1}⟩⟨ψ_{1} ⊗ ψ_{2}|)
= ⟨ψ_{1} ⊗ ψ_{2} | ψ_{2} ⊗ ψ_{1}⟩
= |⟨ψ_{1} | ψ_{2}⟩|^{2} .

Consequence: The flip operator F̂ is an entanglement witness for the mixed Werner states^{11} ρ̂_{W}(β) with β ∈ [−1, −1/ dim(H)] , where

ρ̂_{W}(β) =^{def} \frac{ˆ1 + β F̂}{trace (ˆ1 + β F̂)} ∀ β ∈ [−1, +1] .  (6.9)

DRAFT, June 26, 2009

11Compare Footnote 27 of Chapter 4.

For H_{1} = H_{2} = H the positive linear mapping (6.3) associated with the flip operator on H ⊗ H is the transposition^{12} w.r.t. {ϕ_{1}, …, ϕ_{n}} = {ϕ_{1}^{(1)}, …, ϕ_{n_{1}}^{(1)}} :

𝔏F̂(|ϕ_{ν_{1}}⟩⟨ϕ_{ν_{2}}|) = T (|ϕ_{ν_{1}}⟩⟨ϕ_{ν_{2}}|)
=^{def} |ϕ_{ν_{2}}⟩⟨ϕ_{ν_{1}}|  (6.10)
= 𝔏′ ; F̂(|ϕ_{ν_{1}}⟩⟨ϕ_{ν_{2}}|) ∀ ν_{1}, ν_{2} ∈ {1, …, n} .

While the flip operator cannot be an entanglement witness for, e.g., the F̂-invariant pure states,^{13} a two-qubit state ρ̂ is separable iff its partial transpose (1 ⊗ 𝔗) (ρ̂) is positive. Slightly more generally we have:

Theorem 6.1.5 For j ∈ {1, 2}, let {ϕ_{1}^{(j)}, …, ϕ_{n_{j}}^{(j)}} be a MONS of the Hilbert space H_{j}. If n_{1} + n_{2} ∈ {4, 5} then an arbitrarily given state

ρ̂ = ∑_{ν_{1},μ=1}^{n_{1}} ∑_{ν_{2},μ_{2}=1}^{n_{2}} ρ_{ν_{1}ν_{2},μ_{1}μ_{2}} |ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{ν_{2}}^{(2)}⟩⟨ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{μ_{2}}^{(2)}| ∈ S(H_{1} ⊗ H_{2})

is separable iff it has a positive partial transpose:^{14}

∑_{ν_{1},μ=1}^{n_{1}} ∑_{ν_{2},μ_{2}=1}^{n_{2}} ρ_{ν_{1}ν_{2},μ_{1}μ_{2}} |ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{μ_{2}}^{(2)}⟩⟨ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{ν_{2}}^{(2)}| ≥ 0 .

Proof:^{15} See (Horodecki et al., 1996, Theorem 3).

Another example for H_{1} = H_{2} = H and {ϕ_{1}, …, ϕ_{n}} = {ϕ_{1}^{(1)}, …, ϕ_{n_{1}}^{(1)}} =

{ϕ_{1}^{(2)}, …, ϕ_{n_{1}}^{(2)}} is

Ŵ = ˆ1_{H⊗H} − n P̂_{H}^{+}

= ∑_{ν, μ=1}^{n}(|ϕ_{ν}⟩⟨ϕ_{μ}|) ⊗ (δ_{νμ}ˆ1 − |ϕ_{ν}⟩⟨ϕ_{μ}|)
= m (1 ⊗ 𝔏_{H}^{D}) (P̂_{H}^{+}) ,

DRAFT, June 26, 2009

12The transposition was considered at the end of 4.2.1.

13See, however, (Horodecki and Horodecki, 1996).

14Positivity of the partial transpose, contrary to the partial transpose itself, does not depend on the choice for the MONS {ϕ_{1}^{(2)}, …, ϕ_{n_{2}}^{(2)}}.

15The essential point is that — provided n_{1} + n_{2} ∈ {4, 5} — every positive mapping 𝔏′ : L(H_{2}) ⟶ L(H_{1}) is decomposable; see (Labuschagne et al., 2003) and references given there. For entangled PPT states in higher dimensions see (Ha et al., 2003).

6.1. DETECTING ENTANGLEMENT 155

where

𝔏Ŵ (B̂) =_{(6.1)} 𝔏_{H}^{D}(B̂)  (6.11)
=^{def} trace (B̂) − B̂ ∀ B̂ ∈ L(H) ,  (6.12)

Obviously, 𝔏_{H}^{D} is positive and

(1 ⊗ 𝔏_{H}^{D}) (ρ̂) = trace_{2}(ρ̂) ⊗ ˆ1 − ρ̂ ∀ ρ̂ ∈ L(H) .

Therefore,^{16}

trace_{2}(ρ̂) ⊗ ˆ1 ≱ ρ̂ ⟹ ρ̂ ∉ S_{sep}(H ⊗ H) .  (6.13)

6.1.3 Other Criteria^{17}

Up to now, in view of Corollary 6.1.3, we considered the entanglement criterion

(1 ⊗ \underbrace{𝔏}_{≥0})(ρ̂) ≱ 0 ⟹ ρ̂ ∉ S_{s}(H_{1} ⊗ H_{1}) .  (6.14)

only for positive maps 𝔏 : L(H_{2}) ⟶ L(H_{1}) . But, of course, (6.14) also holds for positive maps 𝔏 : L(H_{2}) ⟶ L(H_{2}) . Typical examples are

1. 𝔏 = 𝔗 : Every separable state ρ̂ ∈ S(H_{1} ⊗ H_{2}) is PPT, i.e. has a positive partial transpose (1 ⊗ 𝔗)(ρ̂) — for every choice of MONS.

2. 𝔏 = 𝔏_{H}^{D} :

ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) ⟹ trace_{2}(ρ̂) ⊗ ˆ1 ≥ ρ̂.  (6.15)

Remarks: 1. Since^{18}

((1 ⊗ 𝔗)(ρ̂))^{†} = (1 ⊗ 𝔗)(ρ̂) ∀ ρ̂ ∈ S(H_{1} ⊗ H_{2})

and

trace ((1 ⊗ 𝔗)(ρ̂)) = 1 ∀ ρ̂ ∈ S(H_{1} ⊗ H_{2}) ,

DRAFT, June 26, 2009

16See (Hiroshima, 2003) in this connection.

17See also (Doherty et al., 2003)

18Note that

(1 ⊗ 𝔗)(Â^{†}) = ((1 ⊗ 𝔗)(Â))^{†} ∀ Â ∈ L(H_{1} ⊗ H_{2}) .

we have^{19}

(1 ⊗ 𝔗)(ρ̂) ≱ 0 ⟺ ∥(1 ⊗ 𝔗)(ρ̂)∥_{1} > 1 ∀ ρ̂ ∈ S(H_{1} ⊗ H_{2}) ,

as exploited in (Vidal and Werner, 2002).

2. Since^{20}

T (τ̂^{ν}) = (−1)^{δ_{ν2}} τ̂^{ν} ∀ ν ∈ {0, …, 3} ,

we have for two-qubit states ρ̂:^{21}

(1 ⊗ 𝔗)(ρ̂) = ρ̂ ⟺ trace ((τ̂^{μ} ⊗ τ̂^{2}) ρ̂) = 0 ∀ μ ∈ {0, …, 3} .

3. For all ρ̂ ∈ S(H_{1} ⊗ H_{2}) :

(1 ⊗ 𝔗)(ρ̂) Ψ = \underbrace{λ}_{<0} Ψ ≠ 0

⟹ \{ \begin{matrix}(1 ⊗ 𝔗)(|Ψ⟩⟨Ψ|) entanglement witness for ρ̂, \\ Ψ non-separable . \end{matrix}

Let us list some other sufficient criteria for entanglement:

1. Range criterion:^{22}

ψ^{(1)} ⊗ (ψ^{(2)})^{∗} ∉ range((1 ⊗ 𝔗)(ρ̂)) ∀ ψ^{(1)} ⊗ ψ^{(2)} ∈ range(ρ̂)

⟹ ρ̂ ∉ S_{sep}(H_{1} ⊗ H_{2}) ,

(6.16)

where

(ψ^{(2)})^{∗} =_{(4.9)} ∑_{ν=1}^{n_{2}} ⟨ψ^{(2)} | ϕ_{ν}^{(2)}⟩ ϕ_{ν}^{(2)}

depends on the MONS {ϕ_{1}^{(2)}, …, ϕ_{n_{2}}^{(2)}} of H_{2} .

Outline of proof: By Theorem A.4.5 (and the spectral theorem), every ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) can be written in the form

ρ̂ = ∑_{k=1}^{N} |ψ_{k}^{(1)} ⊗ ψ_{k}^{(2)}⟩⟨ψ_{k}^{(1)} ⊗ ψ_{k}^{(2)}| .

DRAFT, June 26, 2009

19Recall (5.31). 20Recall (A.22) 21Compare (Altafini, 2003, Sect. II.B., Corollary 1). 22This criterion especially implies that every state ρ̂ ∈ S(H_{1} ⊗ H_{2}) containing no product state

in its range must be entangled. For a non-separable PPT state on ℂ^{2} ⊗ ℂ^{4} containing product states in its range and fulfilling the range criterion see (Horodecki et al., 2001, Eqn. (27)).

Then Lemma A.4.2 implies

ψ_{k}^{(1)} ⊗ ψ_{k}^{(2)} ∈ range(ρ̂) ∀ k ∈ {1, …, N}

and, since^{23}

(1 ⊗ 𝔗)(ρ̂) = ∑_{k=1}^{N} |ψ_{k}^{(1)} ⊗ (ψ_{k}^{(2)})^{∗}⟩⟨ψ_{k}^{(1)} ⊗ (ψ_{k}^{(2)})^{∗}| ,

also

ψ_{k}^{(1)} ⊗ (ψ_{k}^{(2)})^{∗} ∈ range((1 ⊗ 𝔗)(ρ̂)) ∀ k ∈ {1, …, N} .

