carolina thesis(1)

8/13/2019 Carolina Thesis(1)

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Detection of quantum entanglement

in physical systems

Carolina Moura Alves

Merton College

University of Oxford

A thesis submitted for the degree of

Doctor of Philosophy

Trinity 2005


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Abstract

Quantum entanglement is a fundamental concept both in quantum mechanics and in

quantum information science. It encapsulates the shift in paradigm, for the descrip-

tion of the physical reality, brought by quantum physics. It has therefore been a key

element in the debates surrounding the foundations of quantum theory. Entangle-

ment is also a physical resource of great practical importance, instrumental in the

computational advantages offered by quantum information processors. However, the

properties of entanglement are still to be completely understood. In particular, the

development of methods to efficiently identify entangled states, both theoretically

and experimentally, has proved to be very challenging. This dissertation addresses

this topic by investigating the detection of entanglement in physical systems.

Multipartite interferometry is used as a tool to directly estimate nonlinear properties

of quantum states. A quantum network where a qubit undergoes single-particle

interferometry and acts as a control on a swap operation between k copies of the

quantum state is presented. This network is then extended to a more general

quantum information scenario, known as LOCC. This scenario considers two distant

parties A and B that share several copies of a given bipartite quantum state.

The construction of entanglement criteria based on nonlinear properties of quantumstates is investigated. A method to implement these criteria in a simple, experimen-

tally feasible way is presented. The method is based of particle statistics effects

and its extension to the detection of multipartite entanglement is analyzed. Finally,

the experimental realization of the nonlinear entanglement test in photonic systems

is investigated. The realistic experimental scenario where the source of entangled

photons is imperfect is analyzed.


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Acknowledgements


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Contents

Abstract i

Acknowledgements ii

1 Introduction 1

1.1 Entanglement as a property of quantum systems . . . . . . . . . . . . . . . . . . 1

1.2 Entanglement as a physical resource . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Detection and characterization of entanglement . . . . . . . . . . . . . . . . . . . 2

1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Basic concepts 52.1 State Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Mathematical properties of density operators . . . . . . . . . . . . . . . . 8

2.2.2 Ensemble interpretation of density operators . . . . . . . . . . . . . . . . 9

2.3 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Superoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Mathematical properties of superoperators . . . . . . . . . . . . . . . . . 10

2.4.2 Jamiolkowski isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Mathematical characterization of bipartite entanglement . . . . . . . . . . . . . . 11

2.5.1 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Experimental detection of entanglement . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.1 Bells inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.2 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 Multipartite entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7.1 Maximally entangled state . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7.2 W State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7.3 Cluster state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Quantum networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8.1 Universal set of gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8.2 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

iii


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CONTENTS iv

3 Direct estimation of density operators 21

3.1 Modified interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Multiple target states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Spectrum estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Quantum communication . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Extremal eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.4 State estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.5 Arbitrary observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Quantum channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Direct estimation of density operators using LOCC 28

4.1 LOCC estimation of nonlinear functionals . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Structural Physical Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 SPA using only LOCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Entanglement detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Channel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Entanglement Detection in Bosons 33

5.1 Nonlinear entanglement inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Estimation of the purities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 Bipartite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.2 Multipartite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Realization of the entanglement detection network . . . . . . . . . . . . . . . . . 36

5.4 Detection of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.5 Degree of macroscopicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5.1 Determination of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Entropic inequalities 41

6.1 Entropic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.1.1 Graphical comparison between Bell-CHSH and entropic inequalities . . . 42

6.2 Experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2.1 Realistic sources of entangled photons . . . . . . . . . . . . . . . . . . . . 46

7 Conclusion 49

Bibliography 51


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List of Figures

2.1 The controlled-Ugate. The top line represents the control qubit and the bottom

line represents the target qubit. U acts on the target qubit iff the control qubit

is in the logical state|1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 The Mach-Zender interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 The quantum network corresponding to the Mach-Zender interferometer ( =

1 0). The visibility of the interference pattern associated withp0 varies as afunction of according to Eq.(2.70). . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 A modified Mach-Zender interferometer with coupling to an ancilla by a controlled-

Ugate. The interference pattern is modified by the factor vei = Tr [U ]. . . . 22

3.2 Quantum network for direct estimations of both linear and non-linear functionsof a quantum state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 A quantum channel acting on one of the subsystems of a bipartite maxi-

mally entangled state of the form|+ =

k |k|k/

d. The output state

= 1d

kl |kl| (|kl|), contains a complete information about the channel. 26

4.1 Network for remote estimation of non-linear functionals of bipartite density op-

erators. Since Tr[V(k)k] is real, Alice and Bob can omit their respective phaseshifters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Network of BS acting on pairs of identical bosons. The two rows of N atoms,

labelled I and IIrespectively, are identical, and the state of each of the rows is

123...N. The total state of the system is 123...N 123...N. . . . . . . . . . . . . . 345.2 In Fig. 4.2(a), we plot the violationVof the inequalities Eq. (5.2),V1= Tr(

2123)

Tr(212) (dashed),V2= Tr(212) Tr(21) (grey) and V3= Tr(212) Tr(22) (solid),

as a function of the phase , forN= 3 atoms. Whenever V >0, entanglement is

detected by our network. In Fig. 4.2(b) we plot different purities associated with

a cluster state of size N, as a function of . B is any one atom not at an end

(dotted), any two atoms not at ends and with at least two others between them

(dashed), any two or more consecutive atoms not including an end (dash-dotted),

any one or more consecutive atoms including one end (solid). The plotted purities

are independent ofN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

v


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LIST OF FIGURES vi

5.3 Plot of the purity Nm for m = 1 (solid black), m= 7 (dashed black), m = 14(solid grey) andm = 20 (dashed grey), as a function of, for N = 300 atoms. . . 40

6.1 A graphical comparison of the Bell-CHSH inequalities with the entropic inequali-

ties (6.2). All points inside the ball satisfy the entropic inequalities and all pointswithin the Steinmetz solid satisfy all possible Bell-CHSH inequalities. NB not all

the points in the outlining cube represent quantum states. . . . . . . . . . . . . . 43

6.2 In a special case of locally depolarized states, represented by points within the

tetrahedron, the set of separable states can be characterized exactly as an octahe-

dron. All states in the ball but not in the octahedron are entangled states which

are not detectable by the entropic inequalities. . . . . . . . . . . . . . . . . . . . 44

6.3 An outline of our experimental set-up which allows to test for the violation of the

entropic inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.4 Possible emissions leading to four-photons coincidences. The central diagram

shows the desired emission of two independent entangled pairs one by sourceS1 and one by source S2. The top and the bottom diagrams show unwelcome

emissions of four photons by one of the two sources. . . . . . . . . . . . . . . . . 47


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CHAPTER1

Introduction

The subject of this dissertation is the detection of quantum entanglement in physical systems.

Quantum entanglement was singled out by Erwin Schrodinger as ...thecharacteristic trait of

quantum mechanics, the one that enforces its entire departure from classical lines of thought. [1].

Indeed, after playing a significant role in the development of the foundations of quantum me-

chanics [1, 2, 3], quantum entanglement has been recently rediscovered as a physical resource in

the context of quantum information science [4, 5, 6, 7]. This set of correlations, to which a clas-

sical counterpart does not exist, arises from the interaction between distinct quantum systems.

Entanglement is instrumental in the improvements of classical computation and classical com-

munication results, of which two particularly important examples are the exponential speedup

of certain classes of algorithms [8, 9] and physically secure cryptographic protocols [4].

1.1 Entanglement as a property of quantum systems

Entanglement was first used by Einstein, Podolski and Rosen (EPR) [2] to illustrate the con-

ceptual differences between quantum and classical physics. In their seminal paper published

in 1935, EPR argued that quantum mechanics is not a completetheory of Nature, i.e. it does

not include a full description of the physical reality, by presenting an example of an entangled

quantum state to which it was not possible to ascribe definite elements of reality. EPR defined

an element of reality as a physical property, the value of which can be predicted with certainty,

before the actual property measurement. This condition is straightforwardly obeyed in the con-text of classical physics, but not in the context of quantum mechanics. The predictive power

of quantum mechanics is limited to, given a quantum state and an observable, the probabilities

of the different measurement outcomes. This feature led EPR to deem quantum mechanics as

incomplete. The incompleteness of quantum mechanics, as understood by EPR, was to plague

physicists for decades.

