New Studies in Classical and Quantum Correlations and Their … · 2014. 1. 30. · useful applications were discovered. In modern times, these applications have seen large use in

Studies in Classical and Quantum Correlations and Their

Evolution in Physical Systems

by

Asma Al-Qasimi

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

Copyright c© 2011 by Asma Al-Qasimi

Abstract

Studies in Classical and Quantum Correlations and Their Evolution in Physical Systems

Asma Al-Qasimi

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2011

More than a century ago, starting with Michelson, the field of classical coherence has

developed rapidly. By studying and uncovering the coherence properties of light, many

useful applications were discovered. In modern times, these applications have seen large

use in fields like astronomy, where the properties of light can be used to discover stars

and determine their radius, for example. Another class of correlations, namely quantum

correlations, which were discovered in the beginning of the twentieth century, have gained

much attention from the scientific community in the last two decades. In particular,

the field of quantum information developed, promising great computational power by

using quantum correlations to build computers. Currently, quantum computation is a

very active field bringing together physicists, mathematicians, engineers, chemists, and

computer scientists to find solutions to the problems encountered in building quantum

computers.

I consider some classical coherence effects of the degree of cross polarization (DCP) on

the Hanbury-Brown Twiss effect, with a specific focus on Gaussian Schell-model beams.

I show that the DCP is necessary, in general, to determine the correlations in intensity

fluctuations of a beam at two different points. As for quantum correlations, I consider

entanglement in realistic systems: one in two-qubit systems, and the other in continuous

variable quantum systems. In the former case, when the temperature of the system is

finite, entanglement always decays in a finite time. However, in the latter case, entan-

ii

glement is long-lived, although in the long run it is not of much practical use. Finally,

I unravel the relationship between quantum discord and quantum entanglement, as well

as quantum discord and entropy for the most general two-qubit systems, and I identify

the states that define the boundaries of these relationships.

iii

Dedication

To the memory of my very dear friend Lyn Zhu (1983-2004).

iv

Acknowledgements

Thanks to Professor Dylan Jones, Professor Ted Shepherd, and Professor Daniel James

for having faith in me and for supporting my application to graduate school.

Thanks to my supervisor Professor Daniel James for all the useful things he taught

me during these years.

Thanks to Professor Heidi Fearn for teaching me some very useful skills that I greatly

benefited from as a student.

Thanks to Professor Emil Wolf, Professor Paul Scott Carney, Professor Joseph Eberly,

Dr. Bill Munro, Professor Andrew White, Dr. Joseph Altepeter, Dr. Peter Milonni,

Professor John Sipe’s Group and Professor Aephraim Steinberg’s Group for insightful

discussions. Special thanks to the Physics Department and the Institute of Optics at the

University of Rochester for the friendly and productive environment that they provide,

which made scientific discussions and collaborations a great pleasure to me during my

graduate years.

Thanks to My Platinum Set of Officemates: James, Jean-Sebastian, Navin, So, and

Philip for the great respect and kindness that they have shown me.

Thanks to My Gold Set of Officemates: Julien, Ania, Kuljit, Temok, and Kiran for

their kindness and for setting a wonderful example in diligence and productivity for me.

Thanks to Temok who helped me with some latex issues I had as well as with im-

proving some of the plots (Figures 1.6, 4.2, 4.3, 5.3) in this thesis using GNU.

Thanks to my students in the First Year Labs, in the Third Year Quantum Mechanics

course, and in the First Year Physics course for their great contribution to my education

while in grad school.

Thanks to Teresa Baptista and to Valerie Kolesnikow for their support and help,

especially when I had an accident in which my finger was badly hurt.

Thanks to Galina Velikova and to Julian Comanean for helping me with most of the

the computer issues I had in these years.

v

Thanks to my friends Nela, Isabel, Mushtari, Lenka, Somayeh, and Elham whose

support during these years meant a lot to me.

Thanks to my group members: Charles, Max, Mark, Ariela, Nachum, Arghavan,

Rebecca, Ardavan, Omar, Bassam, Petar, and Chris for their positive and educational

interactions.

Thanks to my committee members: Professor Joseph Thywissen, Professor John Sipe,

and Professor Henry van Driel for being on my committee, for their support, and for their

invaluable advice.

Thanks to my external committee member Professor Andrew Jordan for coming from

the University of Rochester to test me during my oral exam and for asking me very

interesting and stimulating questions about my work.

Thanks to NSERC, the University of Toronto, and the Burton Fellowship Program

for funding me during my graduate years.

Thanks to my mother, who when I was a 5-months-old baby and the doctors predicted

that I would survive for a maximum of 6 more months, she stormed at them and told

them that her baby will become a scientist.

Thanks to my father who supported me during these years whenever I asked him for

help, for always being kind to me, and for always having faith in my abilities.

Thanks to those from my family and from among my relatives who have been sup-

portive to me during these years.

vi

Preface

The three quantum optics projects that I describe in this thesis are: Sudden Death of

Entanglement at Finite Temperature, Sudden Death at High NOON, and A Comparison

of the Attempts of Quantum Discord and Quantum Entanglement to Capture Quantum

Correlations. They all represent my own work, done under the supervision of Professor

Daniel James.

As for the classical project Intensity Fluctuations and Cross-Polarization in Gaussian

Schell-Model Beams, the problem was proposed by Professor Emil Wolf, I solved the

problem, under the supervision of Professor Daniel James, for Gaussian Schell-model

beams at the source and came up with physical examples to illustrate the result. However,

the effect that we were trying to demonstrate was not significant in these examples

when we look at them at the source. Dr. David Kuebel suggested introducing beam

propagation into the problem, which we did and found that the effect was enhanced for

propagated beams. Mayukh Lahiri helped with producing and preparing the manuscript

describing the work for publication. However, the plots presented in this thesis were

produced independently by myself.

During my PhD, the projects I have worked on were centered around Classical and

Quantum Correlations. That is why in the first chapter, I give a brief overview of these

correlations from an angle that applies to my work. In the chapters following that, I

discuss my work on quantum correlations: Entanglement Sudden Death in Two-Qubit

Systems in Chapter 2, Entanglement Sudden Death in Two Harmonic Oscillator Systems

in Chapter 3, and the relationship between Quantum Discord, Quantum Entanglement,

and Linear Entropy for Two-Qubits in Chapter 4. Finally, I discuss my classical correla-

tion work in Chapter 5, which describes the effect of the degree of cross-polarization on

the Hanbury-Brown Twiss effect.

My graduate work has led to the following publications:

1) Asma Al-Qasimi, Olga Korotkova, Daniel F. V. James and Emil Wolf, Definitions of

vii

the degree of polarization of a light beam, Optics Letters 32, 1015-1016 (2007).

2) Asma Al-Qasimi and Daniel F. V. James, Sudden death of entanglement at finite

temperature, Physical Review A 77, 012117 (2008).

3) Asma Al-Qasimi and Daniel F. V. James Nonexistence of entanglement sudden death

in dephasing of high NOON states, Optics Letters 34, 268-270 (2009).

4) Asma Al-Qasimi, Mayukh Lahiri, David Kuebel, Daniel F. V. James, and Emil Wolf

The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect,

Optics Express 18, No. 16, 17124-17129 (2010).

5) Asma Al-Qasimi and Daniel F. V. James Comparison of the attempts of quantum

discord and quantum entanglement to capture quantum correlations, Physical Review A

83, 032101 (2011).

viii

Contents

1 Introduction 1

1.1 Classical Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Random Variables and Averages . . . . . . . . . . . . . . . . . . . 2

1.1.2 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Young’s Interference Experiment and Second-Order Coherence . . 5

1.2 Quantum Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Why do we know that quantum correlations exist and how can we

quantify them? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Entanglement in Two-Qubit Systems . . . . . . . . . . . . . . . . 16

1.2.3 Entanglement in Bipartite Harmonic Oscillator Systems . . . . . . 19

1.2.4 Quantum Discord, MID, and Ameliorated MID . . . . . . . . . . 23

1.2.5 Other methods and final words . . . . . . . . . . . . . . . . . . . 28

2 Sudden Death of Entanglement at Finite Temperature 30

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Two-qubit model system . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Solution for the Sudden Death Time . . . . . . . . . . . . . . . . . . . . 39

2.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

ix

3 Sudden Death at High NOON 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Dynamics of Two-Mode-N-Photon States Undergoing Dephasing . . . . . 47

3.3 Measure of Entanglement for Photon States . . . . . . . . . . . . . . . . 48

3.4 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Example: NOON States . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Is long-lived entanglement practical? . . . . . . . . . . . . . . . . . . . . 55

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Capturing Correlations by Discord and Entanglement 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Discord, Entanglement, and Linear Entropy for Two-Level Bipartite Systems 61

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 The Degree of Cross-Polarization and the HBT Effect 72

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 The HBT Effect, Taking into Account the Vector Nature of the Field . . 74

5.3 Results: The Dependence of the HBT Effect in Gaussian Schell-Model

Beams on the Degree of Cross-Polarization . . . . . . . . . . . . . . . . . 76

5.4 Example: Realizable Gaussian Schell-model beams . . . . . . . . . . . . . 78

5.5 Beam Propagation Enhances the Dependence of the HBT Effect on the

Degree of Cross-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Conclusions 85

Bibliography 90

x

List of Figures

1.1 Ergodicity. Taken from figure 2.4 in [1]. Here is a visual statement for

the concept of ergodicity in stationary random processes; i.e., for processes

that do not depend on the time coordinate. Symbolically that means:

〈x(t)〉 = 〈x(t+ τ)〉. When a process is ergodic, it means that the time

average and the ensemble average are equal. Here is why. Notice in that

in (a) there is only one realization of x(t), and by integrating over time as

in (1.1), the time average is obtained. However, one can also divide the

curve in (a) into several segments, as shown in (b), and treat each as a

different realization. After averaging over those, one obtains the ensemble

average. Here, the two averages are only different in the way one looks at

the random variable x(t), but in the end, they amount to the same quantity. 3

1.2 Young’s Interference Experiment. Adapted from figure 3.1 in [1]. . . 6

xi

1.3 Clauser Experiment. Taken from figure 2 in [2]. This figure shows three

experimental setups of a source of two photons (from parametric down-

conversion, for example) with three polarization analyzer loops to its left

and two to its right. The analyzers labeled with x and y take an incoming

beam, split it into the two orthogonal components polarized in the x and

y directions, and then recombines them before they exit the loop. The

same argument applies for the loops labeled θ and θ(= π − θ), and φ and

φ(= π − φ), where θ and φ corresponds to the angles of the polarization

with respect to the x-axis. The vertical thick line that can be seen at some

ends of the analyzers block the corresponding component so that in the

recombining process just before exiting the loop that component is absent.

For example, in the top most set up, the second analyzer loop to the left

of the photon source has its x-polarized component blocked. This means

that the component that will be allowed to exit will be in the y-direction 12

1.4 Quantum Harmonic Oscillator. The energy spectrum is not continu-

ous, but discrete. However, there is an infinite number of energy levels. . 20

1.5 Quantum Computation Experiment in the Absence of Entan-

glement. Taken from figure 2 in [3]. In this experiment, the DQC1

(Deterministic Quantum Computation with One Pure State) algorithm

[4] is implemented using photons. The control qubit, ρc, is a pure state,

and the register qubit ρr represents a maximally mixed state, which is

intially pure, but is rendered mixed by introducing a phase delay between

the perpendicular components of the field. Both qubits are encoded in

polarization states of single photons. The purpose of this algorithm is to

estimate the normalized trace of the rotation operator Zθ, where θ is the

rotation angle. The power of this algorithm comes from the purity of ρc. 26

xii

1.6 Discord and MID versus the Rotation Angle θ. Notice that discord

is always finite, except for the few cases when θ = 2πn, where n is an

integer, as was shown in [3]. It was also found in [3], that these points

have no quantum advantage and can be stimulated efficiently classically.

However, it is interesting to note that for some of the points where discord

is zero, specifically for those with θ = 2πm, where m is zero or an even

integer, MID is maximal, which raises questions with regard its definition

since some of these points have already been shown to have no quantum

advantage. This is a work done in collaboration with Andrew White, but

we found out that an equivalent observation about MID was made earlier

in [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1 Disentanglement by spontaneous emission of a two-qubit system

in a heat bath. The reservoir is modelled by different harmonic oscillator

modes. Each qubit, here depicted by a two-level atom, interacts with its

reservoir. The only interaction between the qubits that ever exists is the

one that leads to their entanglement at time t = 0. Following this, however,

the only interaction that remains is that with the corresponding reservoir.

This leads to decoherence, which causes the qubits to disentangle. Here I

include the effect of heat by studying the system at T 6= 0. . . . . . . . . 32

2.2 Plot of F(X) vs X. This is the plot of the first quartic equation in (2.22);

i.e., F (X) = |z(t)|2 − a(t)d(t), for n = 0.8, a0 = 0.1, d0 = 0.05, z0 = 0.3. . 41

2.3 Plot of C (concurrence) vs X (= e−Γ(2n+1)t) vs α. C = 0 corresponds

to no entanglement. X = 1 corresponds to t = 0, while X = 1 corresponds

to t = ∞. Notice that as soon as n becomes finite, for all values of α, C

becomes zero at X < 0; i.e., entanglement decays in a finite time. As n

becomes bigger, all states disentangle at approximately X = 0.5. . . . . . 43

xiii

3.1 Plot of W (Wigner Function) vs q1 vs p1 for a NOON state with

N = 3. This relation is for the case when q1 = −q2 and p1 = −p2. Notice

that although in (b), the state is completely dephased, a case in which no

entanglement is believed to be in the system, the Wigner function can still

take negative values. This shows that, generally speaking, the negativity

of the Wigner function cannot be used as an indication of the existence of

entanglement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Interference pattern formation adapted from Fig. 1 of [6] Two

photon beams pass through a beam splitter and get reflected off the upper

and lower mirrors to form an interference pattern on the screen. The upper

beam passes through a phase shifter before reaching the screen. The phase

aquired depends on the number of photons N that pass through the upper

path, and it equals eiNφ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1 The Discord-Entanglement Horn. Discord increases as entanglement

increases. In the case of pure states, the two quantities are identical. While

in the mixed state case the relationship broadens. However, notice that

this relationship narrows in the high quantum correlated regime and is the

broadest in the low correlation regime. This gives the plot its ‘horn’ shape.