2. Entropic inequalities:^{24}

\begin{matrix}ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) \\ α ∈ \{0, 1, 2, ∞\} \end{matrix} \} ⟹ S_{α}(trace_{j}(ρ̂)) ≤ S_{α}(ρ̂) ∀ j ∈ \{1, 2\} , (6.17)

(more information for the total state than for the partial states), where^{25}

S_{α}(ρ̂) =^{def} \frac{log_{2}(trace (ρ̂^{α}))}{1 − α} ∀ α ∈ (0, 1) ∪ (1, ∞)  (6.18)

and^{26}

S_{k}(ρ̂) =^{def} lim_{α→k} S_{α}(ρ̂) ∀ k ∈ {0, ∞} .

3. Greatest cross norm criterion (Rudolph, 2000, Thm. 5):

ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) ⟺ \left∥{ \atop ρ̂}\right∥_{γ} = 1 ,  (6.19)

where

∥Â∥_{γ} =^{def} inf\{∑_{k=1}^{N} ∥Â_{k}^{(1)}∥_{1} ∥Â_{k}^{(2)}∥_{1} : ∑_{k=1}^{N} \underbrace{Â_{k}^{(1)}}_{∈L(H_{1})} ⊗ \underbrace{Â_{k}^{(2)}}_{∈L(H_{2})} = Â , N ∈ ℕ\} (6.20)

for Â ∈ L(H_{1} ⊗ H_{2}) .

DRAFT, June 26, 2009

23Recall (4.14). 26Note also that

S_{1}(ρ̂) = lim_{1<α→1} S_{α}(ρ̂) .

24See (Vollbrecht and Wolf, 2002, Sect. III) and references given there.

25The normalization factor 1/1 − α guarantees

max_{ρ̂∈S(H)} S_{α}(ρ̂) = log_{2}(dim(H))

for the so-called Renỳi quantum entropies S_{α}; see also (Lavenda and Dunning-Davies, 2003).

4. Computable cross norm criterion^{27} (Rudolph, 2002, Prop. 19):

ρ̂ ∈ S_{sep}(H ⊗ H) ⟹ ∥𝔄(ρ̂)∥ ≤ 1 ,  (6.21)

where the linear mapping 𝔄 : L(H ⊗ H) ⟶ L(L(H)) is characterized by

\begin{matrix}𝔄(|ϕ_{ν_{1}} ⊗ ϕ_{ν_{2}}⟩⟨ϕ_{μ_{1}} ⊗ ϕ_{μ_{2}}|)Â \\ =^{def} ⟨ϕ_{ν_{2}} | Â ϕ_{μ_{2}}⟩ |ϕ_{ν_{1}}⟩⟨ϕ_{μ_{1}}| ∀ ν_{1}, ν_{2}, μ_{1}, μ_{2} ∈ \{1, …, n\} , Â ∈ L(H) \end{matrix} (6.22)

w.r.t. the fundamental MONS {ϕ_{1}, …, ϕ_{n}} of H.

5. Reduction criterion^{28} (Horodecki and Horodecki, 1999):

ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) ⟺ \{ \begin{matrix}trace_{2}(ρ̂) ⊗ ˆ1 − ρ̂ ≥ 0 , \\ ˆ1 ⊗ trace_{1}(ρ̂) − ρ̂ ≥ 0 . \end{matrix}  (6.23)

6.2 Local Operations and Classical Communication (LOCC)

6.2.1 General Aspects

By local operations and classical communication (LOCC) state transformations of the form^{29}

ρ̂ ⟼ ρ̂′ = 𝔏_{LOCC}(ρ̂) = ∑_{k=1}^{N} p_{k} (ℭ_{k}^{(1)} ⊗ ℭ_{k}^{(2)})(ρ̂)  (6.24)

can be implemented on a bipartite system with state space H_{1} ⊗ H_{2} , where

N ∈ ℕ ∑_{j=1}^{N} \underbrace{p_{k}}_{>0} = 1

and the ℭ_{k}^{(1)} resp. ℭ_{k}^{(2)} are trace preserving elements of Q(H_{1}, H_{1}) resp. Q(H_{2}, H_{2}) . Obviously,

\begin{matrix}ψ^{(1)} ∈ H_{1} , ψ^{(2)} ∈ H_{2} \\ ∥ψ^{(1)}∥ = ∥ψ^{(2)}∥ = 1 \end{matrix} \} ⟹ S_{LOCC}(ψ^{(1)} ⊗ ψ^{(2)}) = S_{sep}(H_{1} ⊗ H_{2}) ,  (6.25)

DRAFT, June 26, 2009

27Note that for dim(H) ≥ dim(H_{1}), dim(H_{2}) there is a canonical mapping of S(H_{1} ⊗ H_{2}) into S(H ⊗ H) respecting separability. Therefore, it is sufficient to consider the case H_{1} = H_{2} .

28See (Hiroshima, 2003), in this connection.

29Obviously, the LOCC quantum operations (6.24) form a group.

where

S_{LOCC}(ρ̂) =^{def} {states ρ̂′ of the form (6.24)} .

In order to characterize the possible transformations of pure states by LOCC we need the following definition:

Definition 6.2.1 Let Ĥ, Ĥ′ be Hermitian operators on the Hilbert space H with spectral decompositions

Ĥ = ∑_{ν=1}^{n} E_{ν} |ψ_{ν}⟩⟨ψ_{ν}| , Ĥ′ = ∑_{ν=1}^{n} E′_{ν} |ψ′_{ν}⟩⟨ψ′_{ν}| .

Then we write Ĥ ⪯ Ĥ′ iff^{30} there p_{1}, …, p_{n} ≥ 0 and permutations π_{1}, …, π_{n} ∈ S_{n} with

E_{μ} = ∑_{ν=1}^{n} p_{ν} E′_{π_{ν}(μ)} ∀ μ ∈ {1, …, n} , ∑_{ν=1}^{n} p_{ν} = 1 .

Theorem 6.2.2 Let Ĥ, Ĥ′ be Hermitian operators on the Hilbert space H. Then Ĥ ⪯ Ĥ′ iff there are p_{1}, …, p_{n} ≥ 0 and unitary Operators Û_{1}, …, Û_{n} on H with

Ĥ = ∑_{ν=1}^{n} p_{ν} Û_{ν} Ĥ′ Û_{ν}^{†} , ∑_{ν=1}^{n} p_{ν} = 1 .

Proof: See (Nielsen and Chuang, 2001, Theorem 12.13).

For pure states and H_{1} = H_{2} , concerning LOCC, nothing is lost if the operations on one side are restricted to be unitary:

Lemma 6.2.3 Let H be a Hilbert space, 0 ≠ Ψ ∈ H ⊗ H and M̂ ∈ L(H) . Then there are N̂, Û ∈ L(H) with

(ˆ1 ⊗ M̂) Ψ = (N̂ ⊗ Û) Ψ , Û^{†} = Û^{−1} .

Outline of proof: According to Theorem A.4.7 there are non-negative numbers s_{1}, …, s_{n} and orthonormal bases {ϕ_{1}^{(1)}, …, ϕ_{n}^{(1)}} and {ϕ_{1}^{(2)}, …, ϕ_{n}^{(2)}} of H with

Ψ = ∑_{α=1}^{n} s_{α} ϕ_{α}^{(1)} ⊗ ϕ_{α}^{(2)} , n =^{def} dim(H) .

DRAFT, June 26, 2009

30Obviously, ⪯ does not depend on the choice of orthonormal eigenbases for Ĥ, Ĥ′ and gives a semi-ordering of S(H) .

160 CHAPTER 6. HANDLING ENTANGLEMENT

Then, writing

M̂ = ∑_{ν,μ=1}^{n} M_{νμ} |ϕ_{ν}^{(2)}⟩⟨ϕ_{μ}^{(2)}|

and defining

N̂′ =^{def} ∑_{ν,μ=1}^{n} M_{νμ} |ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}|

we get

Ŝ (N̂′ ⊗ ˆ1) Ψ = (ˆ1 ⊗ M̂) Ψ ,

where Ŝ denotes the linear swap operator characterized by

Ŝ ϕ ⊗ ψ =^{def} ψ ⊗ ϕ ∀ ϕ, ψ ∈ H .

Therefore (N̂′ ⊗ ˆ1) Ψ and (ˆ1 ⊗ M̂) Ψ have the same Schmidt coefficients, i.e. there are unitary Û, V̂ ∈ L(H) with

(ˆ1 ⊗ M̂) Ψ = (V̂ ⊗ Û) (N̂′ ⊗ ˆ1) Ψ .

Defining N̂ =^{def} V̂ N̂′ we get the statement of the lemma.

The state 𝔏_{LOCC}(ρ̂) in (6.24) can be pure only if it coincides with (ℭ_{k}^{(1)} ⊗ ℭ_{k}^{(2)})(ρ̂) for all k ∈ {1, …, N} with p_{k} ≠ 0 . This, together with Theorem 6.2.2 allows us to prove the following: Theorem 6.2.4 Let H be a Hilbert space. Then

ρ̂′ ∈ S_{LOCC}(ρ̂) ⟺ trace_{2}(ρ̂) ⪯ trace_{2}(ρ̂′) ∀ ρ̂, ρ̂′ ∈ S_{pure}(H ⊗ H) .

Outline of proof: Assume that (6.24) holds. Then, by Lemma 6.2.3, 𝔏_{LOCC} can be chosen such that

ρ̂′ = ∑_{j=1}^{N′} \underbrace{(M̂_{j} ⊗ Û_{j}) ρ̂(M̂_{j}^{†} ⊗ Û_{j}^{†})}_{∝ρ̂′}

holds with N′ ∈ ℕ and M̂_{j}, Û_{j} ∈ L(H) fulfilling

∑_{j=1}^{N′} M̂_{j}^{†}M̂_{j} = ˆ1 , Û_{j} = Û_{j}^{†} ∀ j ∈ {1, …, N′} .  (6.26)

This implies

M̂_{j} trace_{2}(ρ̂) M̂_{j}^{†} = p′_{j} trace_{2}(ρ̂′) ∀ j ∈ {1, …, N′} ,  (6.27)

where

p′_{j} =^{def} trace (M̂_{j} trace_{2}(ρ̂) M̂_{j}^{†}) ∀ j ∈ {1, …, N′} .  (6.28)

6.2. LOCAL OPERATIONS AND CLASSICAL COMMUNICATION 161

Using the polar decomposition

+ \sqrt{trace_{2}(ρ̂)} M̂_{j}^{†} = Û_{j} ^{+} \sqrt{( ^{+} \sqrt{trace_{2}(ρ̂)} M̂_{j}^{†})^{†} ^{+} \sqrt{trace_{2}(ρ̂)} M̂_{j}^{†}}
= Û_{j} ^{+} \sqrt{M̂_{j} trace_{2}(ρ̂) M̂_{j}^{†}}
=_{(6.27)} Û_{j} ^{+} \sqrt{p′_{j} trace_{2}(ρ̂′)}

and multiplying from the right with its adjoint gives

+ \sqrt{trace_{2}(ρ̂)} M̂_{j}^{†}M̂_{j} ^{+} \sqrt{trace_{2}(ρ̂)} = p′_{j} Û_{j} trace_{2}(ρ̂′) Û_{j}^{†} ∀ j ∈ \{1, …, N′\} .