On one hand the quantum mechanical formalism explained the behaviour of microscopical

systems to a great degree of accuracy. On the other hand, it was conceptually unsatisfactory

as a fundamental theory of Nature and the EPR argument seemed a valid one. It was not

until John Bell published his seminal paper in 1964 [3], where he discussed the validity of the

EPR assumptions, that light was shed into the matter. In his paper Bell does not make any

assumption about quantum mechanics. It does, however, assume that our classical common senseview of the world is true. He considered a thought experiment where two causally disconnected

1


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CHAPTER 1. INTRODUCTION 2

observers share many identical pairs of physical systems and are allowed to perform two different

types of measurements on their respective systems. The measurements performed in each pair

are chosen at random and correspond to elements of reality. The expectation values of these

observables depend of the probability associated with a given outcome and the actual value of

the outcome. Bell then derived a set of inequalities that bound the expectation value of a linearcombination of the observables. It turns out that certain entangled states theoretically violate

these inequalities, which means that either quantum mechanics is an incomplete description of

Nature or the EPR assumptions are incorrect. The only way to decide which is the case was

by performing an experimental test of Bells inequalities. This test was realized with entangled

pairs of photons in 1982 [10] and it shown the violation of the Bells inequalities, as predicted

by quantum mechanics. This type of experimental test has subsequently been used to detect

entanglement experimentally in physical systems [11].

1.2 Entanglement as a physical resource

Fundamental quantum effects, such as quantum tunnelling or stimulated emission, have yieldedover the last century important technological breakthroughs, of which semiconductors or lasers

are two examples. Entanglement too has proved to be a physical resource capable of revolution-

izing the theories of computation and information. Within quantum information science, the

logical unit of information is the qubit, a two-level quantum system. The qubit differs from the

bit in that is can be any superposition of 0 and 1. In particular, a set of qubits can be in

an entangled state. The possibility of exploiting these quantum correlations between qubits, for

realizing computations faster than it would be possible classically, was first realized by Deutsch

in 1985 [12]. The development of quantum algorithms that ensued culminated with a result by

Shor for the efficient factoring the primes of a number [8]. The best classical algorithms for

this task scale exponentially with the size of the number to be factored, which means that it iseffectively impossible to factor large numbers. However, Shors algorithm can factor the primes

in a time that scales polynomially with the number size, i.e. efficiently. This result is particularly

relevant since the security of currently used cryptographic protocols is based on the difficulty of

factoring large numbers. Therefore a quantum factoring machine would render these protocols

useless.

Ironically, entanglement turns out to be the key resource in one of the possible solutions

to the security of cryptographic protocols. This solution, proposed by Ekert in 1991 [4], uses

entangled states as the carrier of protected information. The security of the protocol comes from

the fact that any attempt to gain access to the encrypted information, via a measurement on the

state, will necessarily disturb the quantum correlations. As mentioned earlier, the amount of

entanglement in a given state can be measured by checking for the violation of Bells inequalities.Therefore, any tampering of the carriers of information can be detected and the protocol aborted.

1.3 Detection and characterization of entanglement

We have seen how entanglement is not only a key concept in quantum mechanics, but also

a physical resource of great practical importance. It is therefore no wonder that it has been

extensively researched, both as a mathematical concept and as a property of physical systems.

In particular the experimental detection of entanglement is of paramount relevance for both

probing the limits of validity of quantum mechanics, as a physical theory, and for the monitoring

of quantum information processes. Its success is intimately related to the successful development

of theoretical tools that not only help us to further understand the properties of entanglement,


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but also provide practical experimental methods of detection.

There have been so far two different approaches to investigating the concept of entanglement.

One approach, the mathematical one, treats quantum states as mathematical objects and tries to

define entanglement as a mathematical property. It considers the density matrix representation

of quantum states and attempts to derive conditions that the matrices must obey in order torepresent an entangled state. This approach enabled the derivation of necessary and sufficient

conditions for entanglement in systems of two or three qubits. These results were obtained by

Peres [13] and the Horodeckis [14]. They pointed the way to a more general strategy of identifying

the mathematical properties of entanglement, based on the theory of positive maps. I will return

to this statement in more detail in the next chapter. However, a full characterization of the set

of entangled states for high-dimensional bipartite systems is yet to be found. In particular the

understanding of entanglement between more than two systems, multipartite entanglement, is

at present quite limited. Here, additional problems arise in the classification of entanglement,

since it is possible for states to exhibit multipartite entanglement while being separable with

respect to some of the subsystems. A general framework for the classification of entanglement

is yet to be developed and researchers have so far concentrated in studying specific classes ofmultipartite entangled states. I will present some examples of these classes in the next chapter

that we believe illustrate simultaneously the complexity of multipartite entanglement and its

great potential for quantum information processing.

The second approach to entanglement research, the physical one, treats quantum states as

properties of physical systems, that either exist in Nature or can be experimentally generated in

the laboratory. This approach differs fundamentally from the mathematical one in that it focuses

on the types of states actually generated in a given physical setting. The characterization or

detection of entanglement in this case is accomplished by via tests that are tailored for the specific

class of states considered. In the next chapter we will present the two most commonly used

experimental entanglement tests. Rather than aiming at a full characterization of entanglement,

this approach aims at developing techniques and methods for entanglement detection that are

experimentally accessible. In particular, it tries to identify which properties of a given quantum

system are relevant for entanglement detection. Providing a solution for this question will have

important consequences on the realization of experiments in quantum information processing,

since it will direct the experimentalists to a more efficient, and possibly easier, detection of

entanglement in the laboratory.

Despite all the effort devoted in recent years to the characterization of entanglement, the full

understanding of entanglements properties still eludes researchers. My doctoral research aimed

to contribute to our knowledge about entanglement by pursuing the physical approach. I have

developed new methods for not only the detection of both bipartite and multipartite entangle-

ment but also the characterization of certain properties of quantum states. These methods areexperimentally realistic and one of them was in particular realized experimentally.

1.4 Outline of thesis

When writing this thesis, I was faced with the difficult choice of which of my doctoral research

results to include. I decided to include the results that were not only the most directly relevant

to the subject of the dissertation, entanglement detection, but also the results that formed

the most chronologically coherent set. It will become apparent that these results were obtained

sequentially and that they are different instances of one research program. This program started

from a rather abstract setting of quantum networks, specifically designed to measure state

properties, and ended in the development of tailor-made experimental methods for the detection


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of entanglement in photons. However, I also pursued other research projects, such as the study

of the computational complexity of quantum languages [15], the development of methods to

generate classes of bound entangled states [16] and the investigation of methods to efficiently

generate graph states [17].

1.5 Chapter outline

I will now present the outline of remainder chapters of the thesis. Chapter 2 introduces the basic

concepts underlying the research results of the thesis. In particular it provides a mathematical

description of entanglement and discusses in more detail the general methods to detect and

characterize entanglement. Chapter 3 addresses the problem of estimating nonlinear functionals

Trk, k= 1, 2,... of a general density operator. The estimation method we proposed allows the

direct estimation of these nonlinear functionals. Our method uses an interferometric network

where a qubit undergoes single-particle interferometry and acts as a control on a swap operation

betweenkcopies of. Chapter 4 extends the above result to a more general quantum information

scenario, known as LOCC. In this scenario we consider two distant parties A andB that shareseveral copies of a given bipartite quantum state AB and are only allowed to perform local

operations and communicate classically. Chapter 5 investigates entanglement criteria based

on nonlinear functionals of that could be implemented in a simple, experimentally feasible

way. Our method is based of particle statistics effects and uses the fact that measuring the

purity of is tantamount to measuring the probability of projecting the state of two copies of

in its symmetric or antisymmetric subspaces. We extend of the nonlinear inequalities to the

detection of multipartite entanglement. Chapter 6 investigates the experimental realization of

the nonlinear entanglement test. We consider two copies of a polarization entangled pair of

photons AB. We also analyze the realistic experimental scenario where the source of entangled

photons is imperfect. Chapter 7 presents a conclusion to the thesis, with a summary of the mainresearch results presented.