The upper bound of this relationship is given by the α-states eq.(4.10)

(for 0 ≤ EoF ≤ 0.620, and 0 ≤ Q ≤ 0.644), the Werner states [7] (for

0.620 ≤ EoF ≤ 0.746, and 0.644 ≤ Q ≤ 0.746), and the pure states (for

0.740 ≤ EoF,Q ≤ 1). The lower bound is given by the β-states eq.(4.11). 67

4.2 The Relationship Between Discord and Linear Entropy. The most

general relation between discord and linear entropy for a two-level bipartite

system would look like this plot. For the states that define the boundaries

of this relationship, see Fig. 4.3. . . . . . . . . . . . . . . . . . . . . . . . 69

xiv

4.3 The Boundaries on the Relationship Between Discord and Lin-

ear Entropy. To easily illustrate the boundaries, this plot only includes

the states that are involved in defining them. The two-parameter states,

eq.(4.14), bound the curve from above up to Q = 1/3 and SL = 8/9,

after which the Werner states take over. Discord and Linear Entropy, as

expected, display an inverse relationship: more randomness implies less

quantum correlations. One of the interesting phenomena occurs at the

‘pimple’, where there exists states in which an increase in their entropy

results in an increase in their discord. This is also the point that defines

the value of linear entropy after which no entanglement can exist in the

system (See [8]). Unlike entanglement, states exist that are very close to

the maximally mixed states, but still have non-zero discord. In fact the

only value for entropy such that discord cannot be finite is for it being

equal to 1, in the case when the system is maximally mixed. . . . . . . . 70

5.1 Simple Correlation Experiment. The intensity of an electromagnetic

beam at points P1 and P2 fluctuates in realistic systems. In Ref. [9]

and Ref. [10], the correlations in intensity fluctuations was shown to be

dependent on the degree of coherence and the newly discovered statistical

parameter, the degree of cross-polarization. . . . . . . . . . . . . . . . . . 75

xv

5.2 Correlation in Intensity Fluctuations versus Separation Distance.

The degree of coherence and the degree of polarization are the same for

both beams. However, their degree of cross-polarization is different; for

the solid red curve, it is given for the value of Q = 0, while for the dashed

blue curve Q = 716

. This has an effect on the correlation in intensity

fluctuations of the beams, which is no longer the same for the two beams.

This can be seen in this graph: as the separation distance between point

P1 and P2 (illustrated in Fig. 5.1) becomes finite, the correlations for the

two beams are different. Note that the area between the curves, which

corresponds to the difference between the correlations for the two curves,

is given by 27256

√π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Correlations in Intensity Fluctuations of Two Propagated Beams.

Here we have two beams with the same degree of coherence and the same

degree of polarization, but with different degrees of cross-polarization that

are allowed to propagate. This is a plot of the correlations in intensity

fluctuations of the beams, 10km away from the source, of two points at

r and -r (diametrically opposite) versus r. Notice how the profile of the

correlations are different. This is due to the difference in the degree of

cross-polarization of the two beams. As in Fig. 5.2, the solid curve corre-

sponds to Q = 0 at the source, while for the dashed curve Q = 716

at the

source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xvi

Chapter 1

Introduction

In the last century or so, the field of classical coherence has developed rapidly with the

discovery of many applications. In modern times, these applications have seen large

use in fields like astronomy, where the properties of light can be used to discover stars

and determine their radius, for example. Another class of correlations, namely quantum

correlations, which were discovered in the beginning of the twentieth century, have gained

much attention from the scientific community in the last two decades. In particular, the

field of quantum information developed, promising great computational power by using

quantum correlations to build computers. Currently, quantum computation is a very

active area bringing together researchers from various fields together to find solutions to

the problems encountered in building quantum computers.

In this chapter, I give a brief overview of the status of classical and quantum correla-

tions in the literature, focusing on the aspects that are most relevant to my work, which

I present in the following chapters.

1

Chapter 1. Introduction 2

1.1 Classical Correlations

1.1.1 Random Variables and Averages

In classical optics, realistic field variables fluctuate [1]. That is why an important quantity

to look at is the average of these variables. Let us say we have the field variable x(t) that

can, for example, represent the cartesian component of the electric field as a function of

time. In general, one can think of x(t) as a random variable. Different realizations of

x(t) can be created by performing the same experiment several times. I will label each

realization by k. There are two types of averages that are of interest here: the time

average and the ensemble average, described, respectively, as follows:

⟨kx(t)

⟩= lim

T→∞

1

2T

∫ T

−T

k x(t)dt, (1.1)

where (1.1) gives the time average of one realization of x(t), namely kx(t); and

〈x(t)〉e =∫xp1(x, t)dx, (1.2)

where p1(x, t) is the probability density, and p1(x, t)dx is the probability that x takes a

value in the range (x, x+dx) at time t. In the classical systems that I study, which I will

elaborate more on in Chapter 5, I assume the system is ergodic, which means that the

time average and the ensemble average are equal. See Fig. 1.1 for details. Throughout

this thesis, I will use 〈...〉 to represent the average. It can be taken as the time average,

the ensemble average, or the expectation value. I will not be distinguishing between these

quantities.

1.1.2 Correlation Functions

As I mentioned in the previous section, since a random variable x(t) fluctuates in time,

a useful quantity to study is its average value 〈x(t)〉. Another important quantity is the


kx(t)

t

t

2T 2T 2T 2T 2T

(a)

(b)

Figure 1.1: Ergodicity. Taken from figure 2.4 in [1]. Here is a visual statement for

the concept of ergodicity in stationary random processes; i.e., for processes that do not

depend on the time coordinate. Symbolically that means: 〈x(t)〉 = 〈x(t+ τ)〉. When a

process is ergodic, it means that the time average and the ensemble average are equal.

Here is why. Notice in that in (a) there is only one realization of x(t), and by integrating

over time as in (1.1), the time average is obtained. However, one can also divide the curve

in (a) into several segments, as shown in (b), and treat each as a different realization.

After averaging over those, one obtains the ensemble average. Here, the two averages

are only different in the way one looks at the random variable x(t), but in the end, they

amount to the same quantity.


expectation value of the product of x at time t with x at time t + τ . This quantity is

called the autocorrelation function and is given by:

R(τ) = 〈x(t)x(t+ τ)〉 . (1.3)

The width of R(τ) is a measure of how statistically similar x(t) is with itself after a

certain period of time τ . Equation (1.3) is used in the case when the random variable is

real. However, if instead, it is a complex random variable z(t) (for example, the analytic

signal associated with a real process x(t)), then the autocorrelation function is given by:

R(τ) = 〈z∗(t)z(t+ τ)〉 . (1.4)

The Wiener-Khintchine theorem tells us that when a random process has zero mean,

R(τ) and the spectral density S(ω) (which tells us what frequency components the field

contains), where ω is the frequency, form a Fourier transform pair.

These concepts can be extended to two random processes z1(t) and z2(t). In this case

the analogue of R(τ) and S(ω) are, respectively, the cross-correlation function R12(τ) =

〈z∗1(t)z2(t+ τ)〉 and the cross-spectral densityW12(ω). Again, these quantities are Fourier

transforms of each other.

The idea of a complex quantity such as z(t), and the fact that when it shows up

as a product in (1.4), the complex conjugate has to be used, is important in classical

optics because of the following. Physical electric fields are real quantities. A fluctuating

real quantity of the field, such as its cartesian component, can be represented by a real

function U(t), which can be represented by the following Fourier integral:

U(t) =∫ ∞

−∞u(ω)e−iωtdω. (1.5)

The fact that U(t) is real implies that the information contained in the negative values

of ω is already contained in its positive values. In other words, it is redundant to include


these values. Therefore, U(t) can be represented by V (t), which is called the complex

analytic signal as follows:

V (t) =∫ ∞

0u(ω)e−iωtdω. (1.6)

The complex analytic signal can be written as V (t) = 12

[V (r)(t) + iV (i)(t)

], where r labels

the real component of V (t) and i its imaginary component. V (r)(t) and V (i)(t) happen

to form a Hilbert-transform pair. If V (r)(t) is stationary, which means that its average

value does not depend on the origin of time, then:

⟨[V (r)(t)

]2⟩=

1

2〈V ∗(t)V (t)〉 . (1.7)

As I mentioned earlier, the field variable is described by a real quantity. Therefore, here,

the real quantity V (r)(t) is ultimately what we are interested in, and from (1.7), we can

conclude that the average intensity of the field, given by the square of its amplitude,

is expected to be proportional to 〈V ∗(t)V (t)〉. This explains why the complex quantity

appears in the expressions for expectation value of the field.

1.1.3 Young’s Interference Experiment and Second-Order Co-

herence

The concept of coherence described in the previous section can be applied to interference

phenomena such as Young’s interference experiment depicted in Fig. 1.2. As can be

seen in the figure, a quasi-monochromatic light beam travels toward two pinholes at Q1

and Q2, and from these points, travels a distance R1 and R2, respectively, to form an

interference pattern on a screen at point P . The interference pattern will depend on the

correlation between the light vibrations at Q1 and Q2 as well as the difference in time,

τ , that it takes these vibrations to reach P . In other words, interference will depend on

a form of (1.4) known as the mutual coherence function given by:


A B

PR1

R2

Q1

Q2

Figure 1.2: Young’s Interference Experiment. Adapted from figure 3.1 in [1].

Γ(Q1, Q2, τ) = 〈V ∗(Q1, t)V (Q2, t+ τ)〉 , (1.8)

where V (Qi, t) corresponds to vibrations coming from Qi. Γ(Q1, Q2, τ) is considered a

second order coherence function because of the fact that a product of two quantities is

involved in its definition. A normalized version of (1.8) is:

γ(Q1, Q2, τ) =Γ(Q1, Q2, τ)√

Γ(Q1, Q1, 0)√

Γ(Q2, Q2, 0). (1.9)

γ(Q1, Q2, τ) is known as the complex degree of coherence and obeys the relation 0 ≤

|γ(Q1, Q2, τ)| ≤ 1. If the vibrations arriving at P are highly correlated, then γ(Q1, Q2, τ)

is large and the interference pattern will be sharp. Similarity, small γ(Q1, Q2, τ) implies

that the fringes will not be very sharp and will be more difficult to distinguish. That is

why the visibility of fringes is defined by:

V = γ(Q1, Q2, τ). (1.10)


An application of the concept of the degree of coherence, described here, is in Michel-

son Interferometers, for example. This class of interferometers can be used to measure

the diameter of stars. It is built in a way that allows it to receive radiation from a star,

filter through a single frequency component (of wavelength λ), and then use a set of

mirrors to focus the beam into a point. The focusing can be achieved by changing the

distance d between a set of mirrors. The far-zone van Cittert-Zernike theorem tells us [1]

that the equal time degree of coherence, which as I stated earlier can be determined by

the visibility of fringes, is the Fourier transform of the intensity distribution across the

source, in this case the star. Applying this result to the quasi-monochromatic source,

filtered star radiation, of radius a, it can be shown [1] that:

d =0.61Rλ

a(1.11)

where R is the distance to the star.

Another example in which the concept of coherence is used in in the Hanbury-Brown

Twiss Effect, which I elaborate on in the introduction of chapter 5. Briefly, it can be

shown that the correlation of intensity fluctuations of a beam at two points is equal to

the absolute value of the correlation. In other words, it depends on the absolute value

of the degree of coherence. Then using a very similar formula to (1.11), the radius of

the star can be obtained. However, the advantage is that since the absolute value of the

degree of coherence is required this means that phases from jiggling of equipment or from

atmospheric factors have no effect on the fringes, and will, therefore not affect the results

negatively, as can happen in a Michelson Interferometer.

The concepts discussed in this section are from Emil Wolf’s latest book [1]. More

details can be found there.


1.2 Quantum Correlations

1.2.1 Why do we know that quantum correlations exist and

how can we quantify them?

The job of theorists, as I view it, is to take the acceptable mathematical description of

the physical world, to make realistic assumptions, and then to see what effects can be

predicted. It is a very clean way of studying the world around us and can be done at

the comfort of one’s home. It also saves time and effort for the experimentalist who can

get an idea of what to expect in the lab and what is the optimal way of setting up an

experiment. At the same time, theorists need experimentalists to keep them honest and

also to give them a reason to justify the assumptions they make about the physical world.

In fact, in the cases when experimentalists observe effects that were not predicted by the

theory, not only do they discover a new field, but they also make theorists realize that

they need to reconsider their descriptions. This is a common way in which revolutions in

science occur. Science is there to help us understand the world we live in and to explore

ways to use the laws of nature to make our lives better. That is why I believe that

theorists should try as much as it is possible to come up with experimentally verifiable

ideas.

Roughly a century ago, certain observations such as those regarding the Black Body

Radiation and the Photoelectric Effect alerted the world that the accepted laws of nature

formulated by Isaac Newton can be violated. This led to the belief that a more general

formalism is required to describe the world; Newton’s Classical laws only applied to huge

macroscopic objects. This led to the formalism of Quantum Mechanics by scientists such

as Planck, Einstein, Bohr, Born, Schrodinger, Heisenberg, and the rest. The formalism

justified the observations made in the experiments. It also was a starting point for build-

ing quantum theory. Some of the most intriguing observations was regarding quantum

correlations.


Quantum Correlations, dominated in the last century by Quantum Entanglement,

define quantum properties. Only systems in which these correlations can exist can be

classified as being quantum [11]. Being in an entangled state in a multi-component

quantum system implies being in a superposition of possible states that cannot be factored

into a product of states corresponding to each of the components. Quantum theory tells

us that this attribute of quantum systems can be used to perform operations that cannot

be performed classically [12]. That is why demonstrating them can be thought of as being

equivalent to the assertion that quantum correlations exist. Experimental achievements

that demonstrate this are to date numerous. An example is the quantum teleportation

in atoms [13] in which the existence of entanglement was essential for the operation to be

performed. Also, determining a trace of a certain class of matrices was accomplished [3],

and again this was shown [4] to only occur if a correlation stronger than a classical one

is present in the system. Therefore, experiments so far are reassuring us that quantum

correlations exist, but we are still not very clear what these correlations are or what the

optimal way to study them is.

In this section, I give few examples of how quantum correlations can be quantified,

commenting where appropriate on limitations. These include: Violation of Bell’s Inequal-

ities, Partial Transpose and Separability, Wootters’ Concurrence, Conjugate Variable

Criterion for Separability, Discord, and MID.

Entanglement using Bell’s Inequality

To motivate the idea of defining entanglement using Bell’s Inequalities, let us focus on a

system that consists of three coins: one nickel, one dime, and one loonie 1, an example

inspired by a talk [14] given by Joseph Eberly. The idea is to shake the three coins in

one’s hand and then to look at the outcome; i.e., whether heads or tails were obtained for

each of the coins. Note that no assumptions are made about the individual probabilities

1The loonie is a colloquial term for the Canadian one dollar coin.


of the outcomes for each coin; throughout the experiment, the only assumption is that the

probabilities are between 0 and 1. There are many different probabilities one can choose

to study in such an experiment, but here we will only be interested in three probabilities

(each involving two coins at a time):

1) Prob(Hn, Td), the probability of obtaining heads for the nickel and tails for the dime,

2) Prob(Hd, Tl), the probability of obtaining heads for the dime and tails for the loonie,

and

3) Prob(Hn, Tl) the probability of obtaining heads for the nickel and tails for the loonie,

where H stands for heads, T for tails, n for nickel, d for dime, and l for loonie.

All these probabilities describe the outcomes for two of the coins only, which can be

described in the three-coins probability functions as follows:

Prob(Hn, Td) = Prob(Hn, Td, Hl) + Prob(Hn, Td, Tl),

P rob(Hd, Tl) = Prob(Hn, Hd, Tl) + Prob(Tn, Hd, Tl),

P rob(Hn, Tl) = Prob(Hn, Td, Tl) + Prob(Hn, Hd, Tl). (1.12)

If we add the first two equations in (1.12), we obtain:

Prob(Hn, Td) + Prob(Hd, Tl) = Prob(Hn, Td, Hl) + Prob(Tn, Hd, Tl)

+ Prob(Hn, Td, Tl) + Prob(Hn, Hd, Tl).