Finally, summing over j gives

trace_{2}(ρ̂) =_{(6.26)} ∑_{j=1}^{N′} p′_{j} Û_{j} trace_{2}(ρ̂′) Û_{j}^{†}  (6.29)

and hence trace_{2}(ρ̂) ⪯ trace_{2}(ρ̂′) , by Theorem 6.2.2, since (6.26) and (6.28) imply

∑_{j=1}^{N′} \underbrace{p′_{j}}_{≥0} = 1 .  (6.30)

Conversely, if ρ̂_{1} ⪯ ρ̂′_{1} , where

ρ̂_{1} =^{def} trace_{2}(ρ̂_{1}) , ρ̂′_{1} =^{def} trace_{2}(ρ̂′_{1}) ,

then Theorem 6.2.2 implies (6.29) for suitable unitary Û_{1}, …, Û_{N′} ∈ L(H) and p′_{1}, …, p′_{N′} > 0 fulfilling (6.30). Then, if we define

M̂_{j} =^{def} ^{+} \sqrt{p′_{j} ρ̂′_{1}} Û_{j}^{†} ^{+} \sqrt{(ρ̂_{1}/^{\}ρ̂H)^{−1}} P̂_{ρ̂H} + \frac{ˆ1 − P̂_{ρ̂H}}{\sqrt{p′_{j}}} ∀ j ∈ \{1, …, N′\}

(6.29) and (6.30) imply

∑_{j=1}^{N′} M̂_{j}^{†}M̂_{j} = ˆ1  (6.31)

and

trace_{2}((M̂_{j} ⊗ ˆ1) ρ̂(M̂_{j}^{†} ⊗ ˆ1)) = p′_{j} trace_{2}(ρ̂′) ∀ j ∈ {1, …, N′} ,  (6.32)

Since

\begin{matrix}trace_{2}(ρ̂′′) = trace_{2}(ρ̂′) \\ ρ̂′, ρ̂′′ ∈ S_{pure}(H ⊗ H) \end{matrix} \} ⟹ TheoremA.4.7 \{ \begin{matrix}(ˆ1 ⊗ Û) ρ̂′′ (ˆ1 ⊗ Û^{†}) = ρ̂′ \\ for some unitary Û ∈ L(H) , \end{matrix}

(6.32) shows that there are unitary V̂_{1}, … V̂_{N′} ∈ L(H) with

(M̂_{j} ⊗ V̂_{j}) ρ̂(M̂_{j}^{†} ⊗ V̂_{j}^{†}) = p′_{j} ρ̂′ ∀ j ∈ {1, …, N′} .

This, together with (6.30) and (6.31), shows that ρ̂ can be transformed into ρ̂′ by LOCC.

Lemma 6.2.5 Let Ĥ, Ĥ′ be Hermitian operators on the Hilbert space H with spectral decompositions

Ĥ = ∑_{ν=1}^{n} E_{ν} |ψ_{ν}⟩⟨ψ_{ν}| , Ĥ′ = ∑_{ν=1}^{n} E′_{ν} |ψ′_{ν}⟩⟨ψ′_{ν}| .

Then Ĥ ⪯ Ĥ′ iff the conditions

∑_{ν=1}^{n} E_{ν} = ∑_{ν=1}^{n} E′_{ν}  (6.33)

and

max_{π∈S_{n}} ∑_{ν=1}^{n′} E_{π(ν)} ≤ max_{π∈S_{n}} ∑_{ν=1}^{n′} E′_{π(ν)} ∀ n′ ∈ {1, …, n − 1}  (6.34)

are fulfilled.

Proof: See (Nielsen and Chuang, 2001, Proposition 12.11).

Remarks:

1. Obviously,

\frac{1}{dim H} ˆ1 ⪯ ρ̂_{1} ∀ ρ̂_{1} ∈ S(H) .

2. According to Theorem 6.2.4, therefore,^{31}

S_{pure}(H ⊗ H) ⊂ S_{LOCC}\binom{}{P̂_{H}^{+}} .

3. For qubit states ρ̂, ρ̂′ we always have either ρ̂ ⪯ ρ̂′ or ρ̂′ ⪯ ρ̂ or both.

4. However, for higher dimensional H neither ρ̂ ⪯ ρ̂′ nor ρ̂′ ⪯ ρ̂ need be true for ρ̂, ρ̂′ ∈ S(H) as application of Lemma 6.2.5 to, e.g., the case

ρ̂ = \frac{1}{15} (7 ϕ_{1} + 7 ϕ_{2} + ϕ_{3}) , ρ̂ = \frac{1}{15} (11 ϕ_{1} + 2 ϕ_{2} + 2 ϕ_{3})

shows if {ϕ_{1}, ϕ_{2}, ϕ_{3}} is a MONS of H.

Note that LOCC would be much more powerful if intermediate use of nonlocally entangled ancillary pairs, restoring their original states, could be made:

DRAFT, June 26, 2009

31Recall (6.2).

For j = 1 resp. j = 2 let {ϕ_{1}^{(j)}, …, ϕ_{n_{j}}^{(j)}} be a MONS of the Hilbert space H_{j}, let

Ψ = ∑_{νμ=1}^{n_{1}} λ_{νμ} ϕ_{ν}^{(1)} ⊗ ϕ_{μ}^{(1)} , Ψ′ = ∑_{ν,μ_{1}}^{n_{1}} λ′_{νμ} ϕ_{ν}^{(1)} ⊗ ϕ_{μ}^{(1)}

be pure states of the bipartite system with state space H_{1} ⊗ H_{1} and let

Φ = ∑_{ν_{1}}^{n_{1}} ∑_{α,β=1}^{n_{2}} λ_{νμ} λ_{αβ}^{anc} (ϕ_{ν}^{(1)} ⊗ ϕ_{α}^{(2)}) ⊗ (ϕ_{μ}^{(1)} ⊗ ϕ_{β}^{(2)}) ,

Φ′ = ∑_{ν_{1}}^{n_{1}} ∑_{α,β=1}^{n_{2}} λ′_{νμ} λ_{αβ}^{anc} (ϕ_{ν}^{(1)} ⊗ ϕ_{α}^{(2)}) ⊗ (ϕ_{μ}^{(1)} ⊗ ϕ_{β}^{(2)})

be pure states of the bipartite system with state space (H_{1} ⊗ H_{2}) ⊗ (H_{1} ⊗ H_{2}). Then it may happen that^{32}

|Ψ⟩ ⪯̸ |Ψ′⟩ but |Φ⟩ ⪯ |Φ′⟩

— an effect of the ancillary system in the state ∑_{α,β=1}^{n_{2}} λ_{αβ}^{anc} ϕ_{α}^{(2)} ⊗ ϕ_{β}^{(2)} that

is called entanglement catalysis.

Finally, given ρ̂, ρ̂′ ∈ S_{pure}(H ⊗ H) , let us note that ρ̂ can be transformed into ρ̂′ by local operations without communication iff

trace_{2}(ρ̂′) ∝ K̂ trace_{2}(ρ̂) K̂^{†} , K̂^{†}K̂ ≤ ˆ1  (6.35)

holds for some K̂ ∈ L(H) . Let

trace_{2}(ρ̂) = ∑_{ν=1}^{n} p_{ν} |ψ_{ν}⟩⟨ψ_{ν}| , trace_{2}(ρ̂′) = ∑_{ν=1}^{n} p′_{ν} |ψ′_{ν}⟩⟨ψ′_{ν}|

be spectral decompositions of ρ̂, ρ̂′. Then, exploiting unitary transformations and the polar decomposition of K̂ one can show that (6.35) is equivalent to existence of real numbers k_{1}, …, k_{n} and a permutation π ∈ S_{n} with:

∑_{ν=1}^{n}(k_{ν})^{2} ≤ 1 , p′_{ν} = (k_{ν})^{2} p_{π(ν)} ∀ ν ∈ {1, …, n} .

DRAFT, June 26, 2009

32See (Nielsen and Chuang, 2001, Exercise 12.21) for an explicit example with (n_{1}, n_{2}) = (4, 2) .

6.2.2 Entanglement Dilution

...

6.2.3 Entanglement Distillation

See, e.g., (Bowmeester et al., 2000, Section 8.4) and (Devetak and Winter, 2005).

6.3 Quantification of Entanglement^{33}

For pure states |Ψ⟩⟨Ψ| of a bipartite system S with state space H = H_{1} ⊗ H_{2} there is a generally accepted measure of entanglement, namely the entropy of entanglement^{34}

E_{pure}(|Ψ⟩⟨Ψ|) =^{def} S_{1}(trace_{2}(|Ψ⟩⟨Ψ|))  (6.36)
=_{Th. A.4.7} S_{1}(trace_{1}(|Ψ⟩⟨Ψ|)) ,

i.e. the von Neumann entropy of the partial states. Since

E_{pure}(∑_{ν=1}^{n′} \sqrt{p_{ν}} ϕ_{ν}^{(1)} ⊗ ϕ_{ν}^{(2)}) = − ∑_{ν=1}^{n′} \underbrace{p_{ν}}_{>0} log_{2}(p_{ν})

holds for Schmidt decompositions, E_{pure}(|Ψ⟩⟨Ψ|) becomes maximal^{35} for

n′ = min\{dim(H_{1}), dim(H_{2})\} , p_{ν} = \frac{1}{n′} ∀ ν ∈ \{1, …, n′\} .