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CHAPTER2

Basic concepts

2.1 State Vectors

Statistical predictions of quantum mechanics are based on two main concepts, quantum states

and quantum observables. With everyisolatedphysical system S, we associate a complex HilbertspaceHS of a suitable dimension, so that quantum states are represented by time-dependentunit vectors | (t) HS, and quantum observables by Hermitian operators acting in this space.Given a observable represented by the operatorA, there is a set of vectors{|i} such that

A

|i

=ai

|i

, ai

R. (2.1)

The vectors|i are called the eigenvectors of A, with respective eigenvalues ai. The set ofvalues{ai}is called the spectrum ofA. The time evolution of state vectors is unitary, i.e.

| (t) =U(t, t0) | (t0) , (2.2)whereU(t, t0) is a unitary operator, U U

= 11.Given a quantum system described by a state vector | and any observable A, represented

by a Hermitian operator, we can calculate all statistical properties ofA from the relation

A = | A | , (2.3)

whereA stands for the average value of A. In particular, when A is a projection operator,projecting on a one dimensional subspace spanned by vector |, A =||. In this caseA =|||2 represents the probability, for a system in state|, to pass a test for being inthe state|.

Quantum states can be equally well represented by projectors on the state vectors. Namely,

if instead of states|we consider the corresponding projectors ||, then the time evolutionof the state of the system will be given by

| (t) (t) | =U | (t0) (t0) | U, (2.4)and the average value of observable observable A will be written as

A = Tr A, (2.5)

5


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CHAPTER 2. BASIC CONCEPTS 6

where =|| and the trace Tr A stands for the sum of the diagonal elements ofA. Thetrace operation is linear, Tr (A+B) =Tr A+Tr B, and is basis-independent. The operator

is called density operator.

2.1.1 SubsystemsConsider a quantum systemScomposed of two subsystemsA andB. The Hilbert space asso-ciated with systemSis the tensor product of the Hilbert spaces of sub-systemAandB

HS= HA HB. (2.6)The dimension ofHS is dim HS= dim HA dim HB and any state|S of the system Scan beexpressed as a linear superposition of elements of the type |a|b, where |a HAand |b HB .Whenever convenient, well also write |a |b as |a|b or as |a, b. If we introduce orthonormalbases, i.e. maximal sets of vectors{|ak} inHA and{|bm} inHB, such thatak| al = kl,

bm

|bn

=mn, then any vector in

HScan be written as,

|S =k,l

ckl|ak|bl ,kl

|ckl|2 = 1. (2.7)

A particular subset of the states inHS can be written as a tensor product of state vectorsofHA andHB,

|S = |A |B =

k

k|ak

l

l|bl

(2.8)

= kl

ll

|ak

|bl

, (2.9)

where

k |k|2 =

l |l|2 = 1. This requires (comparing Eq.(2.8) and Eq.(2.7)) that

ckl = kl. (2.10)

The states for which this holds are called separable states. Note that this decomposition is

basis-independent. Thus, if |S is separable, we can associate state |A with the subsystem Aand state |B with the subsystem B. Otherwise we need to resort to density operators in orderto represent quantum states in subsystemsAandB.

2.2 Density OperatorsAny linear operator Sacting inHScan be written as a superposition of operators of the typeA B, whereA acts onHA andB acts onHB. We can choose operators bases,{Ak}acting onHA,{Bk} acting onHB, such that

S=k,l

SklAk Bl. (2.11)

The most common operator bases are formed from operators of the type |i j |. In our casewe have|ak al |, for operators acting onHA, and|bm bn |, for operators acting onHB (recallthat

|ai

and

|bj

are, respectively, orthonormal bases in

HA and

HB). This means that S can

be expressed as


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S=

k,l,m,n

Skmln | ak al | | bm bn | . (2.12)

Any operatorApertaining only to sub-system

Acan be trivially extended to systemSthrough 1

A A 11. (2.13)The average value of an observable S= A B acting onSis given by

S | S| S = S | (A B) | S (2.14)=

k,l,m,n

clnckm al | bn | (A B) | ak | bm

=

k,l,m,nclnckm(al | A | ak)(bn | B | bm).

In the special case of an observable pertaining to one of the subsystems, i.e. if either A = 11 or

B = 11, we obtain (we choose B = 11),

S|S|S =

k,l,m,n

clnckm(al | A | ak)(bn | 11 | bm),

=

k,l,m,n

clnckmal|A|aknm,

=

k,l,mclmckmal|A|ak,

=Tr

k,l,m

clmckm|akal|A, (2.15)

=TrAA, (2.16)

whereA=

k,l,m clmckm | ak am | is called the reduced density operator and is associatedonly

with sub-systemA. Recall that the density operator associated with the total system is

AB = |SS| =

k,l,m,n

ckmcln(|akal|) (|bmbn|) . (2.17)

Given AB, the density operator of a bipartite system, we obtain A, the reduced densityoperator of the subsystem A, by taking the partial trace over the subsystem B. Mathematicallythe partial trace operation

AB A, (2.18)is defined as

Tr B(A B) =ATr B. (2.19)Thus,

1The procedure for sub-system B is analogous.


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Tr B(AB) =

k,l,m,n

ckmcln | ak al | Tr| bm bn | (2.20)

=

k,l,m,nckmcln | ak al | mn

=k,l,m

ckmclm | ak al |

=A.

2.2.1 Mathematical properties of density operators

Density operators provide a description of quantum states. They can be defined as such without

any reference to state vectors. LetH be a finite-dimensional Hilbert space. A density operator, on

H, is a linear operator such that

is positive semi-definite, that is|| 0, for any| H. Tr= 1.

Any linear positive semi-definite operator X onH is always Hermitian, with non-negativeeigenvalues, and can be written as X=YY for some Y[18]. Many inequalities regarding pos-itive operators can be derived directly from|X| 0 by special choices of|. In particular,if| has only two non-zero components, labelled by i and j, then the submatrix of X withthe elements labelled by the indices i and j is also positive semi-definite. More generally, any

submatrix of a positive semi-definite matrix, obtained by keeping only the rows and columns

labelled by a subset of the original indices, is itself a positive semi-definite matrix and as suchmust have a nonnegative determinant (because all its eigenvalues are nonnegative).

To make a connection with the state vectors, let us consider a particular state (a pure state)

which can be described by a state vector| H. The density operator of any pure statecorresponds to a projection operator on that particular state, defined as

= ||, (2.21)which, like any projection operator, is idempotent:

2 =. (2.22)

For example, the state of a qubit|0 + |1 is described by the density operator= (|0 + |1) (1|+ 0|) = ||2 |00| + |01| + |10| + ||2 |11|, (2.23)

or, in the matrix form,

=

||2 ||2

. (2.24)

The diagonal elements 00 =||2 and 11 =||2 correspond, respectively, to the expectationvalues0||0 and1||1, giving the probabilities of observing bit values 0 and 1 respectively.


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2.2.2 Ensemble interpretation of density operators

Consider a quantum source which emits particles in states|1, |2...|n with a priori prob-abilities p1,p2...pn. We will write it as an ensemble{pi,|i} . In this case

S = ni=1

pii|S|i = ni=1

piTrS|ii| = TrS

ni=1

pi|ii|

= TrS. (2.25)

The result depends on the observable Sand on the quantum state, which appears in the expres-

sion above only as the combination

=n

i=1

pi|ii|. (2.26)

We call this operator the density operator that describes a mixture of pure states|1, |2...

|n

with weights p1,p2...pn. The operator is not a projector any more,

2

= , but it has

all the properties we require for density operators (self-adjoint, semi-positive, unit-trace). If werefer to a single particle, we are uncertain as to which particular pure state|i it is preparedin. However, it makes perfect sense to say that the particle is in the state . Please note that

many different mixtures may lead to the same density operator:

=n

i=1

pi|ii| =n

i=1

qi|ii|. (2.27)

Note the sets of pure states{|i, |},{|i, |} are not in general orthonormal. In fact, unlessthere is any degeneracy in the values pi, only one such set can be orthonormal.