(1.13)

Notice that the right hand side of (1.13) is almost the same as the right hand side of the

third equation in (1.12), except for the extra terms Prob(Hn, Td, Tl) and Prob(Hn, Hd, Tl).

These two quantities are always greater than or equal to zero, which implies the following

relation:

Prob(Hn, Td) + Prob(Hd, Tl) ≥ Prob(Hn, Tl). (1.14)


This inequality is an example of the sort of inequalities that Bell introduced in his famous

1964 paper [15], known as Bell’s inequalities, named after the physicist John Bell, and

it tells us that the sum of the probabilities of obtaining H for the nickel with T for the

dime and H for the dime with T for the loonie is always going to be greater than the

probability of obtaining H for the nickel with T for the loonie. Although this is always

going to be true in the classical systems of coins, in quantum systems when analogous

equations are constructed, a similar inequality, which also falls under the class of Bell’s

inequalities, can be violated. Physicists justify this by the existence of other stronger

non classical correlations; i.e., quantum correlations. To elaborate on this, an example

given by Eberly in [2] will be used, from which the analysis, with minor modifications is

taken.

The first experimental work to be published in which the violation of Bell’s inequality

was demonstrated is by Clauser and Freedman using photon polarization [16]. In the

years following this work, the violation was demonstrated again in improved experimental

setups such as the one done by Aspect et al. [17]. The experiment I focus on is called

the Clauser experiment, as it was inspired by his work [16]. See Fig. 1.3. Here the three

classical coins are replaced by three setups involving photon polarization detection. In

each set up, a photon pair is created with each moving in the opposite direction toward

a set of analyzer loops. As described in the figure caption, each analyzer takes a beam,

splits into into two orthogonal components, and then recombines the components before

they exit. Blocking one of the components just before exiting ensures that only the

component orthogonal to it leaves the loop. The three possible orthogonal pairs in this

setup are in the: x and y directions, θ and θ directions, or φ and φ directions.

The photon pairs are emitted at the same time in opposite directions and with or-

thogonal polarizations. An example would be the following:

1√2

(|xy〉+ |yx)〉 , (1.15)


Figure 1.3: Clauser Experiment. Taken from figure 2 in [2]. This figure shows three

experimental setups of a source of two photons (from parametric down-conversion, for

example) with three polarization analyzer loops to its left and two to its right. The

analyzers labeled with x and y take an incoming beam, split it into the two orthogonal

components polarized in the x and y directions, and then recombines them before they

exit the loop. The same argument applies for the loops labeled θ and θ(= π − θ), and φ

and φ(= π−φ), where θ and φ corresponds to the angles of the polarization with respect

to the x-axis. The vertical thick line that can be seen at some ends of the analyzers block

the corresponding component so that in the recombining process just before exiting the

loop that component is absent. For example, in the top most set up, the second analyzer

loop to the left of the photon source has its x-polarized component blocked. This means

that the component that will be allowed to exit will be in the y-direction


where |xy〉 is the state in which the first photon (traveling to the left) is polarized in the

x-direction, and the second photon (traveling to the right) is polarized in the y-direction.

The term |yx〉 is defined in a similar way. The state in (1.15) is the state that is used as

the initial state of the photons in the following discussion.

Let us look at the setup labeled (a). If our state is prepared in (1.15), then by

blocking the x in the loop to the left of the photon means that when the detector fires

it is the y-component that gets detected. Here, θ and θ do not matter since none are

blocked. The important point is that when the left detector does actually fire signaling

that the left moving photon is y-polarized, we know that the photon moving to the right

of the source must be x-polarized. An interesting question to ask is for each time the left

detector fires, what is the fraction of times that the right detector fires. This quantity

is labeled f(x, φ) and refers to the fact that the x-polarized photon is detected in the φ

direction (since φ is blocked). Since by leaving both θ and θ unblocked, we did not ask

through which the photon passed, these two polarization directions do not matter, and

f(x, φ) = f(x, θ, φ) + f(x, θ, φ). (1.16)

where f(x, θ, φ) refers to the probability of the photon going through ports x, θ, and φ

before getting detected, and f(x, θ, φ) is defined likewise. With similar arguments one

obtains the following relations for setups in (b) and (c), respectively:

f(y, θ) = f(y, θ, φ) + f(x, θ, φ), (1.17)

f(θ, φ) = f(x, θ, φ) + f(y, θ, φ). (1.18)

Adding (1.16) and (1.17), comparing the sum to (1.18), and using the same argument as

in the three-coins experiment, the following Bell inequality is obtained:

f(x, φ) + f(y, θ) ≥ f(θ, φ). (1.19)


By applying Malus’ law, which tells us that the fraction of light that passes through a

polarizer is cos2(θ), where θ is the angle between the direction of the polarization of light

with respect to the polarizers’ angle, each term in (1.19) is given as follows:

f(x, φ) = cos2φ,

f(y, θ) = cos2(π − θ) = sin2θ,

f(θ, φ) = cos2(φ− θ), (1.20)

Substituting (1.20) into (1.19) results in the following:

cos2φ+ sin2θ ≥ cos2(φ− θ). (1.21)

If we look at the special case when φ = 2θ, (1.21) becomes:

cos22θ ≥ cos2θ. (1.22)

Mathematics tells us that (1.22) not true for 0 < θ < π/4. This means that there exists

a range of values for θ such that the Bell Inequality in (1.19) is violated !

The violation comes from the fact that we are trying to describe these quantum

system of entangled photons (moving in opposite directions) in an analogous way, in fact

in a one-to-one correspondence, to the classical coin experiment I described above. For

example, by writing down a statement such as (1.16), which involved separate terms for

θ and θ, we are asserting that there is a distinction between the paths (θ or θ) and that

the events that occur in these two paths are independent of each other. However, in the

general case; i.e., beyond the classical case, we cannot rule out that when dealing with

probability amplitudes of different events, their sum involves cross terms. We cannot

assume that the probability associated with moving into the θ path is independent of

the one associated with the θ path by writing an expression like (1.16) in which the two


cases are assumed to be independent of each other. That is why using the classical eye

to view the quantum problem can lead to contradictions such as in (1.22).

Generally speaking, many different Bell’s Inequalities can be constructed to study

quantum correlations in systems, and we know with confidence that when the inequality

is violated, this is due to the presence of nonclassical properties. However, if the inequality

is not violated, that does not necessarily imply the absence of nonclassical correlations.

Partial Transpose Separability Criterion for Entanglement–necessary, but not

sufficient

A density matrix, usually represented by ρ, is a common and general representation of

states in quantum systems. Some of its properties include unit trace and positivity of

eigenvalues. The method [18] I describe here detects entanglement between two subsys-

tems by performing an operation over one of them and checking the physical properties

of the density matrix of the whole system.

The operation performed is called partial transpose, which involves transposing the

density matrix of ρ over one subsystem only. The idea is that if the two systems are not

quantum mechanically correlated, then transposing over one of them, while ignoring the

presence of the other, is an acceptable action, which should not affect the properties of

ρ. However, if these correlations exist, then there is a chance that such an action can

lead to spoiling the properties of ρ, specifically the positivity of the eigenvalues.

Using equations, here is a brief description of the criterion. The density matrix of a

bipartite system (System C = System A + System B) may be written as:

ρ =∑

i

ciρAi ⊗ ρB

i , (1.23)

where ρAi and ρB

i are density matrices involved in describing subsystem A and subsystem

B, respectively, and ⊗ is the symbol for tensor product. Taking the partial transpose

over one of its subsystems one obtains:


σ =∑

i

ci(ρAi )T ⊗ ρB

i . (1.24)

If σ has at least one negative eigenvalue, then we know with certainty that the system

is entangled. However, if none of the eigenvalues are negative, then the system could be

entangled or separable.

This method can be used for any bipartite quantum system, but one has to keep in

mind that positive eigenvalues of (1.24) provide a necessary, but not sufficient criterion

for separability.

1.2.2 Entanglement in Two-Qubit Systems

Concurrence–the solution?

A very common approach in quantifying quantum correlations in a system is to tamper

with it, look at how this action affects the system, and then take the worse case scenario

and define the correlation using that case. The meaning of this statement will become

clear with the example of defining concurrence in this section and quantum discord in

the following one.

If we are given a system of two qubits (labeled A and B, for instance); i.e., a bipartite

system of two-level quantum systems, then its density matrix representation is given by

a 4 x 4 matrix, ρ. If we wanted to focus on one subsystem only, say A, then all we have

to do is trace over subsystem B and we will end up with the 2 x 2 density matrix that

represents subsystem A, also known as the reduced density matrix of A, ρAred. The same

argument can be applied if we want to focus on subsystem B only. Mathematically the

relationship between ρ and ρred is given as follows:

ρred =1∑

m,n,i=0

(|m〉〈n|) 〈m, i| ρ |n, i〉 , (1.25)

where i labels the basis for the system to be traced over, and m and n the remaining


system. However one has to keep in mind that this tracing over a basis is equivalent to

throwing away information. When does this discarding of information about subsystem

B matter to subsystem A and how can we find out if it matters?

It is when systems A and B are quantumly correlated that they matter to each

other, and one cannot just forget about the existence of one system while studying the

other without losing some information. Now, the loss of this information introduces a

randomness to a system if we trace over a system to which it is entangled (in our example,

trace over B when A is entangled to it). This can be quantified simply by calculating the

von Neumann entropy of the density matrix representing system A, defined as follows:

S = −∑

i

pi ln(pi), (1.26)

where pi are the eigenvalues of the density matrix of A. Note that this formula can

be used to find the entropy of any density matrix of interest. If the two systems are

correlated, then the entropy of the reduced density matrix of A, which is also known as

the Entanglement Entropy (of the whole system: A + B), is going to be greater than zero.

However, if the systems are not correlated, then the entropy will be zero. The entropy

S of ρred is a monotonic function of Det(ρred), the determinant of ρred. Concurrence, in

the pure case, is defined in terms of this determinant as follows:

C = 2√

Det(ρred). (1.27)

In fact with some algebra, one can show that for a pure state given by:

|ψ〉 = α |00〉+ β |01〉+ γ |10〉+ δ |11〉 , (1.28)

equation, (1.27) gives:

C = 2|αδ − βγ|. (1.29)


It is clear from the proceeding discussion that concurrence is a monotonic function of

the Entanglement Entropy, except that the way it is defined allows it to have values

between 0 and 1 only, which corresponds to the two extreme cases of no entanglement

and maximal entanglement, respectively.

The concept described above only applies if the density matrix represents a pure state,

which are states with zero entropy. In [19, 20], the definition of concurrence is extended

to the mixed state case using an optimization I hinted about in the introduction of this

section. It extends the result in the pure state case to the mixed state case by using the

connection between them, which is that mixed states have pure state decompositions:

ρ =∑

i

pi|ψi〉〈ψi|. (1.30)

Here ρ is the mixed state and |ψi〉 are the pure states involved in the decomposition

of ρ. However, it is important to remember that these decompositions are not unique.

Therefore, if one is not careful, different results for concurrence can be obtained for the

same system. To overcome the problem, the entanglement of a mixed state is defined to

be the average of the entanglement of the pure states involved in the decomposition of

ρ, minimized over all decompositions of ρ:

E(ρ) = min∑

i

piE(ψi), (1.31)

Here, E(ψi) is the Entanglement Entropy of the pure state |ψi〉〈ψi| involved in the de-

composition of ρ. E(ρ) is also known as the Entanglement of Formation.

Based on this idea, Wootters constructed a recipe to find concurrence for the most

general density matrices, the details of this construction can be found in [19]. The starting

point is to look at the R matrix defined as follows:

R = ρ(σy ⊗ σy)ρ∗(σy ⊗ σy), (1.32)


where σy is the Pauli spin matrix given by

σy =

0 −i

i 0

, (1.33)

and ⊗ is the symbol for tensor product. Then, λi, the eigenvalues of R are found.

Concurrence is defined to be:

C = max0,√λ1 −

√λ2 −

√λ3 −

√λ4

, (1.34)

where the square roots of the eigenvalues in the difference start from the largest to the

smallest. As in the earlier case of pure states, 0 ≤ C ≤ 1.

Notice that the definition of concurrence in the general mixed state case depends on

a minimization which results in the worst possible value for concurrence to be taken in

order to be on the safe side, but this does not necessarily mean that we are getting all

the information possible about the correlations we are seeking to quantify.

1.2.3 Entanglement in Bipartite Harmonic Oscillator Systems

Conjugate Variables Criterion–necessary and sufficient for entanglement in

Gaussian States

A quantum harmonic oscillator (QHO) is a good example of the continuous variable

quantum systems that I discuss here. In this case, the energy levels are discrete and are

separated by the same amount, but there are an infinite number of them (See Fig. 1.4).

How does the QHO show up in the treatment of the electric field vector ~E that describes

a light beam?

This happens when ~E is quantized in free space. Here Maxwell’s equations are given

by the following:

~∇× ~H =∂ ~D

∂t,


h=

h=

Figure 1.4: Quantum Harmonic Oscillator. The energy spectrum is not continuous,

but discrete. However, there is an infinite number of energy levels.

~∇× ~E = −∂~B

∂t,

~∇ · ~B = 0,

~∇ · ~D = 0, (1.35)

where ~B = µ0~H and ~D = ε0 ~E. The symbols ~B, µ0, and ε0 correspond to the magnetic

field, the permeability of free space, and the permittivity of free space, respectively.

Mathematical manipulation of the equations in (1.35) yields the following wave equation

for the field ~E(~r, t):

~∇2 ~E − 1

c2∂2 ~E

∂t2= 0,

~∇. ~E = 0. (1.36)

In general the solution of (1.36) can be written as a linear combination of plane waves

as follows:


~E(~r, t) =∑j

~Ajqj(t)sin(~kj ·~r), (1.37)

where ~k stands for the wavenumber, and the index j labels the j-th mode in the sum.

The vector ~r labels a location in space, while the time-dependent function qj(t) maps

out how the field amplitude changes as a function of time. Field quantization takes qj(t),

treats it as a position operator, and uses the relation between position and the creation

and annihilation operators, which is given as follows:

X =

√h

2mω

(a† + a

), (1.38)

where a† and a, are the creation and annihilation operators, respectively. In other words,

quantization of the field takes qj(t) and treats it as a linear combination of a and a†.

Notice that this QHO description is motivated by the fact that the time-dependent part

of (1.37) obeys the classical harmonic oscillator equation (See (1.43)). It can be shown

(See for example [21], from which the description of field quantization here is adapted,

for more details) that the quantized electric field ~E(~r, t) takes the form:

~E(~r, t) =∑~k

ε~kE~kake−iωkt+i~k·~r + c.c., (1.39)

where k labels the k-th mode and is the wavenumber vector for that mode, ε~k is the

unit polarization vector, E~k is the amplitude associated with the k-th mode, ω~k is the

frequency of the k-th mode, and c.c. stands for complex conjugate.