Hence

max_{Ψ∈H_{1}⊗H_{2}_{∥Ψ∥=1}} E(|Ψ⟩⟨Ψ|) = log_{2}( min{dim(H_{1}), dim(H_{2})})  (6.37)

holds for E = E_{pure} .

Strict requirements for every entanglement measure E on S(H_{1} ⊗ H_{2}) :

1.

E(ρ̂) ≥ 0 ∀ ρ̂ ∈ S(H_{1} ⊗ H_{2}) .

DRAFT, June 26, 2009

33See also (Řeháček and Hradil, 2002).

34The unit of entanglement is one ebit.

35Recall Remark 3 in the beginning of Section 5.1.

6.3. QUANTIFICATION OF ENTANGLEMENT 165

2.

E(ρ̂) = 0 ∀ ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) .

3.

E(ρ̂) = 0 ⟹ ρ̂ ∈ S_{sep}(H_{1} ⊗ H_{2}) ∀ ρ̂ ∈ S(H_{1} ⊗ H_{2})

(could be relexed by cosideration of additional entanglement measures). 4.

ρ̂′ ∈ S_{LOCC}(ρ̂) ⟹ E(ρ̂′) ≤ E(ρ̂) ∀ ρ̂, ρ′ ∈ S(H_{1} ⊗ H_{2}) .

5. For all pure states |Ψ_{1}⟩⟨Ψ_{1}| , |Ψ_{2}⟩⟨Ψ_{2}| ∈ S(H_{1} ⊗ H_{2}) : E_{pure}(|Ψ_{1}⟩⟨Ψ_{1}|) < E_{pure}(|Ψ_{2}⟩⟨Ψ_{2}|) ⟹ E(|Ψ_{1}⟩⟨Ψ_{1}|) < E(|Ψ_{2}⟩⟨Ψ_{2}|) .

Desirable for entanglement measures E on S(H_{1} ⊗ H_{2}) :

1.

E(ρ̂) ≥ E(ρ̂′) ⟹ ρ̂′ ∈ S_{LOCC}(ρ̂) ∀ρ̂, ρ̂′ ∈ S(H_{1} ⊗ H_{2})

(not possible).

2. Continuity

3. Additivity

4. Subadditivity

5. Convexity

Standard entanglement measures:^{36}

1. Entanglement of formation^{37} is the convex continuation

E_{F}(ρ̂) =^{def} inf ; ρ̂=∑_{ν} ^{p_{ν}}_{0} ^{ρ̂_{ν}}_{pure} ∑_{ν} p_{ν} S_{1}(trace_{2}(ρ̂_{ν}))

of the entropy of entanglement (6.36) to all of S(H_{1} ⊗ H_{2}) .

Additivity of the entanglement of formation is generally conjectured but not yet proved.

2.

DRAFT, June 26, 2009

36There cannot be a unique entanglement measure (Morikoshi et al., 2003). 37For pairs of qubits the entanglement of formation coincides with the concurrence (Wootters,

2001, Section 3.1).

Squashed Entanglement (Christandl and Winter, 2004)

• In general, Werner states are not maximally entangled, i.e. their entanglement of formation (tangle) is not maximal for given a fixed degree of mixedness (linear entropy) (White et al., 2001).

• Compare with (Eckert et al., 2002). (see also quant-ph/0210107)

• What about local superselection rules? (Verstraete and Cirac, 2003; Bartlett and Wiseman, 2003)

• Exploit the notion of truncated expectation values. (Lee et al., 2003)

• What is the generalization of the latter for mixed states?

• Existiert ein Abstandsmaß a la (Lee et al., 2003)?

• Warum verwendet man nicht den Hilbert-Schmidt-Abstand von Zuständen, der sich leicht mithilfe von Erwartungswerten von (Produkten von) Pauli-Operatoren ausdrücken läßt?

• Besteht ein Zusammenhang mit (Lee et al., 2003)? Der Abstand zur Menge der separablen Zustände sollte doch ein Maß für Verschränktheit sein...

Remarks:

1. For (dim(H_{1}), dim(H_{1})) either (2,2) or (3,2) PPT w.r.t. the second factor is necessary and sufficient for separability (Horodecki et al., 1996).

2. Otherwise states with bound entanglement, i.e. entangled PPT states, exist (Horodecki et al., 1996, Appendix).

3. In the 2-qubit case every entangled mixed state can be represented as a convex combination of a separable (in general mixed) state with a pure entangled state (Lewenstein and Sanpera, 1998). The representation with minimal norm of the pure state is unique.

4. Also this shows that, for the 2-qubit case, separability is equivalent to PPT.

Appendix A

A.1 Turing’s Halting Problem

The halting problem is the following:

There is no algorithm by which one may decide for every program and every finite input to the program whether the program will halt or loop forever.

The proof given by Turing is essentially as follows:

Since every program may be encoded into a finite sequence (b_{1}, …, b_{n}) of bits the programs may be indexed by the corresponding numbers ∑_{ν=1}^{n} b_{ν} 2^{n−ν} . The same holds for all finite inputs. Now assume that there is an algorithm telling us for all (j, k) ∈ ℤ_{+}^{2} whether program j will halt on input k. Then this algorithm may be used to write a program P with the following property:

For every j ∈ ℤ_{+} , program P will hold on input j iff program j will not.

Obviously, program P is different from program j for every j ∈ ℤ_{+} — a contradiction.

A heuristic explanation is the following:

There are uncountably many possibilities for infinite loops which, therefore, cannot be checked in a systematic way. But we cannot be sure whether a given program will halt on a given input or not unless

• an infinite loop is found by chance or

• the program was actually tested on the input and found to halt.

Let us finally note that the halting problem is a solution to Hilbert’s 23rd problem (see http://aleph0.clarku.edu/ djoyce/hilbert/).

A.2 Some Remarks on Quantum Teleportation

Even though quantum teleportation, described in 1.2.2, seems to indicate some kind of quantum nonlocality, there is a naive ‘explanation’ relying on locality and some kind of realism:

There is a set of four compatible relations, corresponding to the Bell states, between a pair of qubits. These correlations are so strong that the state (predicting ensemble averages) of the second qubit is fixed by that of the first qubit and the Bell relation (considered as an element of reality). Therefore, Alice need only inform Bob about the Bell relation of qubit 1 to some qubit 2 with known Bell relation to Bob’s qubit 3 in order to enable Bob to transform qubit 3 into the unknown state of qubit 1. If qubits 1 and 2 are accessible to Alice and far apart from Bob, then Alice can access this information without influencing Bob’s qubit (thanks to locality) , although disturbing qubits 1 and 2 in an uncontrollable way.

Strictly speaking, of course, this picture is inconsistent:

Two qubits may be in a factorized 2-qubit state with factors meeting none of the Bell relations. Nevertheless we claim that one of the Bell relations is an element of reality which we find by measuring w.r.t. to the Bell basis.

This inconsistency, however, is typical for our talking about quantum systems:

Even when we know the state Φ of a system (since its preparation is well specified), we may ask for the probability |⟨Ψ | Φ⟩|^{2} to find it in another state Ψ .

Concerning the Bell relation the situation is even less disturbing:

The 1-qubit states give no information about the actual relation between the partners of the individual pairs. Selection into subensembles corresponding to the 4 Bell relations has to be expected to change the partial 1-qubit states – even from a classical point of view.

In order to test for the Bell relations it seems necessary to get the qubits into contact — not necessarily into interaction (Resch et al., 2002; Hofmann and Takeuchi, 2002). This way they loose their identity – a natural reason for the change of the total (internal) state through measurement.

It seems that the Bell relations may be taken as elements of reality, but they can be applied to only one (freely chosen) set of 1-particle ‘properties’.^{1}

DRAFT, June 26, 2009

1More generally, see (Griffiths, 2002).

A.3 Quantum Phase Estimation and Order Finding

Exercise 25 Show that the following quantum network acts as indicated if Ψ̆_{φ} is a n-qubit eigenstate of the unitary Operator Û with eigenvalue e^{i φ} :

|0⟩ H  ⋯  ^{\frac{1}{\sqrt{2}} (|0⟩ + e^{i 2^{m}φ} |1⟩)}
⋮  ⋮  ⋮  ⋮  ⋮
|0⟩ H  ⋯  ^{\frac{1}{\sqrt{2}} (|0⟩ + e^{i 2^{1}φ} |1⟩)}
|0⟩ H  ⋯  ^{\frac{1}{\sqrt{2}} (|0⟩ + e^{i 2^{0}φ} |1⟩)}
⋯
Ψ̆_{φ} {  ⋮ Û^{2^{0}}  ⋮ Û^{2^{1}}  ⋮ Û^{2^{m}}  ⋮  } Ψ̆_{φ} .
⋯

First, let us consider the case

φ = 2π \frac{I(b)}{2^{m+1}} for some b ∈ \{0, 1\}^{m+1} .  (A.1)

Since

(A.1) ⟹ e^{i 2^{(m+1)−ν}φ} = exp (i 2π ∑_{μ=(m+1)−ν+1}^{m+1} b_{μ} 2^{(m+1)−ν−μ}) ,

(2.22) tells us that

(A.1) ⟹ |b⟩ = F̂^{−1} (2^{− \frac{m+1}{2}} ⨂_{ν=1}^{m+1} (|0⟩ + e^{i 2^{(m+1)−ν}φ} |1⟩)) .  (A.2)

Therefore, the phase φ considered in Exercise 25 can be determined by applying the inverse quantum Fourier transform to the state

\begin{aligned}Φ̆_{φ} & =^{def} & 2^{− \frac{m+1}{2}} ⨂_{ν=1}^{m+1} (|0⟩ + e^{i 2^{(m+1)−ν}φ} |1⟩) \\ & = & 2^{− \frac{m+1}{2}} ∑_{j=0}^{2^{m+1}−1} e^{i φ j} |j⟩_{m+1} \end{aligned}  (A.3)

of the first m + 1 qubits of the output produced by the described network and measuring the result — if (A.1) holds exactly and everything works perfectly.