Now take, for example, this particular density operator of a qubit:

=

34 00 14

. (2.28)

It can be viewed as the mixtures of|0 and|1 with the probabilities 34 and 14 , or as a mixtureof|1 =

32|0 + 12 |1 and|2 =

32|0 12 |1 with probabilities p1 = 12 and p2 = 12 . Even

though states|0and|1are clearly different from states |1and|2, according to Eq.(2.25),these mixtures behave identically under any any physical investigation, i.e. we are not able to

distinguish between different mixtures described by the same density operator.

2.3 EntanglementWe have previously introduced the concept of separable sates. However, there are states inHSwhich are not separable, i.e. they cannot be written as a simple tensor product of two states

|A and|B (states for which ckl= kl). These states are referred to as entangled states.Entanglement is a set of quantum correlations arising from the interaction between two or more

quantum systems that does not have a classical counterpart. An example of an entangled state

is the singlet state of two spin-half particles

| = 12

(||||) , (2.29)

where | and | denote respectively spin up and spin down with respect to a chosen quantizationaxis.


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As we mentioned in the previous chapter, entanglement is a very important physical resource

in quantum information science and both its mathematical characterization and experimental

detection have been subjected to extensive research. Unfortunately, the only general mathe-

matical definition of an entangled state is a negative one. A state is entangled if it cannot be

written as a convex sum of product states [19]

123...N =

C1 2 3 . . . N, (2.30)

where j is a state of subsystem j, and

C = 1. This fact means that in order to test

whether a given unknown state is entangled, we have in principle to check whether the state

can be decomposed in any of all the possible convex sums of product states. We will discuss

in a later section the most important results concerning the characterization and detection of

entanglement. But first, we will introduce the concept of superoperators, since they have proved

particularly relevant in the construction of entanglement criteria.

2.4 Superoperators

As we pointed out before, the time evolution of a state of systemS is unitary and obeysEq.(2.4). Suppose now thatS is composed of two sub-systems,A andB , and that we areinterested in the time evolution of sub-system A only. We can, without loss of generality, choosethe state ofA to be A and the state ofB to be the pure state|0. The time evolution of thestate of systemS, A |00|, is given by

= U A |00|U, (2.31)which is still a density operator describing systemS. The time evolution of the state of sub-systemA is then obtained by performing the partial trace, on sub-systemB, of the state ofsystemS:

A= Tr B(U A |00|U). (2.32)If we now consider an orthonormal basis |i,i = 0, 1,..., for sub-system B, Eq.(2.32) becomes

A=i

i|U|0A0|U|i i

EiAEi , (2.33)

whereEi=

i

|U

|0

are operators, acting on sub-system

A, and are trace-preserving:

i

Ei Ei=i

0|U|ii|U|0 = 0|UU|0 =11. (2.34)

Eq.(2.33) defines a linear mapL that takes linear operatorsA to linear operators A. Such

a map, if the property in Eq.(2.34) is satisfied, is called a superoperator. The representation of

the superoperator given in Eq.(2.33) is called the operator-sum representation.

2.4.1 Mathematical properties of superoperators

A superoperator L : that takes density operators to density operators has the followingproperties [18]:

L is trace-preserving, that is Tr= TrL() = 1.


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L is linear, that is L(11+ 22) =1L(1) + 2L(2), 1+ 2= 1. L is a is completely positive map, that is, if is positive, then = L() is positive and

the extension ofL to a larger sub-system (11 L) is also positive.

All the mathematical properties originate from physical requirements. The first and thirdproperties originate from the requirement that, assuming to be a density operator, will alsobe a density operator. The second property originates from our desire to reconcile the density

operator time evolution and its ensemble interpretation.

The first property of superoperators is quite straightforward to accept, since any density

operator has, by definition, trace equal to one. The third property is perhaps less obvious.

Clearly, L must be a positive map to assure that will be a positive operator (necessarycondition for to be a density operator). But why must L be completely positive? The answeris: in order to assure that, if we decide to consider the action of the superoperator on an extended

system,ext , the resulting operator = ext L() will still be a density operator.

2.4.2 Jamiolkowski isomorphism

The Jamiolkowski isomorphism [20] establishes an equivalence between quantum states and

superoperators. Consider the action of a superoperator on half of the maximally entangled

state|J = 1NN

i |i|i:

1 |JJ| ij

|ij| (|ij|) =. (2.35)

The bipartite state encodes all the properties of the superoperator , as from it we can learnhow each density matrix element is transformed by

|ij| (|ij|) . (2.36)This establishes the equivalence between a completely positive map acting on density operators

pertaining to a Hilbert spaceH of dimension d2 1 and a density operator pertaining to aHilbert spaceH H of dimension 4d2 1.

2.5 Mathematical characterization of bipartite entanglement

When studying the existence of entanglement in bipartite states, it is very useful to distinguish

between pure states of the form Eq.(2.7) and mixed states. Pure bipartite states are entangled iff

the number number of terms of their Schmidt decomposition is greater than one. The Schmidtdecomposition of|Sis defined as:

|S =k,l

ckl|ak|bl =i

i|ai|bi, (2.37)

where|aiand|biare orthonormal bases forHA andHB, respectively, and i are non-negativereal coefficients such that

i

2i = 1. Any state of the form Eq.(2.7) admits a Schmidt de-

composition [18]. Hence, given a pure bipartite state, the computation of the coefficients i in

the Schmidt decomposition is sufficient for entanglement detection. However, there are not any

known efficient methods to determine experimentally the Schmidt coefficients of an unknown

state Eq.(2.7). Therefore, other more accessible entanglement criteria were developed. An ex-

ample is the entropic inequalities. Entropy measures uncertainty or our lack of information


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about a particular physical property. Entropic inequalities, which quantify relations between

the information content of a composite quantum system and its parts, are of the form

S(A) S(AB) , S(B) S(AB), (2.38)

whereAB is a density operator of a composite quantum system and A andB are the reduceddensity operators pertaining to individual subsystems. They indicate that no matter which

physical property is measured there is more uncertainty in the composite system than in any

of its parts. Here S stands for several different types of entropies, including the regular von

Neumann entropy S() =Tr log and the Renyi entropy S() = log Tr2 [21]. Theseinequalities depend on the spectrum of both the state of the composite system and the states

of each individual subsystem, and provide necessary conditions for separability of bipartite pure

states. We will introduce in a later chapter of the thesis an efficient method for the determination

of the spectrum of unknown density operators.

2.5.1 Mixed statesHowever, not all bipartite states are of the form Eq.(2.7). In fact, for more general bipartite

states such as Eq.(2.26), the Schmidt decomposition is no longer valid [18]. Therefore new

methods to identify entangled states were developed. These methods are based on the theory

of positive maps.

Positive, but not completely positive maps are the most powerful tool in the detection of

entanglement. These maps are not physical, that is, they cannot be directly implemented in the

laboratory, but they provide the best mathematical criteria for the existence of entanglement

in a given state. In fact, they provide a necessary and sufficient condition for the existence of

entanglement [22]: a bipartite state AB HA HB is entangled iff (11 L)AB 0, for allL HB

2. Unfortunately, very little is known about the structure of positive maps, even for

small dimensional spaces like C. It is therefore very difficult to extract practical entanglementcriteria from the above condition.

Still, Peres [13] and the Horodeckis [14] have shown that the positive partial-transposition

map provides a necessary and sufficient condition for systems of two or three qubits. This map

preserves the eigenvalues of , so its clearly positive and trace preserving. For example, let

consider a generic density operator of a qubit. This is a 2 2 matrix of the form

, (2.39)

where the coefficients ,, are chosen such that Eq.(2.39) is a valid density operator. It is

sometimes convenient to represent the density operators of qubits as

= 1 + r

2 =

1 +

i=x,y,zrii

2 , (2.40)

where 1 is the identity operator,r is a three dimensional vector of length smaller or equal toone and

x=

0 11 0

, y =

0 i

i 0

, z =

1 00 1

, (2.41)

are the Pauli operators. The action of the transposition map on the density operator of the

qubit is

2Or conversely, (L 11)AB 0.