Let us say that we are interested in only one mode of the field and only in its amplitude

(i.e., we ignore the r-dependence), then (1.39) takes the form:

~E(t) = E ε(ae−iωt + a†eiωt

). (1.40)

Now, let us introduce the (dimensionless) operators:


X1 =1

2

(a+ a†

),

P1 =1

2i

(a− a†

). (1.41)

From the relation[a, a†

]= 1, it can be shown that [X1, P1] = i

2. In other words, X1 and

P1 are conjugate operators, like position and momentum. In terms of these operators,

(1.40) takes the form:

~E(t) = 2E ε (X1cos(ωt) + P1sin(ωt)) . (1.42)

The solution for the general harmonic oscillator equation x(t) + ω20x(t) = 0 (where ω0

corresponds to the frequency) is given by:

x(t) = Acos(ω0t) +Bsin(ω0t), (1.43)

where A and B are constants determined by the initial conditions. They can also be

thought of as the amplitude of x(t) that are multiplying the cos(ω0t) and the sin(ω0t)

components in the sum, respectively.

What (1.42) tells us is that the amplitudes in the quantum mechanical treatment

is given by the two conjugate variables X1 and P1 whose variance obeys Heisenberg’s

Uncertainty Relation. This means that if we try to know more about one amplitude,

then we would know less about the other, and vice versa.

The motivation behind giving this description of field quantization is to discuss how

entanglement can be defined for harmonic oscillator systems, also known as continuous

variable quantum systems. Specifically, I discuss the case that involves two of those

(quantum) oscillators as described in [22].

In their paper [22], Duan et al. propose a measure of entanglement for a bipartite

continuous variable quantum system, which involves looking at conjugate variables of

the two systems. Let us say the conjugate variables of system i are given by xi and pi,


without the implication that they have a position or momentum meaning associated with

them. Then, two new conjugate variable can be defined in terms of the xi’s and pi’s of

the two systems as follows:

u = |a| x1 +1

ax2,

v = |a| p1 +1

ap2, (1.44)

where a is an arbitrary nonzero real number. For the states to be separable, the variances

of the conjugate variables in (1.44) have to obey the following:

⟨(∆u)2

⟩ρ+⟨(∆v)2

⟩ρ≥ a2 +

1

a2, (1.45)

where ρ is the density matrix describing the whole system (system 1 + system 2), and

〈...〉ρ is the average over ρ. However, unless the states are Gaussian, which means that

their Wigner function description is a Gaussian, in general the violation of the inequality

is only a necessary, but not a sufficient condition for separability. This means that there

are states that are entangled and that satisfy (1.45).

1.2.4 Quantum Discord, MID, and Ameliorated MID

Discord

In quantum mechanics, we know that measurements disturb systems. In the most general

case, we do not know the state of a system. The result we obtain by measuring a quantum

system merely tells us the state that we collapsed the system into by allowing it to interact

with the measuring apparatus. This on the other hand is not true in the classical case.

For example, when we measure the position or velocity of a classical system, we know

that the values we obtain accurately describe the state; i.e., the interaction with the

measuring apparatus does not disrupt the state in any way.


The idea behind the definition of quantum discord, which was introduced by Zurek

and Ollivier [23], is to use this very attribute of quantum systems, of being disturbed by

measurements, as a test to whether quantum correlations in a bipartite two-level quantum

system (System C = System A + System B) are present or not. To monitor the effect

of measurements, the mutual information function is used as the witness. This function

serves as a measure of how much information is shared between the two subsystems. In

other words, it gives an indication of how correlated the systems are.

To test for the existence of quantum correlations, a measurement is performed on

one of the subsystems (say system B). Then, by looking at what happens to the mutual

information function, one can tell whether these quantum correlations are present or

not. This is achieved by comparing the mutual information function before and after

the measurement (by taking the difference). However, since the set of measurements

that can be performed on the subsystem is not unique, the definition of quantum discord

is finalized only after optimizing the expression for the measurement-induced mutual

information function over all possible measurements.

In Chapter 4, a detailed mathematical description of quantum discord is given with an

explicit way of how to define it to numerically calculate it for the most general bipartite

two-level quantum system, but here I give a very brief description to introduce the

variables involved in defining the quantities of interest in studies made in this thesis.

Given the mutual information function, I(ρ), and the measurement-induced mutual

information function maximized over all possible measurements Bk on B, C(ρ), the

quantum discord, Q(ρ), is defined as Q(ρ) = I(ρ) − C(ρ). Quantum discord, defined

in this manner, can take numerical values between 0 and 1, where the former boundary

corresponds to the case with no discord and the latter case to that with maximal discord.


MID and Ameliorated MID

Quantum Discord, as described in the previous subsection will give different results if

the measurements were performed on system A rather than system B. This means that

the discord of system A with respect to system B is not, in general, the same as that of

system B with respect to system A. To give an example, I use the experiment described

in [3] which is shown in Fig. 1.5. In this experiment, the purity of the control qubit,

which is related to the amount of quantum correlations present in the whole system, is

what makes the algorithm more powerful than any classical algorithm. If one were to

calculate quantum discord in this system by performing the measurement described in

the previous subsection on ρr, one can show that discord in this case is persistently equal

to zero. However, performing this measurement on ρc yield an oscillating discord (as a

function of θ) as can be see in Fig. 1.6. In this case, one can think of the correlations being

encoded in ρc and not the highly mixed ρr, so it makes sense to induce a measurement

disturbance on the latter to see how that affects quantum correlations. As this example

shows, to define discord one has to think each time a new case is presented.

To eliminate the step of deciding which subsystem one should perform a measurement

on when defining discord, MID seems to provide the solution. MID, which stands for

Measurement-Induced Disturbance, was introduced by Luo [24] as a more symmetrized

version of discord. Now, the measurement is performed on both subsystems. One way to

express this mathematically is given in [25], which gives MID as follows:

M(ρ) = I(ρ)− I(P (ρ)), (1.46)

where

P (ρ) =M∑i=1

N∑j=1

(ΠA

i ⊗ ΠBj

)ρ(ΠA

i ⊗ ΠBj

), (1.47)

where ΠAi and ΠB

j are the projectors constructed using the eigenvectors of system A and


Figure 1.5: Quantum Computation Experiment in the Absence of Entangle-

ment. Taken from figure 2 in [3]. In this experiment, the DQC1 (Deterministic Quan-

tum Computation with One Pure State) algorithm [4] is implemented using photons. The

control qubit, ρc, is a pure state, and the register qubit ρr represents a maximally mixed

state, which is intially pure, but is rendered mixed by introducing a phase delay between

the perpendicular components of the field. Both qubits are encoded in polarization states

of single photons. The purpose of this algorithm is to estimate the normalized trace of

the rotation operator Zθ, where θ is the rotation angle. The power of this algorithm

comes from the purity of ρc.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−π −π/2 0 π/2 π

Quantu

mC

orr

elations

θ

MIDDiscord

Figure 1.6: Discord and MID versus the Rotation Angle θ. Notice that discord is

always finite, except for the few cases when θ = 2πn, where n is an integer, as was shown

in [3]. It was also found in [3], that these points have no quantum advantage and can

be stimulated efficiently classically. However, it is interesting to note that for some of

the points where discord is zero, specifically for those with θ = 2πm, where m is zero or

an even integer, MID is maximal, which raises questions with regard its definition since

some of these points have already been shown to have no quantum advantage. This is

a work done in collaboration with Andrew White, but we found out that an equivalent

observation about MID was made earlier in [5].


system B, respectively. Although the idea of symmetrizing discord makes it easier to

define it in general, it is not clear why selecting this specific set of projectors is ideal. In

fact, as can be seen in Fig. 1.6, this way of defining MID results in it having a maximal

value at points which are known to have no quantum advantage. This shows that the

definition of MID has to be revised to make more sense. In [5], the problem was fixed by

using a similar approach as in the definition of discord: a preference was not given to any

set of projectors, but measurements were optimized over all projectors acting on systems

A and B. This new version of MID was called: Ameliorated MID. Using this result the

authors then produced a plot describing the general relation between Ameliorated MID

and Discord for the bipartite two-level quantum system.

1.2.5 Other methods and final words

Until the beginning of this millennium, the most popular way to define quantum cor-

relations was using entanglement. In fact, the matter of whether quantum correlations

existed in a system or did not was dealt with simply as a case of the existence or ab-

sence of entanglement. The latter case corresponding to separable states, with Werner’s

method to determine separability becoming popular [7]. However, ten years ago, with

the introduction of Quantum Discord by Zurek and Ollivier [23], in which they showed

that states that are not entangled can still have quantum correlations, it became clear

that the problem is more complex than was thought originally. It was also shown [5]

that Ameliorated MID can capture more correlations than discord does. Also, Modi

et al. introduced [26] another correlation, which they call dissonance. They show that

states that have discord, but no entanglement have dissonance, and they claim that the

correlations named entanglement and dissonance when put together give us discord.

In the last decade, we have become more aware of how complex the problem of

quantifying quantum correlations is and that we are not close to understanding the

nature of these correlations. Most of the work done so far is based on ideas that we


feel confident about, such as measurements disturbing a system in the case of quantum

discord. However, every work seems to start from one known point, and what is lacking

is how to unify the various efforts that are being put into this problem. If the problem

of quantifying quantum correlations can be thought of as a box, then each one of those

involved in this study is looking at a spot on the box. If we can find ways to connect

all these spots, then we will end up with a rough shape of this box and, hopefully, get a

better understanding of the problem as well as the actual correlations.

Chapter 2

Sudden Death of Entanglement at

Finite Temperature

2.1 Introduction

In the past few years there has been considerable interest in the properties of entangled

quantum systems. Spurred on by the emergence of compelling applications in quantum

information processing, useful methods by which the entanglement of quantum systems

can be established and characterized have emerged. Perhaps the most impactful to

date has been the simple procedure derived by Wootters [19], which was described in

Chapter 1 of this thesis, for quantifying entanglement for an arbitrary mixed state of

a pair of two-level systems. This has provided a very useful tool for measurement of

experimental quantum states [27]. Building on Wootter’s work, recently Yu and Eberly

[28] investigated the time evolution of entanglement (quantified using the concurrence)

of a bipartite qubit system undergoing various modes of decoherence. Remarkably, they

found that, even when there is no interaction, (either directly or through a correlated

environment), there are certain states whose entanglement decays exponentially with

time, while for other closely related states, the entanglement vanishes completely in a

30

Chapter 2. Sudden Death of Entanglement at Finite Temperature 31

finite time. This “entanglement sudden death” (ESD) is an intriguing discovery. Nor

is this effect limited to the case of two qubit systems: prior to Yu and Eberly’s work,

Diosi [29] demonstrated, using Werner’s criteria for separability [7], that ESD occurs in

two-state quantum systems. Further investigations of different systems have been made

by various groups [30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Extending Yu and Eberly’s

model by considering correlated reservoirs and interactions [40, 31, 36, 39, 38, 34], it was

shown that entanglement may be created by spontaneous emission (something which has

been known for some time [41] in a different context). The ESD has also been predicted

for more complicated systems using other entanglement measures [42, 43, 44], and an

attempt to give a geometric interpretation for the phenomena of ESD has also been

made [45]. Very recently, experimental studies have also been carried out to demonstrate

ESD, using carefully engineered interactions between systems and environments: Sudden

death has been observed both in photons [46], the method being proposed in [47], and

in atomic ensembles [48].

The entropy of systems undergoing irreversible dynamics increases; further, as has

been established some time ago, there is a limit in the amount of entanglement that can

be present in a mixed system [49]: the more mixed a state is, the less entangled it can be,

and when the entropy reaches a certain level, entanglement will necessarily disappear.

However, these heuristic arguments do not tell us the time taken for entanglement to

disappear, which cannot be answered without careful study of dynamics.

One might think that, from the quantum technological point of view, states which

exhibit exponential decay of entanglement, and therefore retain some vestige of this all-

important correlation for long periods, are of significance. Although the vanishingly

small entanglement present in the exponential tail will be of little practical importance,

nevertheless it is of interest to identify precisely in what circumstances ESD will occur.

In this chapter, I consider qubits in finite temperature reservoirs: instead of the energy

of the qubits being lost via spontaneous decay to the environment, now additionally the


Reservoir

0

1

A

T≠0

Reservoir

0

1

B

Figure 2.1: Disentanglement by spontaneous emission of a two-qubit system in

a heat bath. The reservoir is modelled by different harmonic oscillator modes. Each

qubit, here depicted by a two-level atom, interacts with its reservoir. The only interaction

between the qubits that ever exists is the one that leads to their entanglement at time

t = 0. Following this, however, the only interaction that remains is that with the cor-

responding reservoir. This leads to decoherence, which causes the qubits to disentangle.

Here I include the effect of heat by studying the system at T 6= 0.

reservoirs can cause excitation of the qubits. For a broad class of mixed quantum states,

which includes all of the states studied by Yu and Eberly and others in connection with

this problem, I demonstrate that all states undergo sudden death of entanglement at finite

temperature.

2.2 Two-qubit model system

As in [28], I study a system of two qubits initially entangled and interacting with uncor-

related reservoirs. However, unlike [28], in which the system is studied at temperature

T=0, I include the effects of heat in our system (Fig. 2.1). Here the dynamics of the

density matrix ρ describing the two qubits is given by:


∂ρ

∂t=

1

ih[H, ρ] + L1[ρ] + L2[ρ], (2.1)

where [H, ρ] is the unitary part of the evolution (which I shall ignore as it has no effect

on our study of decoherence). The Liouvillian of the ith qubit is given by:

Li[ρ] =(n+ 1)Γ

2

[σi−, ρσ

i+

]+[σi−ρ, σ

i+

]+nΓ

2

[σi

+, ρσi−

]+[σi

+ρ, σi−

], (2.2)

where Γ is the spontaneous decay rate of the qubits, σi+ = (|1〉〈0|)i, and σi

− = (|0〉〈1|)i,

where the index i ∈ 1, 2 denotes the qubits. The first term on the right hand side of

(2.2) corresponds to the depopulation of the atoms due to stimulated and spontaneous

emissions, while the second term describes the re-excitations caused by the finite tem-

perature; and n is the mean occupation number of the reservoir (assumed to be the same

for both qubits), and is given explicitly by the following:

n =1

ehω/kBT − 1, (2.3)

where ω is the difference between the energy level of the atoms, kB is the Boltzmann’s

constant, and T is the temperature of the reservoir. Equation (2.3) is also known as the

Bose-Einstein distribution function.

I assume that the system is initially an “X-state” described, in the computational

basis, by the following density matrix:

ρ(t) =

a(t) 0 0 w(t)

0 b(t) z(t) 0

0 z∗(t) c(t) 0

w∗(t) 0 0 d(t)

. (2.4)

Such states are general enough to include states such as the Werner states, the maximally

entangled mixed states (MEMS) [49], the Bell States, and what I will refer to as the ρjoe

states, studied in [28] and which will be described later.


Substituting (2.4) into (2.1), the Master equation of our system, I obtain the following

first order coupled differential equations:

a(t) = Γ [−2(n+ 1)a(t) + b(t)n+ c(t)n] ,

b(t) = Γ [(n+ 1)a(t)− (2n+ 1)b(t) + nd(t)] ,

c(t) = Γ [(n+ 1)a(t)− (2n+ 1)c(t) + nd(t)] ,

d(t) = Γ [(n+ 1)b(t) + (n+ 1)c(t)− 2nd(t)] ,

z(t) = Γ [−(2n+ 1)z(t)] ,

w(t) = Γ [−(2n+ 1)w(t)] .

(2.5)

The solutions for z(t) and w(t) can easily be obtained using elementary rules in solving

differential equations, and are found to be:

w(t) = w0X,

z(t) = z0X,

(2.6)

where X = e−Γ(2n+1)t, w0 =w(0), and z0 =z(0).