170 APPENDIX A.

If φ ∈ [0, 2π) is not of the form (A.1), we have

F̂^{−1}Φ̆_{φ} =_{(A.3),(2.18)} \frac{1}{2^{m+1}} ∑_{j,k=0}^{2^{m+1}−1} e^{i φ j} e^{−i k \frac{2π}{2^{m+1}} j} |k⟩_{m+1}
=  \frac{1}{2^{m+1}} ∑_{j,k=0}^{2^{m+1}−1} (e^{i2π(\frac{φ}{2π} − \frac{k}{2^{m+1}} )})^{j} |k⟩_{m+1}
=  \frac{1}{2^{m+1}} ∑_{k=0}^{2^{m+1}−1} \frac{1 − e^{i2π(2^{m+1} \frac{φ}{2π} −k)}}{1 − e^{i2π(\frac{φ}{2π} − \frac{k}{2^{m+1}} )}} |k⟩_{m+1}
=  \frac{1}{2^{m+1}} ∑_{k=0}^{2^{m+1}−1} \frac{1 − e^{i 2^{m+1}φ}}{1 − e^{i2π(\frac{φ}{2π} − \frac{k}{2^{m+1}} )}} |k⟩_{m+1} .  (A.4)

Then (A.4) implies that the probability p(k) for finding |k⟩_{m+1} when testing F̂^{−1}Φ̆_{φ} fulfills the inequality

p(k) ≤ \frac{1}{2^{2m}} |1 − e^{i2π(\frac{φ}{2π} − \frac{k}{2^{m+1}} )}|^{−2} .  (A.5)

Let us define

D(k) =^{def} min_{ν∈ℤ} |2^{m+1} (\frac{φ}{2π} − ν) − k| ∀ k ∈ ℤ .

Then, given d ∈ {3, …, 2^{m} − 1}, the probability p_{d} for getting any state |k⟩_{m+1} with D(k) > d when testing F̂^{−1}Φ̆_{φ} fulfills the inequality

p_{d} ≤ \frac{1}{2 (d − 2)} .  (A.6)
Proof: Choosing k_{0} ∈ {0, …, 2^{m+1} − 1} such that

Δ =^{def} 2^{m+1} \frac{φ}{2π} − k_{0} ∈ (0, 1)

we get

p_{d} = ∑_{k∈{0,...,2^{m+1}−1}_{D(k)>d}} p(k)
≤_{(A.5)} \frac{1}{2^{2m}} ∑_{k∈\{0,...,2^{m+1}−1\}_{min_{ν∈ℤ}|k_{0}−k−ν 2^{m+1}|≥d}} |1 − e^{i2π \frac{Δ}{2^{m+1}}} e^{−i2π \frac{k−k_{0}}{2^{m+1}}} |^{−2}
= \frac{1}{2^{2m}} ∑_{j∈\{−2^{m}+1,...,2^{m}\}_{min_{ν∈ℤ}|j−ν 2^{m+1}|≥d}} |1 − e^{i2π \frac{Δ}{2^{m+1}}} e^{−i2π \frac{j}{2^{m+1}}} |^{−2}
≤ \frac{1}{2^{2m}} ∑_{j∈\{−2^{m}+1,...,2^{m}\}_{j∉\{−d+1,...,d−1\}}} |1 − e^{i2π \frac{Δ}{2^{m+1}}} e^{−i2π \frac{j}{2^{m+1}}} |^{−2} .

By (A.6), therefore^{2}

p_{d} ≤ \frac{1}{2^{2m}} ∑_{j∈\{−2^{m}+1,...,2^{m}\}_{j∉\{−d+1,...,d−1\}}} |\frac{2}{π} 2π \frac{Δ − j}{2^{m+1}} |^{−2}

= \frac{1}{4} ( ∑_{j=−2^{m}+1}^{−d} (Δ − j)^{−2} + ∑_{j=d}^{2^{m}} (Δ − j)^{−2})
≤ \frac{1}{4} ( ∑_{j=−2^{m}+1}^{−d} j^{−2} + ∑_{j=d}^{2^{m}} (1 − j)^{−2})
≤ \frac{1}{2} ∑_{j=d−1}^{2^{m}−1} j^{−2}
≤ \frac{1}{2} ∫_{d−2}^{∞} \frac{dx}{x^{2}}
= \frac{1}{2 (d − 2)} .

(A.6) tells us that, with probability ≥ 1 − (d − 2)^{−1}/2 , testing F̂^{−1}Φ̆_{φ} w.r.t. the computational (m + 1)-qubit base gives a state |b⟩ for which

min_{ν∈ℤ} |φ − 2π ν − 2π \frac{I(b)}{2^{m+1}} | ≤ 2π \frac{d}{2^{m+1}} .

Now, let x and N be arbitrarily given coprime positive integers. Then the order finding problem is to determine

r =^{def} min {r′ ∈ ℕ : x^{r′} = 1 mod N} ,

the order of x modulo N .

Defining

L =^{def} min {l ∈ ℕ : N ≤ 2^{l}}

and

[j mod N] =^{def} min {k ∈ ℤ_{+} : k = j mod N} ∀ j ∈ ℤ ,

2Note that

\frac{1}{2} |1 − e^{iθ}| = |sin \frac{θ}{2}| ≥ \frac{|θ|}{2\sqrt{2}} ≥ \frac{|θ|}{π} ∀ θ ∈ [−π, +π] ,

since

\frac{d}{dθ} (sin \frac{θ}{2} − \frac{θ}{2\sqrt{2}}) = \frac{1}{2} cos \frac{θ}{2} − \frac{1}{2\sqrt{2}} ≥ 0 ∀ θ ∈ [0, +π] .

DRAFT, June 26, 2009

we have^{3}

r = |{|[x^{j} mod N]⟩_{L} : j ∈ ℕ}|

and

⟨[x^{j} mod N] | [x^{k} mod N]⟩_{L} = δ_{jk} ∀ j, k ∈ {0, 1, …, r − 1} .

Therefore the states

Ψ_{s} =^{def} \frac{1}{\sqrt{r}} ∑_{j=0}^{r−1} e^{−i 2π \frac{s}{r} j} |[x^{j} mod N]⟩_{L} ∀ s ∈ \{0, …, r − 1\}  (A.7)

are normalized and, thanks to (2.17), fulfill the equation

\frac{1}{\sqrt{r}} ∑_{s=0}^{r−1} e^{+i 2π \frac{s}{r} j} Ψ_{s} = |[x^{j} mod N]⟩_{L} ∀ j ∈ \{0, …, r − 1\} .

Especially for j = 0 the latter gives

|1⟩_{L} = \frac{1}{\sqrt{r}} ∑_{s=0}^{r−1} Ψ_{s} .  (A.8)

We do not yet know the states Ψ_{s} explicitly since we do not yet know r. But we know that these states exist and have the nice property that they are eigenstates of the unitary^{4} L-qubit operator Û characterized by

Û |y⟩_{L} =^{def} \{ \begin{matrix}|[x y mod N]⟩_{L} & if y ∈ \{0, …, N − 1\} \\ |y⟩ & else \end{matrix} ∀y ∈ \{0, …, 2^{L} − 1\} .

More precisely, we have

Û Ψ_{s} = e^{i 2π \frac{s}{r}} Ψ_{s} ∀ s ∈ \{0, …, r − 1\} .

Therefore, replacing n by L and Ψ̆_{φ} by |1⟩_{L}, in Exercise 25 we get the total output state^{5}

\frac{1}{\sqrt{r}} ∑_{s=0}^{r−1} Φ̆_{2π \frac{s}{r}} ⊗ Ψ_{s} .

Since the Ψ_{s} form an orthonormal subset of the L-qubit state space, we may assume that the partial state of the system of the first m + 1 qubits is one of the vector states Φ̆_{2π \frac{s}{r}} with equal probability and the results concerning phase estimation show:

DRAFT, June 26, 2009

3Recall (1.12).

4Note that, because gcd(x, N) = 1 ,

y_{1} ≠ y_{2} mod N ⟹ x y_{1} ≠ x y_{2} mod N .

5Recall (A.3).

When F̂^{−1} Φ̆_{2π \frac{s}{r}} is tested w.r.t. the computational (m + 1)-qubit basis the probability for getting a state |k⟩_{m+1} with

|\frac{s}{r} − \frac{k}{2^{m+1}} | ≤ \frac{d}{2^{m+1}} ,

for d ≥ 3 , is not less than 1 − \frac{1}{2 (d − 2)} .

Choosing d and 2^{m+1}/d sufficiently large we obtain, this way, an excellent approximation k/2^{m+1} of s/r for some random s ∈ {0, …, r − 1}.

A.4 Finite-Dimensional Quantum Kinematics

A.4.1 General Description

• The state space of a finite-dimensional quantum system is a finite-dimensional complex Euclidean space H the inner product of which we denote by ⟨. | .⟩.

• The vectors ψ ∈ H with norm 1 correspond to pure states.^{6}

• In the interaction picture, used here, the state vectors ψ do not depend on time as long as the states are not disturbed by additional interaction (e.g. with some ‘measurement’ apparatus).

• The state vectors label equivalence classes of preparation procedures for ensembles of individual systems of the considered type.

• Preparation procedures are called equivalent if the ensembles they provide cannot be distinguished by the statistical outcome of measurements.

• Important measurements performable (in principle) on individual elements of an ensemble are projective measurements:

The individual drawn from an ensemble with state vector

ψ = ∑_{ν=1}^{n} ⟨ϕ_{ν} | ψ⟩ ϕ_{ν}

will be forced to a transition (if necessary) into one of the ϕ_{ν}- ensembles.^{7} According to the rules of quantum theory we have^{8}

|⟨ϕ_{ν} | ψ⟩|^{2} = probability for the transition ψ ↦ ϕ_{ν}

(A.9)

for all ν ∈ {1, …, n}.

DRAFT, June 26, 2009

6We assume that there are no superselection rules. 7I.e., an ensemble formed by a (sufficiently ) large number of individuals for which ϕ_{ν} is ‘mea-

sured’ actually corresponds (sufficiently well) to ϕ_{ν} , unless additional perturbations have appeared. 8Therefore, ⟨ϕ_{n}u | ψ⟩ is called probability amplitude for the transition ψ ↦ ϕ_{ν} .

• Observables are operators Â ∈ L(H) of the form

Â = ∑_{ν=1}^{n} \underbrace{a_{ν}}_{∈ℝ}|ϕ_{ν}⟩⟨ϕ_{ν}| , \{ϕ_{1}, …, ϕ_{n}\} an orthonormal basis of H , (A.10)

with the following interpretation:

In a state with state vector ϕ_{ν} the physical entity A (corresponding to Â) has the definite value a_{ν} .