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T

. (2.42)

Suppose now that we consider a qubit, part of a larger system in the entangled state

|+ = 12

(|0|0 + |1|1) . (2.43)

If we now apply the transposition map to the second qubit, which corresponds to a situation in

which we consider the extension of transposition to a larger system (11T), the density operatorwill suffer a partial transpose of its matrix elements:

1

2

1 0 0 10 0 0 00 0 0 01 0 0 1

T 12

1 0 0 00 0 1 00 1 0 00 0 0 1

. (2.44)

The resulting density matrix has eigenvalues 12 , 12 , 12 and 12 , so its not a valid density operator.The negativity under partial transposition is a signature of entanglement, even for more general

cases. It is in fact a sufficient condition for the existence of entanglement.

2.6 Experimental detection of entanglement

Entanglement tests based on positive maps are not physical, since positive maps cannot be

directly implemented in the laboratory. While this problem can be circumvented, by mathemat-

ically constructing completely positive maps out of the positive maps relevant for entanglement

detection [23], the actual implementation of these tests in the laboratory is yet to be achieved.

Instead researchers have focussed on experimental tests that, albeit less powerful than positivemaps, are within reach of current technology.

2.6.1 Bells inequalities

Bells inequalities [3] were introduced as an attempt to encapsulate the non-locality of quantum

mechanics. While this is a completely different goal from the detection of entanglement, the

fact that they were designed to capture the quantum essence of physical systems meant that

they were also an entanglement test. In fact, they are the most widely used experimental

entanglement test. We will next briefly present the derivation of the Bell-CHSH inequality [24]

and show that it is violated by the maximally entangled singlet state introduced in Eq.(2.29).

If we remember the thought experiment mentioned in the introduction, we have the followingscenario: two distant observers A and B share many identical pairs of particles; A and B can

perform two different types of measurements on their respective particles, XA, YA and XB , YB ,

respectively; Each measurement is chosen randomly and has two possible outcomes: +1 and 1.Let us consider the quantity Q = XAXB+ YAXB+ YAYB XAYB. Note that

XAXB+ YAXB+ YAYB XAYB = (XA+ YA)XB+ (XA YA)YB. (2.45)Since XA, YA =1, it follows that either XA+YA = 0 or XA YA = 0, which in turn meansXAXB+ YAXB+ YAYB XAYB = 2. Hence, the expectation value ofQ is


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E(Q) =

xAyAxByB

p(xA, yA, xB, yB)(xAxB+ yAxB+ yAyB xAyB) (2.46)

xAyAxByBp(xA, yA, xB, yB) 2 = 2, (2.47)where p(xA, yA, xB, yB) is the probability that, before the measurements are performed, XA =

xA, YA = yA, XB = xB, YB = yB. If we further notice that E(Q) = E(XAXB) + E(YAXB) +

E(YAYB) E(XAYB), we obtain the Bell inequality

E(XAXB) + E(YAXB) + E(YAYB) E(XAYB) 2. (2.48)However, if we now compute the expectation value ofQ, with

XA = Az, (2.49)

YA = Ax , (2.50)

XB = Bz +

Bx

2, (2.51)

YB = Bz Bx

2, (2.52)

on the singlet state|, we obtain that

XAXB| = YAXB| = YAYB| = XAYB| = 1

2. (2.53)

Thus,Q| = 22, which is in clear violation of Eq.(2.47) and implies that the state isentangled.

The violation of this and other Bells inequalities has been extensively observed experimen-

tally [10, 11], mostly in systems of photons. While being a very convenient entanglement test,

that requires only the computation of expectation values of linear operators on the state of the

composite system, these inequalities fail to detect many entangled states currently produced in

the laboratory. Hence, researchers have actively looked for other types of experimental entan-

glement tests.

2.6.2 Entanglement witnesses

Entanglement witnessesWwere recently introduced as a tool for experimental entanglement de-

tection [25, 26]. They are particularly well suited to the experimental detection of entanglement,

where quite often the type of entangled state generated is known. They are linear operators

acting on the composite Hilbert spaceHA HB that obey the following properties:

W is Hermitian, that is W = W. Tr(W|a, ba, b|) 0, for all states|a, b inHA HB, that is, the expectation value ofW

on any separable state is greater or equal to zero.

Wis not a positive operator, that is, it has at least one negative eigenvalue. Tr(W)=1.


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Thus, if we have Tr(W ) < 0 for some , then is entangled. In that case we say that W

detects. Every entanglement witness detects something [26], since it detects in particular the

projector on the subspace corresponding to the negative eigenvalues of W. We will next give an

example of an entanglement witness that detects bipartite entangled states.

Consider an experimental setup that, due to the imperfections, produces the mixed ratherthan pure bipartite state of two qubits [27]

= p|| + (1 p) 14

, (2.54)

where || is the pure state generated under ideal experimental circumstances, 0 p 1 and1/4 is the completely mixed state (white noise).

The witness is constructed by first computing the eigenvector corresponding to the negative

eigenvalue of the partially transposed density operator TB . The witness is given by the partially

transposed projector onto this eigenvector. If the Schmidt decomposition of | is | =a|01 +b|10, with a, b 0, the spectrum ofTB is given by

{1 p4

+pa2,1 p

4 +pb2,

1 p4

+pab,1 p

4 pab}. (2.55)

Therefore is entangled iff p > 1/(1 + 4ab). The eigenvector corresponding to the minimal

eigenvalue is given by

| = 12

(|00 |11). (2.56)

Hence the witness W is given by

W = ||TB =1

2

1 0 0 00 0

1 0

0 1 0 00 0 0 1

. (2.57)Note that this witness does neither depend on p, nor on the Schmidt coefficients a, b. It detects

iff it is entangled, since we have that

Tr(||TB) = Tr(||TB) =. (2.58)Note also that in this particular case we just considered, if Tr(W ) 0, is separable. This isnot a general property of witnesses, and indeed if the noise is not white this is not true anymore.

2.7 Multipartite entanglement

Multipartite entanglement, as a set of quantum correlations, is much more complex than bi-

partite entanglement. Hence, we know considerably less about its mathematical structure and

experimental detection. Still, the general approach of the methods described in the previous sec-

tion is equally suited to detect multipartite entanglement. In fact, Bells inequalities have been

derived for multipartite entangled states [28] and so have entanglement witnesses [29]. How-

ever their experimental implementation has proved to be too challenging so far. The approach

to multipartite entanglement detection is similar to the bipartite case. Therefore we will use

this section to try to capture the complexity of multipartite entanglement by presenting three

examples of multipartite entangled states. These states were all introduced in the context of

quantum information and have proved to be useful resources for quantum information tasks.


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The classification of multipartite entanglement differs from the bipartite case in that it

is difficult to compare the different types of multipartite entanglement that are possible in a

given composite system. For example, multipartite states ofNsubsystems can be biseparable,

i.e. admit the decomposition

=i

ciiA iB , (2.59)

where A, B are two disjunct partitions of the composite system. How does one compare this

type of state with a state that is triseparable or non-separable with respect to any partition?

This question is still open and considerable research is being currently devoted to it.

We will next present three classes of states that are representative of different features of

multipartite entanglement. We will also briefly discuss their application to quantum information.

2.7.1 Maximally entangled state

Just as we introduced the concept of maximally entangled state for the case of two qubits, wewill equally define the maximally entangled state ofN qubits:

|N = 12

(|0000....0N+ |1111....1N) , (2.60)

where|iiii....iN =|iN, i = 0, 1. In this case all the qubits are entangled with one another,but the state of any subset m of qubits is separable

m= TrNm(|NN|) =12

(|00...0m00...0|m+ |11...1m11...1|m) . (2.61)These states are particularly useful for multi-party quantum communication protocols, such

as multiparty quantum coin flipping [30].

2.7.2 W State

This class of symmetric states is, after the maximally entangled state, the most widely used

example of multipartite entanglement. Unfortunately, a practical application in the context of

quantum information is yet to be found. The W state is defined as

|WN = 1N

(|1000....0N+ |0100....0N+ |0010....0N+ ... + |0000....1N) . (2.62)

In this case all the qubits are again entangled with one another, but interestingly enough thestate of any subsetm of qubits is not separable. In fact, for the case of three qubits, theW state

retains maximally bipartite entanglement when any one of the three qubits is traced out [31].