To solve for the remaining four variables, I write the set of equations pertaining to

them in (2.5), as follows:

d

dt|ψ(t)〉 = M |ψ(t)〉 , (2.7)

where


|ψ(t)〉 =

a(t)

b(t)

c(t)

d(t)

, (2.8)

and

M = Γ

−2(n+ 1) n n 0

n+ 1 −(2n+ 1) 0 n

n+ 1 0 −(2n+ 1) n

0 n+ 1 n+ 1 −2n

. (2.9)

Note that M is a non-symmetric matrix. By using properties of this class of matrices, a

solution can easily be obtained for |ψ(t)〉.

Given a non-symmetric matrix M , the effect of each of its right and left eigenvectors

are given by the following equations, respectively :

M |ri〉 = λi |ri〉 ,

〈li|M = 〈li|λi,

(2.10)

where λi is the eigenvalue corresponding to the eigenvectors |ri〉 and |li〉. The last equa-

tion in (2.10) implies that M † |li〉 = λ∗i |li〉. The following are some properties of these

eigenvectors of M [50]:

〈li| ri〉 = δij,

I =∑

i

|ri〉〈li| ,

M =∑

i

λi |ri〉〈li| ,

(2.11)


where δij is a Kronecker delta function and I is the identity operator. By using the

relations in (2.11), one can solve the differential equation (2.7) to yield the following:

|ψ(t)〉 =∑j

eλjt |ri〉〈li| ψ(0)〉 . (2.12)

The result in (2.12) tells us that to find |ψ(t)〉, we need to find the eigenvalues, the right

eigenvectors, and the left eigenvectors of the matrix M . I find the eigenvalues to be:

λi ∈ −(2n+ 1)Γ,−(2n+ 1)Γ, 0,−2(2n+ 1)Γ , (2.13)

the corresponding (un-normalized) right eigenvectors are:

|r1〉 =

−n/n+ 1

−1/n+ 1

0

1

,

|r2〉 =

0

−1

1

0

,

|r3〉 =

n2/(n+ 1)2

n/n+ 1

n/n+ 1

1

,

|r4〉 =

1

−1

−1

1

,


(2.14)

and the corresponding (un-normalized) left eigenvectors are given by:

|l1〉 =

−(n+ 1)/n

−1/n

0

1

,

|l2〉 =

0

−1

1

0

,

|l3〉 =

1

1

1

1

,

|l4〉 =

(n+ 1)2/n2

−(n+ 1)/n

−(n+ 1)/n

1

.

(2.15)

Notice that there is a degeneracy in the eigenvalues of M ; two of the eigenvalues are

−(2n + 1)Γ. This results in the corresponding eigenvectors not to behave as described

in the set of properties (2.11). For example, both 〈l1| r1〉 and 〈l2| r1〉 are non-zero. Also,

〈l3| r3〉 6= 1, but some constant C such that |C| > 1.

To solve the first problem, I redefine |r1〉 and |r2〉 as a linear combination of |r1〉 and

|r1〉 such that 〈li| rj〉 = δij.


The second problem described is a normalization issue that can be remedied by finding

〈li| ri〉 = Ci, and then redefining each of the right and left eigenvectors as follows:

|ri〉 =1√Ci

|ri〉

|li〉 =1√Ci

|li〉

(2.16)

Given the following constants:

α1 =2n(n+ 1)

2n(n+ 1) + 1,

α2 = − (n+ 1)

(2n+ 1)2,

α3 =n+ 1

2n+ 1,

α4 =n

2n+ 1,

β1 = − n

2n(n+ 1) + 1,

β2 =2n2 + 2n+ 1

(2n+ 1)2,

C1 =

√2n2 + 2n+ 1

(2n+ 1), (2.17)

the set of redefined right |Ri〉 eigenvectors and left |Li〉 eigenvectors are given in terms

of the old ones (2.14)(2.15) by:

|R1〉 = C1 α1 |r1〉+ β1 |r2〉 ,

|R2〉 = α2 |r1〉+ β2 |r2〉 ,

|R3〉 = α3 |r3〉 ,

|R4〉 = α4 |r4〉 ,

|L1〉 = C1 |l1〉 ,


|L2〉 = |l2〉 ,

|L3〉 = α3 |l3〉 ,

|L4〉 = α4 |l4〉 .

(2.18)

Using these eigenvectors in the result (2.12), I obtain the solutions for the rest of the

elements of the density matrix, which are given by the following expressions (including

the solutions for z(t) ans w(t)):

a(t) =1

(2n+ 1)2

n2 + [2(a0 − d0)n

2 + (a0 − d0 + 1)n]X

+[(2a0 + 2d0 − 1)n2 + (3a0 + d0 − 1)n+ a0]X2,

b(t) =1

(2n+ 1)2n(n+ 1)

−[2(a0 + 2c0 + d0 − 1)n2 + (a0 + 4c0 + 3d0 − 2)n+ (c0 + d0 − 1)]X

−[(2a0 + 2d0 − 1)n2 + (3a0 + d0 − 1)n+ a0]X2,

c(t) =1

(2n+ 1)2

n(n+ 1) + [2(a0 + 2c0 + d0 − 1)n2 + (3a0 + 4c0 + d0 − 2)n]X

−[(2a0 + 2d0 − 1)n2 + (3a0 + d0 − 1)n+ a0]X2,

d(t) =1

(2n+ 1)2

(n+ 1)2 − (n+ 1)[2n(a0 − d0) + (a0 − d0 + 1)]X

+[(2a0 + 2d0 − 1)n2 + (3a0 + d0 − 1)n+ a0]X2,

w(t) = w0X,

z(t) = z0X,

(2.19)

where X = e−Γ(2n+1)t, a0 =a(0), etc.

2.3 Solution for the Sudden Death Time

Using Wootters’ formula [19], the concurrence for a state of the form given by (2.4) is:


C = 2 max0, |z(t)| −

√a(t)d(t), |w(t)| −

√b(t)c(t)

. (2.20)

This implies that the disentanglement time will be the largest positive solutions of the

following equations:

|z(t)| −√a(t)d(t) = 0, |w(t)| −

√b(t)c(t) = 0. (2.21)

We cannot solve Eq. (2.21) in closed form. Further, since they are not polynomial

functions of X, neither can we make any straightforward deductions about the nature

of their roots. However, multiplying both equations in (2.21) by the positive quantities

|z(t)|+√a(t)d(t) and |w(t)|+

√b(t)c(t), respectively, yields:

|z(t)|2 − a(t)d(t) = 0, |w(t)|2 − b(t)c(t) = 0. (2.22)

Substituting from equation (2.22) gives two quartic equations in X, which I will use in

the proof of our main result.

The quantity X is the time-dependent parameter that I use to monitor the evolution

of entanglement in the system. Notice that at t = 0, X = 1, and that at t = ∞, X = 0.

Physically meaningful values for X are, therefore, between 0 and 1. Asymptotic decay of

entanglement implies a solution at X = 0. However, if the entanglement of the system

decays in a finite time (ESD), the solution of (2.22) must lie in the range 0 < X < 1.

Both equations in (2.22) are polynomial equations in X and continuous. At X = 0,

these equations take the following value:

−n2(n2 + 2n+ 1)

(2n+ 1)4. (2.23)

Notice that since n is a positive quantity, (2.23) is negative. On the other hand, if

we evaluate (2.22) for X = 1, which corresponds to t = 0, we obtain |z0|2 − a0d0 and

|w0|2−b0c0 for the first and second equations, respectively. If we assume that our systems

are initially entangled, at least one of these quantities has to be positive, which is the


10−1−2−4

0.75

−3

F(X)

−0.25

1.0

0.5

0.25

0.0

X

32

Figure 2.2: Plot of F(X) vs X. This is the plot of the first quartic equation in (2.22);

i.e., F (X) = |z(t)|2 − a(t)d(t), for n = 0.8, a0 = 0.1, d0 = 0.05, z0 = 0.3.

case here. The fact that the quartic equations are continuous and have a negative value

at X = 0 and a positive one at X = 1 implies that they have at least one root in our

interval of interest 0 < X < 1. (See Fig. 2.2 for an example illustrating this point).

For finite n, and therefore for finite T, the constant term (2.23) is always finite and

nonzero for n > 0. Hence, there is no X = 0 solution; i.e., no asymptotic decay. Thus

eq.(2.22) has at least one solution in the range 0 < X < 1, implying that the entanglement

falls to zero in a finite time.


2.4 An example

As an example, let us consider a special case when w(t) = 0 . In this case, the only quartic

equation that has to be satisfied for C = 0 is the first one in (2.22). This equation can

be easily solved with appropriate reparametrization. The four solutions for X are given

by:

X = (r + s)±√

(r + s)2 + q2,

X = (r − s)±√

(r − s)2 + q2. (2.24)

where

q =

√√√√ n(n+ 1)

(2n+ 1) [a0(n+ 1) + d0n]− n(n+ 1),

r =1 + (a0 − d0)(2n+ 1)

4 (2n+ 1) [a0(n+ 1) + d0n]− n(n+ 1),

s =(2n+ 1)2

√(1− a0 − d0)2 + 4(z2 − a0d0)

4 (2n+ 1) [a0(n+ 1) + d0n]− n(n+ 1).

Further, following (2.4), let us consider states of the form:

ρjoe =1

3

α 0 0 0

0 1 1 0

0 1 1 0

0 0 0 1− α

. (2.25)

In [28], Yu and Eberly have shown that for 0 ≤ α ≤ 13, the entanglement of this state is

long-lived at zero temperature. Here, I will illustrate that as soon as n becomes finite,

the range vanishes and there is no long-lived entanglement for any value of α.

For ρjoe, the physical solutions; i.e., those between 0 and 1, are only for X = (r −

s)±√

(r − s)2 + q2 in (2.24). Here, we have the following expressions for r, s, and q:

q =

√3n(n+ 1)

2αn− n2 + α− 2n,


0.01n=0n=

0.1n= 1n=

α α

α α

Figure 2.3: Plot of C (concurrence) vs X (= e−Γ(2n+1)t) vs α. C = 0 corresponds

to no entanglement. X = 1 corresponds to t = 0, while X = 1 corresponds to t = ∞.

Notice that as soon as n becomes finite, for all values of α, C becomes zero at X < 0;

i.e., entanglement decays in a finite time. As n becomes bigger, all states disentangle at

approximately X = 0.5.


r =1

2

1 + 2αn+ α− n

2αn− n2 + α− 2n

,

s =1

2

(2n+ 1)2

√2 + α2 − α

2αn− n2 + α− 2n

.

(2.26)

The first solution (with the plus sign) is valid for α > n(n + 2)/2n + 1, while the

second solution is valid for α < n(n+ 2)/2n+ 1. Examples are plotted in fig. 2.3.

As a specific physical example (described in fig. 2.2), which is experimentally acces-

sible with current technology, consider trapped ion qubits interacting with a thermal

reservoir of phonons [51]. Using equation (2.3), I find that with a temperature of 60µK,

for a 1 MHz trap, n=0.8. Here the physical solution exists forX = (r−s)−√

(r − s)2 + q2.

The disentanglement time can be calculated and shown to be: Tdis = −lnX0/(2n+1)Γ ≈

0.2/Γ, Γ being the controllable coupling parameter between the bath and the ions; with,

for example, Γ = 103s−1, I find a disentanglement time of 200µs.

2.5 Conclusion

In conclusion, in this chapter I presented a proof that in two-qubit systems interacting

with uncorrelated reservoirs and described by X-states, ESD always occurs at any finite

temperature. Although X-states are quite general states, they are not the most general

ones. Thus, the next question to ask is: do all states exhibit ESD? In other words, for

a state described by a density matrix, without any zero elements, is entanglement still

lost in a finite time? The answer is not straight forward. One has to find the equation

for concurrence in that case, determine its order, study the properties of its coefficients,

and from there maybe be able to comment on the nature of the roots, and, therefore,

be able to predict what will happen in the actual physical system. Alternatively, one

could adopt the approach of considering the general dynamics from the perspective of the

finite domain of separability surrounding the thermal equilibrium state [52]. In this case,


one must find a means for establishing that the evolution to the entanglement-separable

boundary occurs in finite time: in the mathematical language of this paper, this is

equivalent to proving that the concurrence is a single-valued function of the argument

X at X = 1. One important conclusion can nevertheless be drawn from our results: in

any finite temperature reservoir, all states that have been shown to be long-lived in a

zero-temperature bath, will undergo sudden death. Thus to draw the distinction between

sudden-death and exponentially long-lived entanglement seems to be redundant: in all

realistic circumstances, all entanglement disappears in a finite time.

Chapter 3

Sudden Death at High NOON

3.1 Introduction

Entanglement Sudden Death (ESD) [28], as described in the previous chapter, refers to

the loss of entanglement in a finite time. The work done so far, which mostly concerns

two-qubit systems, has shown, in one way, how fragile entanglement is in realistic systems.

Several papers have shown that ESD always occurs in some very general two qubit

systems. Examples include X-states; i.e., states with nonzero parameters (in general) on

the diagonal and antidiagonal of the density matrix of the system. In [30], it is shown

that for dephasing in X-states, there is always ESD as long as none of the parameters of

the density matrix are zero; in [53], which was also the subject of the previous chapter,

it is shown that for these states at finite temperature, and with depopulation going on,

ESD also always occurs; and in [54], it was demonstrated that external driving fields on

the system will enhance ESD. With results like these, one is tempted to make the guess

that ESD is actually a universal phenomena. So far an attempt has been made to prove

this in [52].

With all this work showing how prevalent ESD is in qubit systems, it is interesting

to ask how common it is in other quantum systems. For example, is ESD as common

46

Chapter 3. Sudden Death at High NOON 47

in Continuous Variable Quantum Systems (CVQS) as it is in qubits? Recently, ESD

has been shown to occur in a system of two free harmonic oscillators interacting with

a Markovian bath [55]. In addition, two initially Gaussian states, states with Gaussian

Wigner functions, coupled to the same (ohmic) environment have been studied in [56],

where the existence of three phases where demonstrated: ESD, ESD with revival, and

No ESD. Entanglement Sudden Death in CVQS systems has also been studied taking

into consideration relativistic effects [57].

In the work I present here, I prove the general result that ESD never occurs in two-

mode-N-photon states undergoing dephasing. NOON states, which have been shown

[6] to beat the Rayleigh limit in interferometry, falls under this general class of states.

The resolution of interference patterns improves when the separation between the wave

amplitudes falls down to λ/N compared to the minimum of λ forced by the Rayleigh

limit. The power of these NOON states lies in their entanglement. It is, therefore,

important to study the decay of entanglement in these systems. The approach I describe

here is studying ESD in such systems, trying to get a feel for the fragility of entanglement.

Finally, I touch on the important question: Does the existence of some entanglement for

a very long time have any practical implication on the usefulness of NOON states?

3.2 Dynamics of Two-Mode-N-Photon States Under-

going Dephasing

The system I consider is that of two harmonic oscillators with N photons shared between

them. The density matrix, most generally, describing such a system is given as follows:

ρ(t) =N∑

k=0

ρkk(t) |N − k, k〉〈N − k, k|+N∑

k,m=0,k 6=m

ρkm(t) |N − k, k〉〈N −m,m| . (3.1)

I deal only with dephasing, since applications of such states tend to be post-selective on


photon number: processes in which N changes are filtered away. If the system undergoes

pure dephasing, due to random fluctuation of the mode frequency, one expects that

the off-diagonal terms; i.e., ρkm(t), where k 6= m, in (3.1) will acquire decay terms.