This together with (A.9) implies:^{9}

trace (|ψ⟩⟨ψ| Â) = \{ \begin{matrix}expectation value for A \\ in a state with state vector ψ . \end{matrix}  (A.11)

• Individuals which are only known to be members of an ensemble with state vector ψ_{j} with probability λ_{j} for j ∈ {1, …, N} , ∑_{j=1}^{N} λ_{j} = 1 , form an ensemble to be described by the density matrix^{10}

ρ̂ = ∑_{j=1}^{N} \underbrace{λ_{j}}_{≥0} \underbrace{ψ_{j}}_{normalized} , trace (ρ̂) = 1 ,  (A.12)

in the sense that trace (ρ̂Â) = expectation value for Â.

(A.13)

• In this sense, the set of all states corresponds to

S(H) =^{def} {Â ∈ L(H) : Â ≥ 0 , trace (Â) = 1} .

States^{11} with ρ̂^{2} ≠ ρ̂ are called mixed.

DRAFT, June 26, 2009

9Note that

trace (|ψ⟩⟨ψ| Â) = ∑_{ν=1}^{n} a_{ν} |⟨ϕ_{ν} | ψ⟩|^{2} = ⟨ψ | Â ψ⟩ .

10Such ensembles arise, e.g., from projective measurements on pure states if the individuals are not selected according to the ‘measurement’ results.

11From now on we identify states with their density matrices. Note that

ρ̂ = |ψ⟩⟨ψ| for some ψ ∈ H  ⟺  ρ̂^{2} = ρ̂.

A.4. FINITE-DIMENSIONAL QUANTUM KINEMATICS 175

Lemma A.4.1 Let ρ̂_{1}, ρ̂_{2} ∈ S(H) . Then

0 ≤ trace (ρ̂_{1} ρ̂_{2}) ≤ 1

and

trace (ρ̂_{1} ρ̂_{2}) = 1 ⟺ ∃ ψ ∈ H : ρ̂_{1} = ρ̂_{2} = |ψ⟩⟨ψ| .

Outline of proof: Thanks to the spectral theorem there are an orthonormal basis {ϕ_{1}, …, ϕ_{n}} of H and λ_{1}, …, λ_{n} ≥ 0 with

ρ̂_{1} = ∑_{ν=1}^{n} λ_{ν} |ϕ_{ν}⟩⟨ϕ_{ν}|

and hence

\begin{aligned}trace (ρ̂_{1} ρ̂_{2}) & = & ∑_{ν,μ=1}^{n} λ_{ν} ⟨ϕ_{μ} | ϕ_{ν}⟩⟨ϕ_{ν} | ρ̂_{2} ϕ_{μ}⟩ \\ & = & ∑_{ν=1}^{n} λ_{ν} ⟨ϕ_{ν} | ρ̂_{2} ϕ_{ν}⟩ . \end{aligned}

Therefore, 0 ≤ trace (ρ̂_{1} ρ̂_{2}) ≤ 1 and

\begin{aligned}trace (ρ̂_{1} ρ̂_{2}) = 1 & ⟺ & (0 < λ_{ν} < 1 ⟹ ⟨ϕ_{ν} | ρ̂_{2} ϕ_{ν}⟩ = 1) ∀ μ ∈ \{1, …, n\} \\ & ⟺ & ∃ ν_{0} ∈ \{1, …, n\} : ρ̂_{1} = ρ̂_{2} = |ϕ_{ν_{0}}⟩⟨ϕ_{ν_{0}}| . \end{aligned}

Lemma A.4.2 Let^{12} ψ_{1}, …, ψ_{N} ∈ H. Then

(∑_{k=1}^{N} |ψ_{k}⟩⟨ψ_{k}|) H = span (⋃_{k=1}^{N} {ψ_{k}}) .  (A.14)

Outline of proof: Thanks to the spectral theorem there is an ONS {ϕ_{1}, …, ϕ_{n′}} ⊂ H with

∑_{k=1}^{N} |ψ_{k}⟩⟨ψ_{k}| = ∑_{ν=1}^{n′} λ_{ν} |ϕ_{ν}⟩⟨ϕ_{ν}|  (A.15)

for suitable λ_{1}, …, λ_{n′} > 0 and, consequently,

(∑_{k=1}^{N} |ψ_{k}⟩⟨ψ_{k}|) H = span ( ⋃_{k=1}^{N} {ϕ_{k}}) .  (A.16)

Then

∑_{k=1}^{N} |⟨χ | ψ_{k}⟩|^{2} = ∑_{k=1}^{N} ⟨χ | ψ_{k}⟩⟨ψ_{k} | χ⟩

=_{(A.15)} ∑_{ν=1}^{n′} λ_{ν} ⟨χ | ϕ_{ν}⟩⟨ϕ_{ν} | χ⟩

= ∑_{ν=1}^{n′} λ_{ν} |⟨χ | ϕ_{ν}⟩|^{2} ∀ χ ∈ H

DRAFT, June 26, 2009

12If not stated otherwise, by H , H_{1}, , H_{2} we denote arbitrarily given finite-dimensional complex Euclidean vector spaces.

176 APPENDIX A.

and, consequently,

χ ⊥ ψ_{k} ∀ k ∈ {1, …, N}  ⟺  χ ⊥ ϕ_{ν} ∀ ν ∈ {1, …, n} ,

i.e.

span ( ⋃_{k=1}^{N} {ψ_{k}}) = span ( ⋃_{ν=1}^{n′} {ϕ_{ν}}) .

The latter together with (A.16) implies (A.14).

Corollary A.4.3 Let ψ_{1}, …, ψ_{N}, ψ′_{1}, …, ψ′_{N} ∈ H. Then

∑_{k=1}^{N} |ψ_{k}⟩⟨ψ_{k}| = ∑_{k=1}^{N} |ψ′_{k}⟩⟨ψ′_{k}| .  (A.17)

holds iff there is a unitary N × N-matrix (U_{j}^{k}) with^{13}

ψ_{k} = ∑_{j=1}^{N} U_{k}^{j} ψ′_{j} ∀ k ∈ {1, …, N} .  (A.18)

Proof: Assume that (A.17) holds and, as in the proof of Lemma A.4.2, let us choose an orthonormal basis {ϕ_{1}, …, ϕ_{n}} ⊂ H and λ_{1}, …, λ_{n′} > 0 for which (A.15) holds. Then Lemma A.4.2 implies

span ( ⋃_{k=1}^{N} {ψ_{k}}) = span ( ⋃_{ν=1}^{n′} {ϕ_{ν}}) .

Especially, the ψ_{k}’s can be written as linear combinations

ψ_{k} = ∑_{ν=1}^{n′} c_{k}^{ν} ^{+} \sqrt{λ_{ν}} ϕ_{ν}

of ϕ_{ν}’s. Then we have

∑_{ν=1}^{n′} λ_{ν} |ϕ_{ν}⟩⟨ϕ_{ν}| =_{(A.15)} ∑_{k=1}^{N} ∑_{ν,μ=1}^{n′} (c_{k}^{μ})^{∗} c_{k}^{ν} λ_{ν} |ϕ_{ν}⟩⟨ϕ_{μ}| .

Since the |ϕ_{ν}⟩⟨ϕ_{μ}| form an ONS w.r.t. the Hilbert-Schmidt scalar product

⟨Â | B̂⟩ =^{def} trace (Â^{†}B̂) ∀ Â, B̂ ∈ S(H) ,  (A.19)

this implies

∑_{k=1}^{N} (c_{k}^{ν})^{∗} c_{k}^{μ} = δ_{νμ} ∀ μ, ν ∈ {1, …, n′} ,

DRAFT, June 26, 2009

13In general, of course, the (U_{j}^{k}) are not fixed by (A.18) i.e. the

c_{ν} = ( c_{1}^{ν} ; ⋮ ; c_{N}^{ν} ) , ν ∈ {1, …, n′} ,

form an orthonormal system in ℂ^{N} . Extending this to an orthonormal basis of ℂ^{N} we get a unitary N × N-matrix (c_{k}^{ν}) with

ψ_{k} = ∑_{ν=1}^{N} c_{k}^{ν} ^{+} \sqrt{λ_{ν}} ϕ_{ν} ∀ k ∈ \{1, …, N\} ,  (A.20)

where

λ_{ν} =^{def} 0 for ν > n′ .

Similarly, we get a unitary N × N-matrix (c′_{k}^{ν}) with

ψ′_{k} = ∑_{ν=1}^{N} c′_{k} ^{ν} ^{+} \sqrt{λ_{ν}} ϕ_{ν} ∀ k ∈ \{1, …, N\}

and hence

+ \sqrt{λ_{ν}}ϕ_{ν} = ∑_{l=1}^{N} (c′_{l}^{ν})^{∗} ψ′_{l} ∀ ν ∈ \{1, …, N\} .  (A.21)

Combining (A.20) with (A.21) we get (A.18) for the unitary matrix with components

U_{k}^{l} =^{def} ∑_{ν=1}^{N} c_{k}^{ν} (c′_{l}^{ν})^{∗} ∀ k, l ∈ {1, …, N} .

Conversely, (A.17) follows from (A.18) by straightforward calculation.

A.4.2 Qubits

Qubits are 2-dimensional quantum systems for which some orthonormal computational basis {|0⟩ , |1⟩} of their state space H is chosen. According to the standard convention

\begin{matrix}ψ = ( \begin{matrix}α \\ β \end{matrix} ) w.r.t. (|0⟩ , |1⟩) \\ ⟺^{def} ψ = α |0⟩ + β |1⟩ \end{matrix} \} ∀ α, β ∈ ℂ

we have

|0⟩ = ( \begin{matrix}1 \\ 0 \end{matrix} ) , |1⟩ = ( \begin{matrix}0 \\ 1 \end{matrix} ) .

All Â ∈ L(H) may be identified with their matrix ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ) w.r.t. the compu-

tational basis:

ψ = ( \begin{matrix}α \\ β \end{matrix} ) ⟹ Â ψ = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ) ( \begin{matrix}α \\ β \end{matrix} ) w.r.t. (|0⟩ , |1⟩) .