2.7.3 Cluster state

The cluster state is perhaps the best example of the computational advantage of multipartite

over bipartite entanglement. This class of pure states is represented by a connected subset of a

simple cubic lattice of qubits [32]. The cluster state is defined as the set of states|{k}C thatobey the set of eigenvalue equations

K

(a)

|{k}C= (1)ka

|{k}C, (2.63)with the correlation operators


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K(a) =(a)x

bnghb(a)(b)z . (2.64)

Therein,

{ka

{0, 1

}|a

C

} is a set of binary parameters which specify the cluster state and

nghb(a) is the set of all neighboring lattice sites ofa. This class of states in cubic lattices withtwo or more dimensions is, together with single qubit measurements, sufficient for universal

quantum computation [32]. It is remarkable how a multipartite entangled state is alone the

computational resource required for quantum computation.

2.8 Quantum networks

A quantum computation is nothing but changing the logical values of a set of qubits through a

series of operations, such that the final result has logical meaning. Similarly to classical com-

putations, quantum computations are described through quantum circuits or networks. These

networks are a sequence of quantum gates, unitary operations that change the logical values ofthe qubits, acting on one or more qubits at a time. They are a very useful paradigm to describe

the dynamical evolution of systems of qubits, where the emphasis is on the state of the system

after the implementation of the quantum gate, rather than on the actual physical interaction

that realizes the gate. Deutsch [33] showed the existence of a universal set of quantum gates,

i.e. a set of gates that can approximate any unitary evolution of a set of qubits with arbitrary

accuracy. It was later shown that this set is finite [34].

2.8.1 Universal set of gates

The universal set of quantum gates is constituted by the set of all possible single qubit unitaries

plus an entangling two-qubit gate [18]. Any single qubit unitary operator can be written in the

form

U= exp(i)Rn() = exp(i)exp(in ), (2.65)where , are real numbers and Rn() denotes a rotation by about the n axis. However,

the actual implementation of arbitrary rotations in a given physical qubit can be experimentally

very challenging. Therefore, researchers have instead concentrated in finding a finite set of single

qubit gates that can approximate an any unitary operation Uto arbitrary accuracy , i.e.

(U, V) = max|

||(U V)|| | , (2.66)

whereV is the unitary implemented instead ofU,(U, V) is the unitary error and the maximumis taken over all normalized states|. A possible such set of gates is constituted by theHadamard, /4 and /8 gates [18]:

H= 1

2

1 11 1

, /4 =

1 00 i

, /8 =

1 0

0 ei/4

. (2.67)

As for the two-qubit gate, it is a controlled operation, i.e. it is a quantum gate where the

inputs have different roles. One of the inputs is the control qubit while the other is the target

qubit. The gate acts on the target qubit iff the control qubit is in state |1. A generic controlled-Ugate is depicted in Fig. 2.1. The prototypical example of the entangling two-qubit gate is the

controlled-NOT gate. It has the following matrix representation in the|control, targetbasis:


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U

Truth tableC T

0 0 0 00 1 0 11 0 1 U(0)1 1 1 U(1)

C TC

T

Figure 2.1: The controlled-Ugate. The top line represents the control qubit and the bottomline represents the target qubit. Uacts on the target qubit iff the control qubit is in the logicalstate|1.

C N OT =

1 0 0 00 1 0 00 0 0 10 0 1 0

. (2.68)

TheC N OTgate flips the target qubit iff the control qubit is in state |1, otherwise it acts asthe identity gate. Let us now understand why is it that this gate is an entangling gate. Consider

the case where the control qubit is in state (

|0

+

|1

)/

2 and the target qubit is in state

|0

.

The composite state of the two qubits is clearly separable. After the C N OT,

12

(|0 + |1)|0 = 12

(|00 + |10) CNOT 12

(|00 + |11), (2.69)

which is no longer separable and is in fact the maximally entangled state |+ mentioned earlier.

2.8.2 Interferometry

As we mentioned earlier, the quantum network formalism provides us with a very useful set of

tools to describe the dynamical evolution of physical systems of qubits. A particularly simple

and relevant example that of quantum interferometry. Consider a single particle going througha Mach-Zender interferometer (Fig. 2.2).

The incoming particle enters the interferometer from the lower left, in the path labelled

|0. It encounters a 50:50 beam-splitter that deflects the particle into arm|1 with probabilityp = 0.5. If the particle goes through arm|0, it acquires a phase 0, while if it goes througharm|1 it acquires the phase 1. The two paths are then recombined in a second 50:50 beam-splitter. If we place particle detectors at each of the output ports of the second beam-splitter,

and repeated this experiment many times, we would observe that the probability of the particle

being detected in port|0or|1 after the interferometer is given by


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1

0

0

1

2 1 0

0 cos

2P

2 1 0

1 sin

2P

50/50 Beam Splitter

50/50 Beam Splitter

Figure 2.2: The Mach-Zender interferometer.

p0 = cos2

1 0

2

, (2.70)

p1 = sin2

1 0

2

. (2.71)

This result can be easily understood if we translate the Mach-Zender interferometer into thelanguage of quantum networks (Fig. 2.3). Let us encode our qubit in the two arms of the

interferometer. The qubit is initially is state|0. After the beam-splitter, which is nothing buta Hadamard gate, the state of the qubit becomes (|0 + |1)/2. The qubit then acquires thephases0, 1: (e

i0 |0+ ei1 |1)/2, which is equivalent the the action of a phase gate 2(10).After the second beam-splitter the state of the qubit is

|0 BS|0 + |12

|0 + ei(10)|1

2BS (1 + e

i(10))|0 + (1 ei(10))|12

, (2.72)

hence the probability of finding the qubit in state

|0

or

|1

after the interferometer is simply

given by Eq.(2.70) and Eq.(2.71), respectively.

2.9 Summary

We have now presented the basic concepts underlying this thesis: mixed states, superoperators,

entanglement and quantum circuits. We have discussed the ambiguity in the definition of any

mixed quantum state, which is due to the indistinguishability of state preparations. We have

introduced superoperators, completely positive maps acting on quantum states, as the most

general evolution of quantum systems. We have mentioned the Jamiolkowski isomorphism be-

tween superoperators and quantum states. Entanglement, and in particular its detection, is the

main object of research of this thesis. We gave an overview of the main results concerning the


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VISIB

ILITY

Figure 2.3: The quantum network corresponding to the Mach-Zender interferometer (= 1 0). The visibility of the interference pattern associated with p0 varies as a function of according to Eq.(2.70).

mathematical characterization and experimental detection of entanglement. Finally, we intro-duced the circuit model of quantum computation and we shown its suitability to describe the

dynamical evolution of quantum systems, and in particular interferometric effects.


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CHAPTER3

Direct estimation of density operators

This chapter presents the results that were published in an article written in collaboration with

A. K. Ekert, D. K. L. Oi, M. Horodecki, P. Horodecki, L. C. Kwek: A. K. Ekert, C. Moura

Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, L. C. Kwek, Phys. Rev. Lett. 88, 217901

(2002).

Certain properties of a quantum state , such as its purity, degree of entanglement, or its

spectrum, are of significant importance in quantum information science. They can be quantified

in terms of linear or non-linear functionals of. Linear functionals, such as average values of

observables{A}, given by TrA, are quite common as they correspond to directly measurablequantities. Non-linear functionals of state, such as the von Neumann entropyTr ln , eigen-values, or a measure of purity Tr2, are usually extracted from by classical means i.e. is first

estimated and once a sufficiently precise classical description of is available, classical evalu-

ations of the required functionals can be made. However, if only a limited supply of physical

objects in state is available, then a direct estimation of a specific quantity may be both more

efficient and more desirable [35]. For example, the estimation of purity of does not require

knowledge of all matrix elements of , thus any prior state estimation procedure followed by

classical calculations is, in this case, inefficient. However, in order to bypass tomography and to

estimate non-linear functionals ofmore directly, we need quantum networks [33, 36] performing

quantum computations that supersede classical evaluations.

In this chapter, we shall present and examine a simple quantum network that can be used as

a basic building block for direct quantum estimations of both linear and non-linear functionalsof any . The network can be realized as multiparticle interferometry. While conventional

quantum measurements, represented as quantum networks or otherwise, allow the estimation of

TrA for some Hermitian operator A, our network can also provide a direct estimation of the

overlap of any two unknown quantum states a and b, i.e. Trab.