On the other hand, the population, which is represented by the diagonal elements will

remain intact; i.e., the photon number will be preserved. It can be easily checked that

the Master equation describing the dynamics of such a system, in which there is no

correlation between the two fields interacting with the two harmonic oscillators, is given

by:

∂ρ(t)

∂t= 2 (Γ1 [n1, ρ(t)n1] + [n1ρ(t), n1] +Γ2 [n2, ρ(t)n2] + [n2ρ(t), n2]) , (3.2)

where Γi and ni are the decay rate and the number state operator of the ith harmonic

oscillator, respectively. By assumption there is no depopulation going on. By (3.2), the

evolution of (3.1) with respect to time in the case of pure dephasing is:

ρ(t) =N∑

k=0

ρkk(0) |N − k, k〉〈N − k, k|

+N∑

k,m=0,k 6=m

ρkm(0)e−12(k−m)2(Γ1+Γ2)t |N − k, k〉〈N −m,m| .

(3.3)

3.3 Measure of Entanglement for Photon States

As with other decoherence mechanisms, one side effect of dephasing is the decay of

entanglement. To find out whether this decay results in ESD or not, we need to use a

reliable measure for entanglement. In two-qubit systems, we have good measures such as

Wootter’s concurrence [19], which can tell with certainty whether a system is entangled

or separable. On the other hand, in the general two-CVQS, the best any measure can

do is provide a necessary but not sufficient condition for separability [18]. In the case


of CVQS, the sufficiency has only been proved for Gaussian states [22, 58], and NOON

states, the generalization of which I discuss here, do not belong to this class. Nevertheless,

this weakness in the criterion does not have to disadvantage the study of entanglement,

as I will show in my case.

Before I describe the criterion that I eventually used in my study, let us look at a

failed, but nevertheless interesting, method that I attempted to use in my study. This

method involved studying the Wigner function for the system.

In classical systems, if we are given a phase space distribution PCl(q, p), the average

of a quantity A that depends on p and q is given by:

〈A〉Cl =∫ ∫

dqdpA(q, p)PCl(q, p). (3.4)

In quantum mechanics, an analogous definition exists, but using a quantity known as

the quasi-distribution function PQ(q, p), rather than an actual distribution function like

PCl(q, p). This is due to the Heisenberg Uncertainty Relation: q and p cannot both

be determined at the same time with certainty. An example of PQ(q, p) is the Wigner

distribution function W . In one dimension, the Wigner function takes the following form:

W (q, p) =1

πh

∫ ∞

−∞dy 〈q − y| ρ |q + y〉 e2ipy/h (3.5)

where ρ is the density matrix representing the state of the system. Equation (3.5) can

be thought of as the Fourier Transform of an element of ρ in the position basis. The

transform is defined by y, the increment in position that is added to or subtracted from

q.

We know that the classical distribution function PCl(q, p) cannot be negative. How-

ever, there is no reason why, when quantum correlations are introduced into the system,

that extensions of classical quantities, in this case, the Wigner function, have to demon-

strate the same behaviour. In fact, it is the departure from classical behaviour that

defines quantum properties. Based on this reasoning one can try to use the negativity of


the Wigner function as an indication of the existence of quantum correlations.

To use the Wigner function in our 2-mode-N-photon state, I begin by writing the

expression that describes it:

W (q1, q2; p1, p2) =1

(πh)2

∫ ∞

−∞

∫ ∞

−∞dy1dy2 〈q1 − y1, q2 − y2| ρ |q1 + y1, q2 + y2〉 ∗

∗e2ih

(p1y1+p2y2),

(3.6)

where ρ is given in general by (3.1). If we take the density matrix (3.1) and substitute it

into (3.6), we end up with the following:

Wρ =N∑

k,m=0

ρkm(t)WN−k,N−mWk,m. (3.7)

What (3.7) says is that the two-mode Wigner function can be written in terms of the

Wigner functions of the single modes involved in defining it. This simplifies the problem

because it reduces it to calculating one-dimensional Wigner functions from which the

two-dimensional Wigner function can be obtained. This means that our problem is

easily solved if we evaluate the following integral:

Wm,n(q, p) =1

πh

∫ ∞

−∞dy 〈q − y| n〉〈m| q + y〉 e2ipy/h. (3.8)

Note that the position representation of the number states that appear in the integral in

(3.8) is given by: 〈x| n〉 =(

mωπh

)1/41

2n/2√

n!e−

mωx2

2h Hn

(x√mω/h

), where ω is the frequency

of the harmonic oscillator and Hn(x) represents the Hermite polynomials. Before we

evaluate (3.8), let us write it in a non-dimensional form by defining:

q0 =

√h

mω

p0 =√hmω


q = q0x

p = p0k

y = q0x′

(3.9)

The non-dimensional form of (3.8) is given by:

Wm,n(q, p) =1

πp0

∫ ∞

−∞dx′ 〈q0(x− x′)| n〉〈m| q0(x+ x′)〉 e2ikx′

. (3.10)

The integral in (3.10) can easily be evaluated if the following relation [59] is used:

∫ ∞

−∞e−x2

Hm(x+ y)Hn(x+ z)dx = 2n√πm!zn−mLn−m

m (−2yz), (3.11)

where m ≤ n and Lab (x) are the generalized Laguerre polynomials. I find that:

Wm,n =(−1)m

πp0

√2n

2m

√m!

n!e−(q2

0x2+k2/q20)

[q0

(x+

ik

q20

)]n−m

Ln−mm

[2(x2q2

0 + k2/q20

)](3.12)

when m ≤ n, and that:

Wm,n =(−1)m

πp0

√2m

2n

√n!

m!e−(q2

0x2+k2/q20)

[q0

(ik

q20 − x

)]n−m

Lm−nn

[2(x2q2

0 + k2/q20

)](3.13)

when m ≥ n. To get a feel for how well the Wigner function can serve as a test for the

presence of quantum correlations, the results obtained in (3.12) and (3.13), which by (3.7)

can be used to evaluate (3.6) can be applied to a special case of a two-mode-N-photon

state. Here I choose to look at an N = 3 NOON state, which is given by:

|ψ〉 =1√2|30〉+ |03〉 . (3.14)


W(p,q)

W(p,q)

(a) Maximally Entangled NOON State (b) Fully Dephased NOON State

Figure 3.1: Plot of W (Wigner Function) vs q1 vs p1 for a NOON state with

N = 3. This relation is for the case when q1 = −q2 and p1 = −p2. Notice that although

in (b), the state is completely dephased, a case in which no entanglement is believed to

be in the system, the Wigner function can still take negative values. This shows that,

generally speaking, the negativity of the Wigner function cannot be used as an indication

of the existence of entanglement.


Using my results, I calculate the Wigner function for (3.14) when it is maximally entan-

gled as well as when it is completely dephased; i.e., when the off-diagonal elements are

zero. I look at the specific case when q1 = −q2 and p1 = −p2. The result is shown in the

plots in Fig. 3.1. Notice that the Wigner function can still be negative in the case when

the NOON state is completely dephased; i.e., with no entanglement left in the system.

This example shows us that we need a different way to find if the system we are studying

is entangled or not.

My search for an appropriate measure of entanglement for the photon states I are

studying led me to the Peres’s criterion for separability [18], also described in Chapter 1

of this thesis. I provide a brief description here. The density matrix of a bipartite system

may be written as:

ρ =∑

i

ciρ′

i ⊗ ρ′′

i . (3.15)

Taking the partial transpose over one of its subsystems one obtains:

σ =∑

i

ci(ρ′

i)T ⊗ ρ

′′

i . (3.16)

If σ has at least one negative eigenvalue, then we know with certainty that the system

is entangled. However, if none of the eigenvalues are negative, then the system could be

entangled or separable. The consequence of this weakness to my study of ESD is that the

existence of ESD cannot be proven with certainty, while its nonexistence can be proven

with certainty, as I will show in the next section.

3.4 Result

To find out if our dephased 2-mode-N-photon state in (3.3) is entangled or not, I use the

partial transpose method described in the previous section. Taking the partial transpose

of the state gives us the following:


σ =N∑

k=0

ρkk(0) |N − k, k〉〈N − k, k|

+N∑

k,m=0,k 6=m

ρkm(0)e−12(k−m)2(Γ1+Γ2)t |N − k,m〉〈N −m, k| . (3.17)

Notice that since k and m in the second terms of the right hand side of (3.17) are not

equal, the total number of photons in the kets and bras are never equal to N. In other

words, N −k+m 6= N and N −m+k 6= N . Also, vectors with different photon numbers

are orthogonal. Since the eigenvectors of a matrix are also orthogonal, this simplifies our

quest for eigenvectors. Mathematically, this implies that each pair of the matrix elements

|N − k,m〉〈N −m, k| and |N −m, k〉〈N − k,m| fall in a different subspace than each

other as well as the space of the diagonal elements.

This breaks the problem of finding the eigenvalues of σ into finding the eigenvalues

of N(N+1)2

matrices; the remaining eigenvalues are just the diagonal elements of σ. Each

of these matrices has the following form:

|ρkm(0)| e−12(k−m)2(Γ1+Γ2)t

eiθ |N − k,m〉〈N −m, k| +e−iθ |N −m, k〉〈N − k,m|

,(3.18)

where θ is the phase of the matrix element ρkm. It can be easily shown that the following

state:

|ψ〉 =1√2

eiθ |N − k,m〉 − |N −m, k〉

, (3.19)

is an eigenvector of (3.18) with eigenvalue

− |ρkm(0)| e−12(k−m)2(Γ1+Γ2)t. (3.20)

This means that (3.17), has at least one negative eigenvalue, which only goes to

zero at infinite time, as long as one of the ρkm, and consequently ρmk, is nonzero; i.e.,


this is true as long as there is some coherence in the system. If there is not any other

decoherence mechanism, such as depopulation, going on as well, this is always true;

there will always be entanglement in the system for any finite time. In other words, for

a general two-mode-N-photon state undergoing pure dephasing, there is no sudden death

of entanglement.

3.5 Example: NOON States

I demonstrate my results using the first realized NOON state [60]; i.e., for a 2-mode-3-

photon state, specifically given by:

|ψ〉 =1√2|N0〉+ |0N〉 , (3.21)

where N = 3, but more generally by:

|ψ〉 = a |30〉+ b |21〉+ c |12〉+ d |03〉 , (3.22)

where |a|2 + |b|2 + |c|2 + |d|2 = 1. Applying the arguments above, I find that for the

partial transpose of the density matrix of this system, the negative eigenvalues are:

−|a||b|e− 12(Γ1+Γ2)t, −|a||c|e−2(Γ1+Γ2)t, −|a||d|e− 9

2(Γ1+Γ2)t, −|b||c|e− 1

2(Γ1+Γ2)t, −|b||d|e−2(Γ1+Γ2)t,

and −|c||d|e− 12(Γ1+Γ2)t. Each of them involve a decay term due to dephasing. However,

they only become zero after an infinite amount of time that renders the negative expo-

nential zero. Therefore, for any finite time, there is always entanglement in the system,

so ESD does not occur.

3.6 Is long-lived entanglement practical?

Finally, I consider how practical this long-lived entanglement is in NOON states under-

going dephasing. In Fig. 3.2, a standard setup for interfering two beams to produce


interference fringes is illustrated. The presence of a phase shifter (PS) in the upper

path induces photons travelling there to acquire a phase shift eiφ. When the “N-photon-

NOON-state” is created inside this interferometer, the phase is accumulated N times,

and the state becomes:

|ψ〉 =1√2

|N0〉+ eiNφ |0N〉

, (3.23)

With dephasing occurring, the density matrix of the state is as follows:

ˆρ(t) =1

2

(|N0〉〈N0|+ e−N2ΓteiNφ |0N〉〈N0| +e−N2Γte−iNφ |N0〉〈0N |+ |0N〉〈0N |

),(3.24)

where I assume Γ1 = Γ2 = Γ.

The expectation value of the exposure dosage, 〈δ〉, displays fringes of visibility V. The

exposure dosage operator δ is described in terms of creation and annihilation operators

acting on the two output (C and D paths) operators in Fig. 3.2. If we symbolize the

annihilation operators representing modes C and D, by c and d, respectively, then the

field that ends up on the screen can be represented by: e = c+ d. The exposure dosage

is now given by [6]:

δ =(e†)N(e)N

N !. (3.25)

With simple algebra, it can be shown, in the case of dephasing of NOON states, that

the expectation value of δ is 〈δ〉 = Trρδ

= 1 + e−N2Γtcos(Nφ), where Tr represents

the trace, and ρ is the dephased NOON state given in (3.24). From this the visibility is

found to be:

V =〈δ〉max − 〈δ〉min

〈δ〉max + 〈δ〉min

= e−N2Γt (3.26)

When the visibility becomes vanishingly small, the fringes (and hence the measured

phase) becomes impossible to measure: I will call this visibility, at which measurement

becomes impracticable, the critical visibility Vcrit; its value will depend on the sensitivity


of the fringe measurement techniques employed. For the given decay rate Γ, the time it

takes to reach this critical visibility Vcrit is given by

tcrit =1

ΓN2ln(

1

Vcrit

). (3.27)

Notice that in (3.27), the expression for time depends inversely on N2. This implies that

the larger N is, the faster it takes for visibility to fall down to Vcrit and become worse.

This is completely the opposite of what was earlier hoped to be achieved in improving

resolution of fringes by creating High NOON states; i.e., states with large N .

3.7 Conclusions

I showed that although the criterion for separability has a weakness that can render some

studies of entanglement uncertain, in our case and by using this criterion, I proved with

certainty that for pure dephasing (no photon loss), ESD does not occur in our system.

I also demonstrate our result using the so called NOON states to show that there is no

sudden death at NOON. Although this criterion allows us to prove that, even after a

long time, there is some entanglement left in the system, it does not give us a way to

determine how much is left. Therefore, to answer the question about the usefulness of

the dephased NOON states in interferometry, I study the time it takes to reach critical

visibility. In doing so, I reveal that the presence of some entanglement does not have

much practical implications. In fact, I show that for this realistic decohering system,

increasing N does not improve resolution, but rather allows it to worsen at a faster rate,

which is proportional to N2.


Mirror

Mirror

Screen

BeamSplitter

PhaseShifter (Φ)

CC

DD

Figure 3.2: Interference pattern formation adapted from Fig. 1 of [6] Two

photon beams pass through a beam splitter and get reflected off the upper and lower

mirrors to form an interference pattern on the screen. The upper beam passes through

a phase shifter before reaching the screen. The phase aquired depends on the number of

photons N that pass through the upper path, and it equals eiNφ.