Here an orthonormal basis w.r.t. the scalar product (A.19) is \{τ̂^{0}/\sqrt{2}, …, τ̂^{3}/\sqrt{2}\}, where

τ̂^{0} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) τ̂^{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) τ̂^{2} = ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} ) τ̂^{3} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) (A.22)

are the well-known Pauli matrices. Therefore we have

Â = \frac{1}{2} ∑_{ν=0}^{3} trace (Âτ̂^{ν}) τ̂^{ν} ∀ Â ∈ L(H) ,

especially

ρ̂ = \frac{1}{2} (ˆ1 + ρ · τ̂) ∀ ρ̂ ∈ S(H) ,

where the vector^{14}

ρ =^{def} trace (ρ̂τ̂)

fulfills^{15}

|ρ| ≤ 1 , ρ̂^{2} = ρ̂ ⟺ |ρ| = 1 .

Note that the components of ρ are just expectation values of observables which are sufficient for quantum state tomography.

Remark: Since

(e_{ϑ,φ} · τ̂) χ_{ϑ,φ} = χ_{ϑ,φ}

holds for

e_{ϑ,φ} =^{def} ( \begin{matrix}sin ϑ cos φ \\ sin ϑ sin φ \\ cos ϑ \end{matrix} ) , χ_{ϑ,φ} =^{def} ( \begin{matrix}e^{−i \frac{φ}{2}} cos \frac{ϑ}{1} \\ e^{+i \frac{φ}{2}} sin \frac{ϑ}{1} \end{matrix} ) ,

every pure state corresponds to a definite spin orientation in the spin-\frac{1}{2} case — where \frac{ℏ}{2} τ̂ is the spin vector observable .

A.4.3 Bipartite Systems

For j = 1, 2 let {ϕ_{1}^{(j)}, …, ϕ_{n_{j}}^{(j)}} be an orthonormal basis of the state space H_{j} of a quantum mechanical system S_{j}. If S_{1} and S_{2} are distinguishable then the bipartite system S composed of these two has the state space H = H_{1} ⊗ H_{2} with orthonormal basis

{ϕ_{ν}^{(1)} ⊗ ϕ_{μ}^{(2)} : ν ∈ {1, …, n_{1}} , μ ∈ {1, …, n_{2}}} .

DRAFT, June 26, 2009

14Usually, the vector of coherence ρ is called Bloch vector when associated with electron spin and Stokes vector when associated with photon polarization. More generally see (Altafini, 2003).

15Note that

1 = trace (ρ̂) ≥ trace (ρ̂^{2}) = \frac{1}{2} (1 + |ρ|^{2}) .

Extending ⊗ to a bilinear mapping of H_{1} × H_{2} into H_{1} ⊗ H_{2} we get

L(H_{1} ⊗ H_{2}) = span (Â_{1} ⊗ Â_{2} : Â_{1} ∈ L(H_{1}) , Â_{2} ∈ L(H_{2})) ,

where the linear operators Â_{1} ⊗ Â_{2} are fixed by

(Â_{1} ⊗ Â_{2})(ψ^{(1)} ⊗ ψ^{(2)}) =^{def} (Â_{1} ψ^{(1)}) ⊗ (Â_{2} ψ^{(2)}) ∀ (ψ^{(1)}, ψ^{(2)}) ∈ H_{1} ⊗ H_{2} .

(A.23)

If Â_{1} resp. Â_{2} is the observable of S_{1} resp. S_{1} corresponding to the physical entity A_{1} resp A_{2} then Â_{1} ⊗ Â_{2} is the observable corresponding to the physical entity A_{1}A_{2} .

If one is only interested in the subsystem S_{1} of the total system S in state ρ̂ then it is sufficient to know its partial state^{16}

ρ̂_{1} =^{def} trace_{2}(ρ̂) =^{def} ∑_{μ=1}^{n_{2}} ⟨ϕ_{μ}^{(2)}| ρ̂|ϕ_{μ}^{(2)}⟩ ,  (A.24)

since:

trace (ρ̂(Â_{1} ⊗ ˆ1)) = trace (ρ̂_{1}Â_{1}) ∀ Â_{1} ∈ L(H_{1}) .

Note that

ρ̂ pure ≠⇒^{i.g.} ρ̂_{1} pure .

Example: If S_{1} and S_{2} are qubit systems then for the the Bell state

Ψ^{−} =^{def} \frac{1}{\sqrt{2}} (( \begin{matrix}1 \\ 0 \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ) − ( \begin{matrix}0 \\ 1 \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ))

we have

trace_{2}( |Ψ^{−}⟩⟨Ψ^{−}|) = \frac{1}{2} ˆ1 ,

i.e. the partial states give no information at all:

trace (|Ψ^{−}⟩⟨Ψ^{−}| (|ψ⟩⟨ψ|) ⊗ ˆ1) = \frac{1}{2} ∥ψ∥^{2} ∀ ψ ∈ ℂ^{2} .

But there are strong (non-classical) correlations between the subsystems, since

Ψ^{−} = \frac{1}{\sqrt{2}} (ψ ⊗ ψ_{⊥} − ψ_{⊥} ⊗ ψ)

DRAFT, June 26, 2009

16The so-called partial trace trace_{2} w.r.t. the second factor is the linear mapping of L(H_{1}⊗H_{2}) into L(H_{1}) characterized by

trace_{2}( |ψ_{1} ⊗ ψ_{2}⟩⟨ψ′_{1} ⊗ ψ′_{2}|) = \underbrace{⟨ψ′_{2} | ψ_{2}⟩}_{=trace (|ψ_{2}⟩⟨ψ′_{2}|)} |ψ_{1}⟩⟨ψ′_{1}| ∀ ψ_{1}, ψ′_{1} ∈ H_{1} , ψ_{2}, ψ′_{2} ∈ H_{2} .

The partial trace w.r.t. the first factor, written trace_{1} , is defined similarly.

for all normalized ψ ∈ ℂ^{2} , where

( \begin{matrix}α \\ β \end{matrix} )_{⊥} =^{def} ( \begin{matrix}−β^{∗} \\ α^{∗} \end{matrix} ) for all α, β ∈ ℂ ,

and hence

trace (|Ψ^{−}⟩⟨Ψ^{−}| (|ψ⟩⟨ψ|) ⊗ (|ϕ⟩⟨ϕ|)) = \frac{1}{2} |⟨ϕ | ψ_{⊥}⟩|^{2}

holds for all normalized ψ, ϕ ∈ ℂ^{2} .

Definition A.4.4 A state ρ̂ of the bipartite system S with state space H_{1} ⊗ H_{2} is called separable iff there is a sequence \left\{{ \atop ρ̂_{N}}\right\}_{N∈ℕ} of states of the form

ρ̂_{N} = ∑_{ν=1}^{N} \underbrace{λ_{ν}}_{≥0} \underbrace{ρ̂_{ν}^{(1)}}_{∈S(H_{1})} ⊗ \underbrace{ρ̂_{ν}^{(2)}}_{∈S(H_{2})}  (A.25)

with^{17}

lim_{N→∞} ∥ρ̂ − ρ̂_{N}∥_{1} = 0 .

Theorem A.4.5 (Horodecki) Let S be a bipartite system with state space H_{1} ⊗ H_{2} . Then the separable states of S are exactly those of the form (A.25) with

N ≤ (dim(H_{1} ⊗ H_{2}))^{2} .

Proof: See (Horodecki, 1997).

Lemma A.4.6 Let Ψ be a normalized vector in H_{1} ⊗ H_{2} . Then ρ̂ = |Ψ⟩⟨Ψ| is separable iff

Ψ = ϕ^{(1)} ⊗ ϕ^{(2)}  (A.26)

holds for some normed ϕ^{(1)} ∈ H_{1} and ϕ^{(2)} ∈ H_{2} .

Outline of proof: Let ρ̂ = ρ̂^{2} be separable, hence of the form

ρ̂ = ∑_{ν=1}^{N} \underbrace{λ_{ν}}_{>0} \underbrace{ρ̂_{ν}^{(1)}}_{∈S(H)} ⊗ \underbrace{ρ̂_{ν}^{(2)}}_{∈S(H)} , ∑_{ν=1}^{N} λ_{ν} = 1 .

DRAFT, June 26, 2009

17Recall (5.31) and Footnote 22 of Chapter 5.

Then, by Lemma A.4.1,

1 = trace (ρ̂^{2})
= ∑_{ν,μ=1}^{N} λ_{ν} λ_{μ} \underbrace{trace_{H_{1}}(ρ̂_{ν}^{(1)} ρ̂_{μ}^{(1)})}_{∈[0,1]} \underbrace{trace_{H_{2}}(ρ̂_{ν}^{(2)} ρ̂_{μ}^{(2)})}_{∈[0,1]}

and, therefore,

trace_{H_{1}}(ρ̂_{ν}^{(1)} ρ̂_{μ}^{(1)}) = trace_{H_{2}}(ρ̂_{ν}^{(2)} ρ̂_{μ}^{(2)}) = 1 ∀ ν, μ ∈ {1, …, N} .

By by Lemma A.4.1, again, the latter is equivalent to the existence of normed ϕ^{(1)} ∈ H_{1} and ϕ^{(2)} ∈ H_{2} with

ρ̂_{ν}^{(j)} = |ϕ^{(j)}⟩⟨ϕ^{(j)}| ∀ ν ∈ {1, …, N} , j ∈ {1, 2}

and hence (A.26).

Conversely, (A.26) together with

ρ̂^{(j)} =^{def} |ϕ^{(j)}⟩⟨ϕ^{(j)}| ∀ j ∈ {1, 2}

implies

|Ψ⟩⟨Ψ| = ρ̂^{(1)} ⊗ ρ̂^{(2)}

and hence separability of |Ψ⟩⟨Ψ|.

Theorem A.4.7 Every vector state on H_{1} ⊗ H_{2} allows a Schmidt decomposition,^{18} i.e. for every Ψ ∈ H_{1} ⊗ H_{2} there are a unique Schmidt number n′ ∈ ℕ and orthonormal subsets {ϕ_{1}^{(1)}, …, ϕ_{n′}^{(1)}} resp. {ϕ_{1}^{(2)}, …, ϕ_{n′}^{(2)}} of H_{1} resp. H_{2} with

Ψ = ∑_{ν=1}^{n′} \underbrace{s_{ν}}_{>0} ϕ_{ν}^{(1)} ⊗ ϕ_{ν}^{(2)}  (A.27)

for suitable Schmidt coefficients s_{1}, …, s_{n′}. If the eigenvalues of the partial state trace_{2}(|Ψ⟩⟨Ψ|) are non-degenerated then the Schmidt decomposition of Ψ is unique.