3.1 Modified interferometry

In order to explain how the network works, let us start with a general observation related to

modifications of visibility in interferometry. Consider a typical interferometric set-up for a single

qubit: Hadamard gate, phase shift

21


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CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 22

Figure 3.1: A modified Mach-Zender interferometer with coupling to an ancilla by a controlled-Ugate. The interference pattern is modified by the factor vei = Tr [U ].

=

1 00 ei

, (3.1)

Hadamard gate, followed by a measurement in the computational basis. We modify the interfer-

ometer by inserting a controlled-Uoperation between the Hadamard gates, with its control on

the qubit and withU acting on a quantum system described by some unknown density operator

, as shown in Fig. 3.1. The action of the quantum network is given by

|0| H 12

(|0 + |1) |cU 1

2(|0| + |1U(|))

12

(|0| + ei|1U(|))H 1

2

|0| + eiU(|)

+ |1

| eiU(|)

. (3.2)

The action of the controlled-U on modifies the interference pattern:

P0() = 14

(1 + veiei + veiei + 1) =12

(1 + v cos( + )) , (3.3)

by the factor Tr(||U) = TrU = vei [37], where v is the new visibility and is the shiftof the interference fringes, also known as the Pancharatnam phase [38]. Thus, the observed

modification of the visibility gives an estimate of TrU , i.e. the average value of the unitary

operator U in state . Let us mention in passing that this property, among other things, allows

the estimation of an unknown quantum state as long as we can estimate TrUk for a set of

unitary operators Uk which form a basis in the vector space of density operators.

Let us now consider a quantum state of two separable subsystems, such that = a b.We choose our controlled-U to be the controlled-V, where V is the swap operator, defined as,

V|A|B = |A|B, for any pure states|A and|B . In this case, the modification of theinterference pattern given by Eq. (3.3) can be written as,


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Va

b

a b

Figure 3.2: Quantum network for direct estimations of both linear and non-linear functions of

a quantum state.

v = TrV (a b)=

ij

rs

rsfj|ei| (|fs|erfs|er|) |ei|fj

=ij

rs

rsjsir ei | fs fj | er

= ij

ij

|ei

|fj

|2

= Trab. (3.4)

Since Trab is real, we can fix = 0 and the probability of finding the qubit in state |0at theoutput, P0, is related to the visibility by v = 2 P0 1. This construction, shown in Fig. 3.2,provides a direct way to measure Trab (c.f. [39] for a related idea).

3.2 Multiple target states

There are many possible ways of utilizing this result. For pure statesa= || andb = ||the formula above gives Tr

ab

=|

|

|2 i.e. a direct measure of orthogonality of

|

and

|. If we put a = b = then we obtain an estimation of Tr2. In the single qubit case,this measurement allows us to estimate the length of the Bloch vector, leaving its direction

completely undetermined. For qubits Tr2 gives the sum of squares of the two eigenvalues which

allows to estimate the spectrum of.

3.2.1 Spectrum estimation

In general, the evaluation of the spectrum of any d ddensity matrix requires the estimationofd 1 parameters Tr2, Tr3,... Trd. We can do so directly via the controlled-shift operation,which is a generalization of the controlled-swap gate. Givenk systems of dimensiond we define

the shift V(k) as

V(k)|1|2...|k = |k|1...|k1, (3.5)


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for any pure states|. Such an operation can be easily constructed by cascadingk 1 swapsV. If we extend the network and prepare = k at the input then the interference will bemodified by the visibility factor,

v= Tr V(k)k = Tr k =k

i=1

ik. (3.6)

Thus measuring the average values ofV(k) for k = 2, 3...d allows us to evaluate the spectrum

of [35]. Although we have not eliminated classical evaluations, we have reduced them by a

significant amount. The average values ofV(k) for k = 2, 3...d provide enough information to

evaluate the spectrum of, but certainly not enough to estimate the whole density matrix.

It should be mentioned that other spectrum estimation methods, relying on single collective

measurements of several copies of, have been proposed [40]. These methods essentially project

the initial state = n, which forms an operator on the n-fold tensor product space, ontoorthogonal subspaces corresponding to irreducible representations of the permutation group of

n points. This decomposition is labelled by Young frames, the arrangement ofn boxes into drows of decreasing length. The normalized row lengths of each tableau are taken as estimates

of the ordered sequence of eigenvalues of . The probability that the error in the spectrum

estimation is greater than some fixed decreases exponentially withn [40].

3.2.2 Quantum communication

So far we have treated the two inputs a andb in a symmetric way. However, there are several

interesting applications in which one of the inputs, say a, is predetermined and the other is

unknown. For example, projections on a prescribed vector |, or on the subspace perpendicularto it, can be implemented by choosinga=

|

|. By changing the input state

|

we effectively

reprogram the action of the network which then performs different projections. This propertycan be used for quantum communication, in a scenario where one carrier of information, in state

|, determines the type of detection measurement performed on the second carrier. Note thatas the state|of a single carrier cannot be determined, the information about the type of themeasurement to be performed by the detector remains secret until the moment of detection.

3.2.3 Extremal eigenvalues

Another interesting application is the estimation of the extremal eigenvalues and eigenvectors

ofb without reconstructing the entire spectrum. In this case, the input states are of the form

||band we vary | searching for the minimum and the maximum ofv = |b|. This,at first sight, seems to be a complicated task as it involves scanning 2(d 1) parameters of.The visibility is related to the overlap of the reference state|andb by,

v = Tr

||

i

i|ii|

=i

i | | i |2 =i

ipi, (3.7)

where

ipi = 1. This is a convex sum of the eigenvalues ofb and is minimized (maximized)

when| =|min (|max). For any| =|min (|max), there exists a state,|, in theneighbourhood of

|

such that v < v (v > v). Thus this global optimization problem

can be solved using standard iterative methods, of which steepest decent [41] is an example.


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Estimation of extremal eigenvalues plays a significant role in the direct detection [35] and

distillation [22] of quantum entanglement. As an example, consider two qubits described by the

density operatorb, such that the reduced density operator of one of the qubits is maximally

mixed. We can test for the separability ofb by checking whether the maximal eigenvalue ofb

does not exceed 1

2 [42].

3.2.4 State estimation

Finally, we may want to estimate an unknown state, say a d d density operator, b. Such anoperator is determined byd21 real parameters. In order to estimate matrix elements |b|,we run the network as many times as possible (limited by the number of copies of b at our

disposal) on the input|| b, where| is a pure state of our choice. For a fixed|, afterseveral runs, we obtain an estimation of,

v= |b|. (3.8)

In some chosen basis{|n} the diagonal elementsn|b|n can be determined using the inputstates |nn|b. The real part of the off-diagonal element n|b|k can be estimated by choosing| = (|n + |k)/2, and the imaginary part by choosing| = (|n + i|k)/2. In particular,if we want to estimate the density operator of a qubit, we can choose the pure states,|0 (spin+z), (|0 + |1) /2 (spin +x) and (|0 + i|1) /2 (spin +y), i.e. the three components of theBloch vector.

Quantum tomography can also be performed in many other ways, the practicalities of which

depend on technologies involved. However, it is worth stressing that the strength of our scheme is

the use of a fixed architecture network, controlled only by input data, to perform the estimation

of properties of.

3.2.5 Arbitrary observables

We can extend the procedure above to cover estimations of expectation values of arbitrary

observables A. This can be done with the network shown in Fig. 3.2, since estimations of mean

values ofanyobservable can always be reduced to estimations of a binary two-output POVMs.

We shall apply the technique developed in Refs. [23, 35]. As A= 1 + Ais positive ifis theminimum negative eigenvalue ofA, we can construct the state a= A =

A

Tr(A)and apply our

interference scheme to the pair A b. The visibility gives us the mean value of V,

v= VAb = Tr A

Tr(A)b , (3.9)

which leads us to the desired value,

Ab Tr(bA) =vTrA + (vd 1), (3.10)

where Tr1= d.