Chapter 4

A Comparison of the Attempts of

Quantum Discord and Quantum

Entanglement to Capture Quantum

Correlations

4.1 Introduction

Since the emergence of Quantum Mechanics at the beginning of the twentieth century,

physicists have been intrigued and puzzled by its interpretation and consequences. Char-

acteristics that distinguish quantum from classical systems have been investigated ex-

tensively. To be able to study quantum correlations in a system, it is important to

quantify them. Although this is a challenging task for multi-component quantum sys-

tems, progress has been made in the case of two-level bipartite quantum systems. One

method suitable for pure states [61, 20] involves calculating the entropy of the reduced

density matrix of the system, also known as the Entanglement of Formation (EoF). To

extend this concept to mixed states, the entanglement in defined to be equal to the

59

Chapter 4. Capturing Correlations by Discord and Entanglement 60

weighted sum of the entanglement of the pure states involved in the decomposition of

the mixed state, minimized over all decompositions [62, 63, 19]. The restriction enforced

by this minimization places a bound on the entanglement for mixed states; indeed, when

a certain level of disorder of the state is attained, it is known that entanglement must

disappear [64, 8]. It is, therefore, not surprising that for some systems as they reach a

certain level of mixture, the entanglement is completely lost, a phenomena which in the

study of state dynamics is commonly known as Entanglement Sudden Death (ESD) [28],

which was the subject of the previous two chapters in this thesis.

Another approach to capture quantum correlations was taken by Zurek and Ollivier

[23], where they used the fact that measurement of quantum systems, unlike classical

systems, disturbs their state. To quantify the correlations based on this idea, one looks

at the mutual information function I(ρ), where ρ is the density matrix describing the

state of the whole system. Given a system C composed of two subsystems A and B, I(ρ)

is a measure of how much information is shared between the two subsystems. In other

words, it is can viewed as an indication of the degree of correlation between them. The

correlations between the two subsystems can be classical and/or quantum. However, if

the correlations are quantum in nature, then calculating I(ρ) after a measurement is

performed on one of the subsystems (say B) yields a different result to that calculated

before the measurement is performed. This disagreement is the basis for defining discord;

the definition is finalized after optimizing over all possible measurement bases. One of the

major reasons in the aroused interest [65, 3, 66, 26, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76]

in this novel correlation is that as was shown in [23] even when entanglement is zero

in a system, discord can still be finite. This led to the hope that using discord instead

of entanglement as a resource, in fields like quantum computation, can lead to more

efficient computations. In [65], discord was characterized in the DQC1 (Deterministic

quantum computation with one bit) model [4], calling for an experimental verifications

of its powers, which was demonstrated in [3].


In this chapter, I look at the relationship between discord, entanglement and linear

entropy to investigate the connection between these quantities. I perform the study on

the most general density matrices representing two-level bipartite systems. Since we

lack an analytic expression for discord for these general states, the heart of the work

is numeric in nature, involving optimization over all possible measurements that can

be performed on one of the subsystems under study. My numerical work facilitates the

principal results presented here: the analytic form of the states for which the discord takes

extreme values. As will be shown, one can place definite boundaries on the relationships

between entanglement and discord, and mixture (quantified by the linear entropy (4.8))

and discord.

4.2 Discord, Entanglement, and Linear Entropy for

Two-Level Bipartite Systems

To consider discord, let us look at system C, which is composed of two subsystems A

and B, both of which are two-level quantum systems. The effects of measurements on

one of the subsystems (B in the following analysis) is captured by looking at the mutual

information function defined as follows:

I(ρ) = S(ρA) + S(ρB)− S(ρ), (4.1)

where ρi is the reduced-density matrix of subsystem i and S (ρ) = −Tr ρ log2ρ [77].

Then, defining all set of projectors on B by Bk, the measurement-induced mutual

information function for each of these sets takes the following form:

I (ρ | Bk) = S(ρA)−∑k

pkS (ρk) , (4.2)

where ρk is the density matrix of the system after Bk is applied on B, k ∈ 1, 2 and pk =


Tr(I⊗ Bk

)ρ(I⊗ Bk

), which is the probability to obtain result k from the projective

measurement. To obtain the final form for the measurement-induced density matrix,

Eq.(4.2) is maximized over all possible Bk to obtain the following expression for discord:

Q (ρ) = I(ρ)−maxBk

S(ρA)−∑k

pkS (ρk)

. (4.3)

To start the comparison between discord and entanglement, I compute the EoF and

discord for pure states in two-level bipartite systems |ψ〉 = a |00〉+ b |01〉+ c |10〉+d |11〉,

where |a|2 + |b|2 + |c|2 + |d|2 = 1. It is convenient to use the Schmidt decomposition [78]

to write the state as |ψ〉 =√λ |1A〉 |1B〉 +

√(1− λ) |2A〉 |2B〉, where λ and (1− λ) are

the eigenvalues of the reduced density matrices, and |1i〉 and |2i〉 are the corresponding

eigenvectors of the reduced density matrix of subsystem i. Using local unitary operations,

which do not affect the quantum correlations present in the system, one can show that

this state is equivalent to |ψ〉 =√λ |00〉+

√(1− λ) |11〉. In this case, the EoF as well as

discord, which can be computed analytically, are found to be identical and are given by:

E (ρ) = Q (ρ) = h(λ), where h(x) = −x log2 x− (1− x) log2 (1− x). Therefore, discord

and ‘ amount to the same set of correlations in the case of pure states [71]. In the mixed

state case, there is no explicit analytic expression for discord. The most general analytic

expression so far was presented in a very interesting paper by Luo [79] for the mixed

states with maximally mixed marginals (MMMS) (i.e, the reduced density matrices ρA

and ρB are both maximally mixed).

Going back to system C described above, with A and B each being two-level quantum

systems, I parametrize all the possible measurements that can be performed on B by two

variables: θ and φ. Each complete set of possible measurements, which is composed of

two elements, is defined as follows:

B1 = |ψ〉〈ψ| (4.4)

B2 = |ψ⊥〉〈ψ⊥| ,


where

|ψ〉 = cosθ |0〉+ eiφsinθ |1〉 (4.5)

|ψ⊥〉 = −sinθ |0〉+ eiφcosθ |1〉 .

The resultant of the density operator when such measurements are performed on subsys-

tem B is:

ρk =1

pk

(I ⊗ Bk

)ρ(I ⊗ Bk

). (4.6)

To obtain the final form of Eq.(4.3), I numerically search the θ and φ space for the set of

values that maximizes Eq.(4.2). For a given density matrix ρ, the EoF is given in terms

of concurrence C (ρ) by the formula [19]:

E (ρ) = h

1 +√

1− C2 (ρ)

2

, (4.7)

where C = max√

λ1 −√λ2 −

√λ3 −

√λ4, 0

, where λi are the eigenvalues, in decreasing

order, of the matrix R = ρ (σy ⊗ σy) ρT (σy ⊗ σy).

For a comparison of discord with entropy, I calculate the linear entropy, defined as

follows:

SL =4

3

(1− Tr

(ρ2)). (4.8)

The linear entropy is defined in this way to limit its values to be between 0 and 1,

where the former case corresponds to pure states and the latter case to the maximally

mixed states.


4.3 Results

Figures 4.1 and 4.2 provide the plots that give the relationship between discord and

entanglement, and discord and linear entropy for the two-level bipartite systems, respec-

tively, with Fig 4.3 displaying the boundaries on the latter relation. As noted above, in

the case of pure states, discord and entanglement are identical. For mixed states, the two

quantities are loosely related; generally speaking the higher the entanglement, the higher

the discord. The region with high quantum correlations has a narrower relationship than

the one in a lower correlation regime. This results in a plot that resembles a horn (See

Fig. 4.1). The difference between discord and entanglement that arises in the mixed

state case is due to the optimization that was done to extend the pure state case to the

mixed state case. In the low quantum regime and for a large number of systems, the

minimization that is done over the pure state decomposition in defining entanglement

has a trend of giving a more pessimistic measure for quantum correlations than the max-

imization over all possible projectors on subsystem B that is done in defining discord.

Discord and entanglement, therefore, are not different quantum correlations. They are

two different ways to quantify these correlations. The disagreement between them, in

the general mixed state case, comes from limitations of optimization methods.

The upper bound for the discord-entanglement plot is given for the most part by two

classes of MMMS: the α-states eq.(4.10) and the Werner states eq.(4.9). In the highly

correlated regime it is bound by the pure states. The lower bound is given by another

class of MMMS: the β-states eq.(4.11). The Werner states, the α-states, and the β-states

are given, respectively, as follows:

ρ(ξ) = (1− ξ)I

4+ ξ

∣∣∣ψ−⟩ ⟨ψ−∣∣∣ , (4.9)

where −13≤ ξ ≤ 1 and |ψ−〉 = 1√

2(|01〉 − |10〉),


ρ(α) =

α2

0 0 α2

0 (1−α)2

0 0

0 0 (1−α)2

0

α2

0 0 α2

, (4.10)

where 0 ≤ α ≤ 1, and

ρ(β) =

β2

0 0 β2

0 (1−β)2

(1−β)2

0

0 (1−β)2

(1−β)2

0

β2

0 0 β2

, (4.11)

where 0 ≤ β ≤ 1. The analytic result for quantum discord in these cases can easily be

obtained from the general expression for the MMMS in [79]. For the α- and β-states, it

is given, respectively, as follows:

Q(α, ζ) = (1− α) log2 (1− α) + αlog2 (α) + (1 + α)

−(1− ζ)(log2 (1− ζ))/2

−(1 + ζ)(log2 (1 + ζ))/2, (4.12)

where ζ = max |α|, |2α− 1|, and

Q(β) = βlog2 (β) + (1− β) log2 (1− β) + 1. (4.13)

Since these states also fall under the class of X-states, expressions for their concur-

rence, from which the EoF is calculated, can be found in [53], for example. The con-

currence of the α- and β-states, is given, respectively, by C(α) = max 0, 2α− 1 and

C(β) = |2β − 1|. To prove that these are indeed the boundaries, I performed two numeric

calculations. First, I generated 106 random density matrices to find that the relationship

between their discord and entropy all fall within these bounds. The algorithm involved


creating a complex and random matrix T , and from it obtaining a well-behaved density

matrix ρ by the relation ρ = T T †/TrTT†. I also generated 105 points very close to the

vicinity of each of the boundaries, with the result that none of the points fell outside the

bounds imposed by them.

In Fig. 4.3, where the plot shows how much discord can be present in the system

for a given amount of mixture, the boundaries are given by a different set of states.

Beyond linear entropy being 8/9, it is bound from above by the Werner states. The rest

of the plot is bound from above by a class of two-parameter density matrices described

as follows:

ρ(a, b) =1

2

a 0 0 a

0 1− a− b 0 0

0 0 1− a+ b 0

a 0 0 a

, (4.14)

where 0 ≤ a ≤ 1, and a− 1 ≤ b ≤ 1− a. I find the analytic result for discord in this case

to be:

Q(a, b) = min a, q , (4.15)

where

q = − b2log2

[(1 + b)(1− a− b)

(1− b)(1− a + b)

]

+a

2log2

[4a2

(1− a)2 − b2

]

−

√a2 + b2

2log2

1 +√

a2 + b2

1−√

a2 + b2

+

1

2log2

[4((1− a)2 − b2)

(1− b2)(1− a2 − b2)

],

(4.16)


Figure 4.1: The Discord-Entanglement Horn. Discord increases as entanglement

increases. In the case of pure states, the two quantities are identical. While in the mixed

state case the relationship broadens. However, notice that this relationship narrows in

the high quantum correlated regime and is the broadest in the low correlation regime.

This gives the plot its ‘horn’ shape. The upper bound of this relationship is given by

the α-states eq.(4.10) (for 0 ≤ EoF ≤ 0.620, and 0 ≤ Q ≤ 0.644), the Werner states

[7] (for 0.620 ≤ EoF ≤ 0.746, and 0.644 ≤ Q ≤ 0.746), and the pure states (for

0.740 ≤ EoF,Q ≤ 1). The lower bound is given by the β-states eq.(4.11).


and the expression for concurrence in this case is C(a, b) = max0, |a| −√

(1− a)2 − b2.

This result is derived using the powerful combination of numerical and analytical calcu-

lations. Here is how I did it: Looking at (4.3), it can be seen that we need to find I(ρ),

S(ρA), and X =

∑k pkS (ρk) for ρ(a, b). The first two quantities can be obtained very

easily using analytic methods. To find the expression for discord, we need to minimize

the third quantity X. However, I found that X is a complicated function of a, b, and θ.

So to find the value of θ that minimizes X, I generated a set of values for the parameters

a and b and plotted X versus θ for each set. I found out that the minimization is achieved

for either θ = 0 or θ = π/4, depending on the values of a and b. For the former case, I

found that discord is equal to a, while for the latter case, discord is equal to q given in

(4.16). As a final check, I used my MATLAB code for discord to calculate it for ρ(a, b) for

over 105 points, incrementing slowly over the values of a and b, and obtained consistent

results with the analytic expression described in (4.15).

As can be seen in Fig. 4.3, the Maximally Entangled Mixed States (MEMS) [8], and

the α-states, both of which fall under this class of states, bound the plot at different

regions. There is no lower bound to the relationship between discord and entropy, as

the area below the two-parameter states as well as the Werner states gets filled up when

the whole range of density matrices is included. Also note that the case when a = q is

what gives the bounding line that slopes down from the ‘pimple’. Again, as is the case

with discord and EoF, to verify these boundaries, points representing 106 random density

matrices were generated, as well as 105 points in the near vicinity of these bounds. Other

points to note about the figure is, first, that at the pimple, after which no entanglement

can exist in the system [8], a rise in linear entropy results in a rise in discord. The states

in this region are interesting to investigate, since experimentally speaking, more noise in

the system at that stage enhances the quantum correlations. Also, unlike the case with

entanglement, even for cases where linear entropy is arbitrary close to the maximally

mixed states at unit entropy, discord can still be finite. It is, therefore, not surprising


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Dis

cord

Linear Entropy

Figure 4.2: The Relationship Between Discord and Linear Entropy. The most

general relation between discord and linear entropy for a two-level bipartite system would

look like this plot. For the states that define the boundaries of this relationship, see Fig.

4.3.

that for states in which ESD occurs, similar behaviour is not observed for discord [76, 69].

4.4 Conclusion

In conclusion, this work describes the relationship between discord and entanglement for

the general two-level bipartite system. I have shown that in the general case of mixed

states the two quantum correlations vary, with the relationship broadening in the low

quantum regime. I conclude that they describe the same set of quantum correlations,

as can be seen in the pure state case, and although they vary in the case of mixed

states, this is due to different methods of optimization used to extend the correlations

from the pure state case to the mixed state case. Some questions that arise are: What


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Dis

cord

Linear Entropy

Pimple

2-parameter states

MEMSWerner states

! states

Figure 4.3: The Boundaries on the Relationship Between Discord and Linear

Entropy. To easily illustrate the boundaries, this plot only includes the states that are

involved in defining them. The two-parameter states, eq.(4.14), bound the curve from

above up to Q = 1/3 and SL = 8/9, after which the Werner states take over. Discord and

Linear Entropy, as expected, display an inverse relationship: more randomness implies

less quantum correlations. One of the interesting phenomena occurs at the ‘pimple’,

where there exists states in which an increase in their entropy results in an increase in

their discord. This is also the point that defines the value of linear entropy after which

no entanglement can exist in the system (See [8]). Unlike entanglement, states exist that

are very close to the maximally mixed states, but still have non-zero discord. In fact the

only value for entropy such that discord cannot be finite is for it being equal to 1, in the

case when the system is maximally mixed.


is the optimal method to quantify quantum correlations? The answer to this question

depends on the application of the quantum correlations. There is no universal definition

for these correlations. An operational view is therefore recommended in order to define

them. Also, what is the meaning of the values for discord and EoF that are between the

extreme cases of 0 and 1? I also reveal the general relationship between discord and linear

entropy highlighting interesting differences with a similar analysis done for entanglement

[8]: at the point where entanglement disappears from the system, discord increases in

value, and discord can be nonzero unless linear entropy is identically equal to one.