Outline of proof: Consider any Ψ ∈ H_{1} ⊗ H_{2} . Thanks to the spectral theorem there are an orthonormal basis {ϕ_{1}^{(1)}, …, ϕ_{n_{1}}^{(1)}} of H_{1} , a positive integer n′ ≤ n_{1} , and s_{1}, …, s_{n′} > 0 with

trace_{2}(|Ψ⟩⟨Ψ|) = ∑_{ν=1}^{n_{1}} (s_{ν})^{2} |ϕ_{ν}^{(1)}⟩⟨ϕ_{ν}^{(1)}| ,  s_{ν} =^{def} 0 ∀ ν > n′ .  (A.28)

Then there are ψ_{1}, …, ψ_{n_{1}} ∈ H_{2} with

Ψ = ∑_{ν=1}^{n_{1}} ϕ_{ν}^{(1)} ⊗ ψ_{ν}  (A.29)

DRAFT, June 26, 2009

18For the generalization to n-partite systems see (Carteret et al., 2000) and, therefore,

trace_{2}(|Ψ⟩⟨Ψ|) = ∑_{ν,μ=1}^{n_{1}} ⟨ψ_{μ} | ψ_{ν}⟩ |ϕ_{ν}^{(1)}⟩⟨ϕ_{μ}^{(1)}| .

The latter together with (A.28) implies

⟨ψ_{μ} | ψ_{ν}⟩ = (s_{ν})^{2} δ_{νμ} ∀ ν, μ ∈ {1, …, n_{1}} .

Thus, with

ϕ_{ν}^{(2)} =^{def} \frac{ψ_{ν}}{s_{ν}} ∀ ν ∈ \{1, …, n′\} ,

(A.29) becomes equivalent to (A.27). Since, conversely, (A.27) implies (A.28) the stated uniqueness properties are obvious.

Remarks:

1. Note that the Bell states (4.55) have Schmidt number 2 and hence, by Lemma A.4.6, be separable.

2. The vector state

\frac{1}{2}(|0, 0⟩ + |0, 1⟩ + |1, 0⟩ + |1, 1⟩) ,

however, is separable since equal to (Ĥ |0⟩) ⊗ (Ĥ |0⟩), where

Ĥ |0⟩ =^{def} \frac{1}{\sqrt{2}}(|0⟩ + |1⟩) , Ĥ |1⟩ =^{def} \frac{1}{\sqrt{2}}(|0⟩ − |1⟩)

characterizes the unitary Hadamard operator, strongly used in quantum computing.

3. For mixed states ρ̂ ∈ S(H_{1} ⊗ H_{2}) there is a Schmidt-like decomposition of the Form

ρ̂ = ∑_{ν=1}^{n′} \underbrace{s_{ν}}_{>0} Â_{ν}^{(1)} ⊗ Â_{ν}^{(2)}

with n′ ≤ (dim H_{1})^{2}, (dim H_{2})^{2} . According to (Herbut, 2002, Corollary 1) the operators Â_{ν}^{(j)} may be chosen Hermitian and such that

ν ≠ μ ⟹ trace (Â_{ν}^{(j)} Â_{μ}^{(j)}) = 0 .

However, in general, they cannot be positive.

Pure states are non-separable iff their partial states are mixed.^{19} Therefore, the correlations in non-separable pure states are non-classical.^{20} Obviously, the partial transpose

\begin{matrix}(1 ⊗ 𝔗̄) & ( ∑_{ν_{1},μ_{1}=1}^{n_{1}} ∑_{ν_{2},μ_{2}=1}^{n_{2}} λ_{ν_{1}μ_{1}ν_{2}μ_{2}} |ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{ν_{2}}^{(2)}⟩⟨ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{μ_{2}}^{(2)}|) \\ =_{(4.13)} ∑_{ν_{2},μ_{2}=1}^{n_{2}} λ_{ν_{1}μ_{1}ν_{2}μ_{2}} |ϕ_{ν_{1}}^{(1)} ⊗ ϕ_{μ_{2}}^{(2)}⟩⟨ϕ_{μ_{1}}^{(1)} ⊗ ϕ_{ν_{2}}^{(2)}| \end{matrix}

w.r.t. the orthonormal basis {ϕ_{1}^{(2)}, …, ϕ_{n_{2}}^{(2)}} of H_{2} must be positive for separable states:

(1 ⊗ 𝔗)(∑_{ν=1}^{N} λ_{ν} ρ̂_{ν}^{(1)} ⊗ ρ̂_{ν}^{(2)})^{T_{2}} = ∑_{ν=1}^{N} λ_{ν} ρ̂_{ν}^{(1)} ⊗ 𝔗(ρ̂_{ν}^{(2)}) ≥ 0 .

Therefore, the mixed Werner states Ŵ_{λ} with λ > 1/3 , considered at the end of Section 4.2.1, are non-separable.^{21}

Lemma A.4.8 For every state ρ̂ ∈ H there is a purification in H ⊗ H, i.e. a normalized vector Ψ ∈ H ⊗ H with

ρ̂ = trace_{2}(|Ψ⟩⟨Ψ|) .  (A.30)

Outline of proof: Thanks to the spectral theorem there are an orthonormal basis {ϕ_{1}^{(1)}, …, ϕ_{n}^{(1)}} of H and λ_{1}, …, λ_{n} ≥ 0 with

ρ̂ = ∑_{ν=1}^{n} λ_{ν} |ϕ_{ν}^{(1)}⟩⟨ϕ_{ν}^{(1)}| , ∑_{ν=1}^{n} λ_{ν} = 1 .  (A.31)

For the normalized vector^{22}

Ψ =^{def} ∑_{ν=1}^{n} ^{+} \sqrt{λ_{ν}} ϕ_{ν}^{(1)} ⊗ ϕ_{ν}^{(1)}

then, we get (A.30).

DRAFT, June 26, 2009

19Usually, non-separable states are called entangled; see (Verstraete and Cirac, 2003), however. 20In a way, also the mixed non-separable states are non-classically correlated (Werner, 1989). 21A decomposition of the 2-qubit Werner states, taking the form (A.25) for λ ≤ 1/3 , is

presented on page 6 of: http://www.physik.uni-augsburg.de/weh-school/bruss.pdf

22For the set of all suitable Ψ see (Kuah and Sudarshan, 2003, Lemma 1).

184 APPENDIX A.

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Index

adder, 16

balanced, 22 Bell

measurement, 25, 26 network, 24, 26 states, 24

Bernstein-Vazirani

oracle, 24 binary digit, 11 bit, 11

qu-, 18 Bloch

vector, 178

Campbell-Hausdorff formula, 66 circuit

classical logic, 12, 15 computational, 11

equivalent, 11 graph, 12 logic, 14

classical

computation, 15, 17, 22

network, 18 logic circuit, 12, 15 reversible computation, 15, 16 reversible network, 15

closed quantum system, 105 code

Calderbank-Shor-Steane, 116 linear classical

dual, 111 quantum, 112

words, 112 stabilizer, 115 Steane, 118

word, 110 coding

quantum dense, 24 coherence

vector of, 178 coherent superposition, 18, 23 complexity

computational, 31 computation

-al basis, 18–20 classical, 15, 17, 22 reversible, 15, 16

complexity, 31 concurrence, 165 CSS codes, 116

decoherence, 106 dense coding, 24 Deutsch-Jozsa oracle, 22, 23 problem, 22 dual code, 111

ebit, 164 entangled, 24

pure state, 24, 26, 105 entanglemen

catalysis, 163 entanglement

of formation, 165 swapping, 27 witness, 149, 153 entropic inequalities, 157 entropy

Renỳi, 157 equivalent

classical network, 11 preparation procedures, 173 quantum network, 21

error correction, 18 Euler angles, 30

fast Fourier, 46 flip

operator, 102 Fourier transform

fast, 46 Fredkin gate, 14 function

balanced, 22 decision, 14

gate, 11 Fredkin, 14 non-deterministic, 19 quantum, 19, 20

universal, 27 reversible, 13 Toffoli, 14 universal, 15, 16

graph, 12

halting problem, 11, 167 Hamming

distance, 110

interaction picture, 173

LOCC, 158 logic circuit, 14

measurement, 18, 19 Bell, 25, 26 projective, 173

mixture, 106

network adder, 16 Bell, 24, 26 classical

2-bit, 17 computational, 18 reversible, 15 optimization, 16 quantum

equivalent, 21 quantum computational, 18 reversible, 15 teleportation, 25

open quantum system, 106 operator

positive

support, 138 oracle

Bernstein-Vazirani, 24 Deutsch-Jozsa, 22, 23

order of an integer modulo N, 171

Partial transpose, 154 Pauli

group, 113 Pauli matrices, 106 polarization identity, 149 post selection, 26 problem

Deutsch-Jozsa, 22 halting, 167

projective measurement, 173

quantum closed system, 105 code, 112 code words, 112 computation

network, 18

result, 19 dense coding, 24 gate, 19, 20 measurement, 18 network

equivalent, 21 open system, 106 parallelism, 18 state

entangled, 24, 26, 105 tomography, 178 state collapse, 18 teleportation, 25, 26 wire, 19

qubit, 18

Range criterion, 156 register, 11

n-qubit, 19 state space, 18

Renỳi entropies, 157 result

of a quantum computation, 19 reversible

classical network, 15

Sleator-Weinfurter construction,

31 stabilizer, 115

codes, 115 state

Bell, 24 mixed, 106 pure, 173 quantum

entangled, 24, 26, 105 separable, 153, 180 Werner, 153

state space, 173 Steane code, 118 Stokes

vector, 178 superoperator, 98 superposition

coherent, 23

teleportation, 26 network, 25 of entanglement, 27

tensor product

formalism of quantum mechanics,

19 Toffoli gate, 14 tomography

quantum state, 178 trace norm, 139 partial, 179 transposition, 102, 154 partial, 154

universal gate, 15, 16 quantum gate, 27

Werner states, 153 wire, 15

quantum, 19