3.3 Quantum channel estimation

Any technique that allows direct estimations of properties of quantum states can be also used

to estimate certain properties of quantum channels. Recall that, from a mathematical point

of view, a quantum channel is a superoperator, (), which maps density operators intodensity operators, and whose trivial extensions, 1k do the same, i.e. is a completely positive


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Figure 3.3: A quantum channel acting on one of the subsystems of a bipartite maximallyentangled state of the form |+ =

k |k|k/

d. The output state =

1d

kl |kl| (|kl|),

contains a complete information about the channel.

map. In a chosen basis the action of the channel on a density operator =

kl kl|kl| can bewritten as

() =

kl

kl|kl|

=kl

kl (|kl|) . (3.11)

Thus the channel is completely characterized by operators (|kl|). In fact, with every channel we can associate a quantum state that provides a complete characterization of the channel.

If we prepare a maximally entangled states of two particles, described by the density operator

P+ = 1d

kl |kl| |kl|, and we send only one particle through the channel, as shown in

Fig. 3.3, we obtain

P+ [1 ] P+= , (3.12)where

= 1

d

kl

|kl| (|kl|) . (3.13)

We may interpret this as mapping the|kl|th-element of an input density matrix to theoutput matrix, (|kl|). Thus, knowledge of allows us to determine the action of onan arbitrary state, (). If we perform a state tomography on we effectively performa quantum channel tomography. If we choose to estimate directly some functions of thenwe gain some knowledge about specific properties of the channel without performing the full

tomography of the channel.

For example, consider a single qubit channel. Suppose we are interested in the maximal

rate of a reliable transmission of qubits per use of the channel, which can be quantified as the

channel capacity. Unlike in the classical case, quantum channels admit several capacities [43, 44],

because users of quantum channels can also exchange classical information. We have then the

capacities QC where C = , , , , stands for zero way, one way and two way classicalcommunication. In general, it is very difficult to calculate the capacity of a given channel.

However, our extremal eigenvalue estimation scheme provides a simple necessary and sufficient

condition for a one qubit channel to have non-zero two-way capacity. Namely, Q > 0 iff


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has maximal eigenvalue greater than 12 . Note that this condition is also necessary for the other

three capacities to be non-zero.

This result becomes apparent by noticing that if we traceover the qubit that went through

the channel (particle 2 in Fig. 3.3), we obtain the maximally mixed state. Furthermore, the

two qubit state, , is two-way distillable iff the operator 1

2 1 has a negative eigenvalue(see [42] for details), or equivalently when has the maximal eigenvalue greater than 12 . Thisimplies Q()> 0 because two-way distillable entanglement, which is non-zero iff given stateis two way distillable, is the lower bound for Q() [44].

3.4 Summary

In summary, we have described a simple quantum network which can be used as a basic building

block for direct quantum estimations of both linear and non-linear functionals of any density

operator . It provides a direct estimation of the overlap of any two unknown quantum states

a and b, i.e. Trab. Its straightforward extension can be employed to estimate functionals

of any powers of density operators. The network has many potential applications ranging frompurity tests and eigenvalue estimations to direct characterization of some properties of quantum

channels.

Finally let us also mention that the controlled-SWAP operation is a direct generalization

of a Fredkin gate [45] and can be constructed out of simple quantum logic gates [36]. This

means that experimental realizations of the proposed network are within the reach of quantum

technology that is currently being developed (for an overview see, for example, [46]).


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CHAPTER4

Direct estimation of density operators using LOCC

This chapter presents the results that were published in an article written in collaboration with

D. K. L Oi, P. Horodecki, A. K. Ekert, L. C. Kwek: C. Moura Alves, D. K. L Oi, P. Horodecki,

A. K. Ekert, L. C. Kwek, Phys. Rev. A 68, 32306 (2003).

In the previous chapter we presented a family of quantum networks that directly estimate

multi-copy observables, Tr[k], of an unknown state [47, 35, 23]. As mentioned before, these

nonlinear functionals quantify important properties of, such as the degree of entanglement or

the spectrum. Therefore it would be very useful to be able to estimate them even when is a

bipartite stateAB shared by two distant parties, Alice and Bob, who can perform only local

operations and communicate classically (LOCC). In this chapter we show that the estimationof non-linear functionals of quantum states admit LOCC implementation. We also show that

Structural Physical Approximations [35, 23], an important tool for entanglement detection, can

be constructed locally. This opens the possibility of the direct estimation of entanglement and

some channel capacities using only LOCC.

As a general remark, let us recall that a quantum operation can be implemented using

LOCC if it can be written as a convex sum

=k

pk Ak Bk, (4.1)

whereAk acts on the subsystem at Alices location and Bk on the subsystem at Bobs location,

andpk represent the respective probabilities.

4.1 LOCC estimation of nonlinear functionals

The direct estimation method is extended to the LOCC scenario by constructing two local

networks, one for Alice and one for Bob, in such a way that the global network is similar to

the network with the controlled-shift. Unfortunately, the global shift operation V(k) cannot

be implemented directly using only LOCC, since it does not admit local decomposition (4.1).

Hence, we will implement it indirectly, using the global network shown in Fig. 4.1. Alice and

Bob share a number of copies of the state AB

Hd. They group them respectively into sets

ofk elements, and run the local interferometric network on their respective halves of the stateAB =

kAB. For each run of the experiment, they record and communicate their result.

28


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CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 29

Figure 4.1: Network for remote estimation of non-linear functionals of bipartite density opera-tors. Since Tr[V(k)k] is real, Alice and Bob can omit their respective phase shifters.


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The individual interference patterns Alice and Bob record will depend only on their respective

reduced density operators. Alice will observe the visibility vA = Tr[kA] and Bob will observe

the visibility vB = Tr[kB]. However, if they compare their individual observations, they will be

able to extract information about the global density operator AB, e.g. about

Tr[kAB] = Tr

kAB

V(k)A V(k)B . (4.2)This is because Alice and Bob can estimate the probabilities Pij that in the measurement Alicesinterfering qubit is found in state |iA and Bobs in state|jA for i, j = 0, 1. These probabilitiescan be conveniently expressed as

Pij = 1

4Tr

kAB

1 + (1)iV(k)A 1 + (1)jV(k)B , (4.3)

hence the formula for the basic non-linear functional ofAB reads

Tr[kAB] = P00 P01 P10+ P11. (4.4)In fact, the expression above is the expectation valuez z, measured on Alices and Bobsqubits (the two qubits that undergo interference). Given that we are able to directly estimateTr[kAB] for any integer value ofk , we can estimate the spectrum ofAB without resorting to a

full state tomography.

4.2 Structural Physical Approximations

We next show how to implement Structural Physical Approximations within the LOCC scenario.

Structural Physical Approximations (SPAs) were introduced recently as tools for determining

relevant parameters of density operators (see [23, 35] for more details). Basically the SPA of a

mathematical operation , denoted as , is a physical operation, a process that can be carried

out in a laboratory, that emulates the character of . More precisely, suppose :

Hd

Hd is a

trace preserving map which does not represent any physical process, for example, an anti-unitary

operation such as transposition. Then a convex sum

=D + (1 ), (4.5)whereD is the depolarizing map which sends any density operator into the maximally mixedstate, represents a physical process, i.e. a completely positive map, as long as is sufficiently

large. On top of thisD, with its trivial structure, does not mask the structure of . TheStructural Physical Approximation to is obtained by selecting, in the expression above, the

threshold value = (d2)/(d2 + 1), whereis the lowest eigenvalue of (1 )P(d)+ andP(d)+is a maximally entangled state of ad dsystem 1. Note that we impose the positivity condition

on the map1

to ensure that is a completely positive map.Please note that the physical implementation of SPAs is not a trivial problem as the for-

mula (4.5), which explicitly contains the physically impossible map , is of little guidance here.

Let us also mention in passing that if is not trace preserving then may be implementable

but only in a probabilistic sense e.g. using experimental post-selection.

There are many examples of mathematical operations which, though important in the quan-

tum information contexts, do not represent a physical process. For example, mathematical cri-

teria for entanglement involve positive but not completely positive maps [21] and as such they

are not directly implementable in a laboratory they tacitly assume that a precise description

of a quantum state of a physical system is given and that such operations are mathematical

transformations on the matrix describing the quantum state.

1The threshold value for is obtained from the requirement of complete positivity of, which in this case canbe reduced to P

(d)+ 0


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4.2.1 SPA using only LOCC

If does not represent any physical process then its trivial extension to a bipartite case, 1 ,does not represent a physical process either. Still, its SPA, 1 , does describe a physi

carolina thesis(1)

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