Chapter 5

Intensity Fluctuations and

Cross-Polarization in Gaussian

Schell-Model Beams

5.1 Introduction

The work I present here affects, in general, applications of the Hanbury Brown-Twiss

(HBT) Effect, which describes the correlation in intensity fluctuations of an electromag-

netic (EM) field at two different points. One of the first uses of the HBT effect was to

calculate the diameter of stars. Briefly, this is how it is achieved: The intensity of an

electric field is given by:

I = |E(ρ, t)|2 , (5.1)

where E(ρ, t) represents the electric field. In realistic systems, the intensity of the field

at point ρi fluctuates. In fact at a given time t, the fluctuation is given by the difference

between the total intensity at t and the average intensity over t as follows:

72

Chapter 5. The Degree of Cross-Polarization and the HBT Effect 73

∆I(ρi, t) = I(ρi, t)− 〈I(ρi, t)〉 , (5.2)

where the brackets correspond to the ensemble average. Here, we assume that the system

is ergodic; i.e., the time average and the ensemble average are equal. It can, therefore,

be shown that, in terms of the fields, that the correlation in intensity fluctuations at two

different points ρi and ρj (See Fig. 5.1) is given by:

〈∆I(ρ1, t)∆I(ρ2, t+ τ)〉 =⟨|E(ρ1, t)|2 |E(ρ2, t+ τ)|2

⟩−⟨|E(ρ1, t)|2

⟩ ⟨|E(ρ2, t+ τ)|2

⟩(5.3)

Applying the Gaussian Moment Theorem [1] to (5.3), we get:

〈∆I(ρ1, t)∆I(ρ2, t+ τ)〉 = 〈E∗(ρ1, t)E(ρ2, t+ τ)〉〈E∗(ρ2, t+ τ)E(ρ1, t)〉

= |Γ(ρ1, ρ2, τ)|2, (5.4)

where Γ(ρ1, ρ2, τ) is a second order correlation function, and it can be shown [1] that this

function is proportional to the degree of coherence of the field.

By changing ρ1 and ρ2, the function |Γ(ρ1, ρ2, τ)| can be mapped. From the van

Cittert-Zernike Theorem [1] we know that the Fourier Transform of this correlation func-

tion is proportional to the size of the source. In fact, it is equal to the intensity per

unit area as a function of position. This is a concept that can be used to find the shape

of stars. The applications of the HBT effect is not limited to astronomy, but to other

fields such as quantum optics and particle physics. In fact, the main result of the work

described in this chapter is important in highly coherent systems, such as in cold atom

traps, and not in incoherent light sources such as stars.

As can be seen in (5.4), in the scalar treatment of the field, the correlation in intensity

fluctuations of an electromagnetic beam at two different points depends on Γ, which is

a product of the degree of coherence and the average intensities at the two different


points. In more recent work [9, 10] in which the vector nature of the field is taken

into consideration, it has been shown that these correlations cannot be described by

the degree of coherence and the degree of polarization alone, but rather by the degree

of coherence and the generalized concept of the degree of polarization: the degree of

cross-polarization. The work we present here is an illustration of this result for Gaussian

Schell-model beams.

5.2 The HBT Effect, Taking into Account the Vector

Nature of the Field

We start from the beginning and ask: What is the correlation between the intensity

fluctuations at two different points ρi and ρj (See Fig. 5.1)?

To find out how the intensity fluctuations of an EM fields at two points are correlated,

Ref. [9] and Ref. [10] used↔W , the cross-spectral density matrix, as a tool. Assuming

that the EM field is propagating in the z-direction,↔W is given by:

↔W (ρ1, ρ2, ω) =

〈E∗x(ρ1, ω)Ex(ρ2, ω)〉〈E∗

x(ρ1, ω)Ey(ρ2, ω)〉⟨E∗

y(ρ1, ω)Ex(ρ2, ω)⟩ ⟨

E∗y(ρ1, ω)Ey(ρ2, ω)

⟩ , (5.5)

where Ex(ρi, ω) and Ey(ρi, ω) are, respectively, the x and y components of the EM field

in the frequency domain at position ρi. By using the definition:

〈∆I(ρ1, ω)∆I(ρ2, ω)〉 = tr[↔W

†(ρ1, ρ2, ω).

↔W (ρ1, ρ2, ω)

], (5.6)

and assuming Gaussian statistics of incident light, it was shown that:

〈∆I(ρ1, ω)∆I(ρ2, ω)〉 =1

2

1 + [P (ρ1, ρ2, ω)]2

|η(ρ1, ρ2, ω)|2 〈I(ρ1, ω)I(ρ2, ω)〉 , (5.7)


P1 P2

CorrelatorD DEx = (ρ1, ω) Ex = (ρ2, ω)

Ey = (ρ1, ω)Ey = (ρ2, ω)

Figure 5.1: Simple Correlation Experiment. The intensity of an electromagnetic

beam at points P1 and P2 fluctuates in realistic systems. In Ref. [9] and Ref. [10],

the correlations in intensity fluctuations was shown to be dependent on the degree of

coherence and the newly discovered statistical parameter, the degree of cross-polarization.


where P (ρ1, ρ2, ω) and η(ρ1, ρ2, ω) are the degree of cross-polarization and the degree of

coherence, respectively. In terms of↔W , these three statistical parameters are given by:

η(ρ1, ρ2, ω) =tr

↔W (ρ1, ρ2, ω)√

tr↔W (ρ1, ρ1, ω)

√tr

↔W (ρ2, ρ2, ω)

P (ρ1, ρ2, ω) =

√√√√√√√√2tr

[↔W

†(ρ1, ρ2, ω).

↔W (ρ1, ρ2, ω)

]∣∣∣∣ ↔W (ρ1, ρ2, ω)

∣∣∣∣2 − 1

P (ρ, ω) =

√√√√√√1− 4det↔W (ρ, ρ, ω)[

tr↔W (ρ, ρ, ω)

]2 , (5.8)

where P (ρ, ω) is the degree of polarization. Note that the degree of cross-polarization

reduces to the degree of polarization when ρ1 = ρ2 = ρ. Unlike the degree of polarization

which can only take values between 0 and 1, the degree of cross-polarization can take any

non-negative value. For example, an unpolarized electromagnetic beam has been shown

to have a degree of cross-polarization equal to zero, while a beam of a deterministic

non-uniform polarization has a degree of cross-polarization equal to infinity [10].

The surprising result was that the complete description of the correlation in intensity

fluctuations cannot be given by the degree of coherence and the degree of polarization

alone, but by the degree of coherence and the generalized concept of the degree of polar-

ization: the degree of cross-polarization.

5.3 Results: The Dependence of the HBT Effect in

Gaussian Schell-Model Beams on the Degree of

Cross-Polarization

We further consider the ramifications of the result described in the previous section by

looking at a special class of beams: Gaussian Schell-model (GSM) beams. For GSM


beams, the cross-spectral density matrix is given as follows:

Wαβ(ρ1, ρ2) = Sα(ρ1, ω)Sβ(ρ2, ω)µαβ(ρ1 − ρ2, ω), (5.9)

where

Sα(ρ, ω) = Aα(ω)exp

[− ρ2

4σ2α(ω)

], (5.10)

µαβ(ρ, ω) = Bαβ(ω)exp

[− ρ2

2δ2αβ(ω)

],

and Bαβ, σ, and δαβ are constants. Here, S(ρ, ω) is the spectrum of light at point ρ and

frequency ω, and µαβ(ρ, ω) is the degree of spectral coherence at the separation ρ and

frequency ω.

The expression for the cross-spectral density matrix (5.9) becomes:

Wαβ(ρ1, ρ2) = AαAβBαβexp

(−ρ

21 + ρ2

2

4σ2

)exp

(−(ρ1 − ρ2)

2

2δ2αβ

), (5.11)

where ω is not shown explicitly. Here, assuming δxy = δyx, the degree of coherence is:

η(ρ1, ρ2) =Cxx exp

[− (ρ1−ρ2)2

2δ2xx

]+ Cyy exp

[− (ρ1−ρ2)2

2δ2yy

]Cxx + Cyy

, (5.12)

the degree of polarization is:

P (ρ) =|Cxx − Cyy|Cxx + Cyy

√√√√1 +4 |Cxy|2

(Cxx − Cyy)2, (5.13)

and the degree of cross-polarization is:

P (ρ1, ρ2) =|Cxx − Cyy|Cxx + Cyy

√√√√1 +4 |Cxy|2

(Cxx − Cyy)2exp

[−(

1

δ2xx

− 1

δ2xy

)(ρ1 − ρ2)2

]. (5.14)


Notice, that although the three statistical parameters depend on various constants,

the degree of coherence and the degree of polarization do not depend on δxy, while the

degree of cross-polarization does. An implication of this is that even if two beams have

exactly the same degree of coherence and the same degree of polarization, they can still

have different values for δxy, and, therefore, the degree of cross-polarization for both is

going to be different. This means that the two beams will not have the same correlations

in intensity fluctuations. This example confirms the earlier assertion that the degree of

coherence and the degree of polarization alone are not sufficient to provide a complete

description for these correlations.

5.4 Example: Realizable Gaussian Schell-model beams

To illustrate our result we look at two different realizable EM beam sources. The real-

izability condition is derived, in the frequency domain, using the same arguments as in

[80].

For an EM beam to be physically realizable, the cross-spectral density matrix must

be positive, which implies that the following condition must be met:

|µxy(k, ω)| ≤√µxx(k, ω)µyy(k, ω), (5.15)

where µxy(k, ω) is the Fourier transform of µxy(ρ, ω) given by:

µxy(k, ω) =1

(2π)2

∫ ∞

−∞

∫ ∞

−∞dρ2µxy(ρ, ω)exp(ik.ρ). (5.16)

Condition (5.15) leads to the following realizability condition, which is similar to the one

in [80], for the Gaussian Schell model beam under study:

δ2xx + δ2

yy

2≤ δ2

xy ≤δxxδyy

|Bxy|. (5.17)


To compare the correlation of intensity fluctuations of two numerical examples, we

make the following assumptions: δxx = δyy, σαβ = σ, and Ax = Ay. We also define the

new parameter Q in the following way: Q = 1− δ2xx

δ2xy

.

The degree of polarization, as defined in (5.13) becomes:

P (ρ) = |Bxy| , (5.18)

The realizability condition becomes:

0 ≤ Q ≤ 1− P. (5.19)

With these assumptions, (5.6) gives the following for the correlation of of intensity

fluctuations of the light beams:

〈∆I(ρ1)∆I(ρ2)〉 (5.20)

= 2A4xexp

[− 1

2σ2(ρ2

1 + ρ22)]

exp

[−(ρ1 − ρ2)

2

δ2xx

]+ P 2(ρ)exp

[−(1−Q)

(ρ1 − ρ2)2

δ2xx

].

In order to maximize the difference of (5.20) for two different beams, we need to

consider the extreme values for the quantity between brackets in the right hand side of

the equation. This quantity has the form:

f(x) = e−x2

+ P 2e−(1−Q)x2

. (5.21)

From (5.19), we can see that the extreme values for Q are 0 and 1− P , so we define two

functions f1(x) and f2(x) for the two cases, respectively, as:

f1(x) = e−x2

+ P 2e−x2

, (5.22)


f2(x) = e−x2

+ P 2e−Px2

.

Now, in order to find the maximum difference between f1(x) and f2(x) we define F (x) =

f2(x)− f1(x) and integrate it from −∞ to ∞ to obtain:

∫ ∞

−∞F (x)dx =

∫ ∞

−∞(f2(x)− f1(x)) dx (5.23)

=√π(P 3/2 − P 2

).

The value for P that maximizes (5.24) is P (ρ) = 916

. This gives the possible ranges for

Q:

0 ≤ Q ≤ 7

16. (5.24)

We use the limits on the values of Q, 0 and 716

, to define two different beams. Figure

5.2 shows the result when plotting the correlation in intensity fluctuations versus the

separation distance between two different points for the two beams.

Note that for these two Gaussian beams, whose degrees of coherence and polarization

are the same, the fact that the degree of cross-polarization is not the same results in

having different correlations in intensity fluctuations for the different beams.

However, notice that the curves in Fig. 5.2 are not significantly different. David

Kuebel proposed that introducing beam propagation into the problem could help enhance

the effect, which we show in the next section.

5.5 Beam Propagation Enhances the Dependence of

the HBT Effect on the Degree of Cross-Polarization

To do so, let us go back to see what parameters differ between the two beams, which

happens to be Q. For the first beam, corresponding to Q = 0, δxy = δxx. While for the


x

10

1.25

3

0.25

−3

0.0

−1

0.5

0.75

2−2

1.0

Correlations in Intensity Fluctuations versus

Separation Distance between points P1 and P2

Figure 5.2: Correlation in Intensity Fluctuations versus Separation Distance.

The degree of coherence and the degree of polarization are the same for both beams.

However, their degree of cross-polarization is different; for the solid red curve, it is given

for the value of Q = 0, while for the dashed blue curve Q = 716

. This has an effect on

the correlation in intensity fluctuations of the beams, which is no longer the same for

the two beams. This can be seen in this graph: as the separation distance between point

P1 and P2 (illustrated in Fig. 5.1) becomes finite, the correlations for the two beams

are different. Note that the area between the curves, which corresponds to the difference

between the correlations for the two curves, is given by 27256

√π.


second one, corresponding to Q = 716

, δxy = 43δxx. In order to study the propagated beams,

one needs to use these relations in the description of the cross-spectral density matrix

for the propagated beams. The expression can be found in [1] and is given by:

Wij(ρ1, ρ2; z, ω) =AiAjBij

∆2ij(z)

exp

[−(ρ2 − ρ1)

2

8δ2∆2ij(z)

]exp

[−(ρ1 + ρ2)

2

2Ω2ij∆

2ij(z)

]exp

[− ik(ρ2

2 − ρ21)

2Rij(z)

],

(5.25)

where ∆2ij(z) = 1 +

(z

kσΩij

)2, 1

Ω2ij

= 14σ2 + 1

δ2ij, and Rij(z) =

[1 +

(kσΩij

z

)2]z.

Setting ∆xx = ∆yy = ∆, Ωxx = Ωyy = Ω, and Ax = Ay, the correlations in intensity

fluctuations for the propagated beam is given by:

〈∆I(ρ1)∆I(ρ2)〉 = 2A4x

1

∆4exp

[−(ρ1 + ρ2)

2

4σ2∆2

]exp

[−(ρ2 − ρ1)

2

Ω2∆2

](5.26)

+|Bxy|2

∆4xy

exp

[−(ρ1 + ρ2)

2

4σ2∆2xy

]exp

[−(ρ2 − ρ1)

2

Ω2xy∆

2xy

].

For a comparison between two beams, for both we choose: Ax = 1, z = 10km, k =

2πλ

= 2π500nm

, σ = 0.01m, σxy = σ = 0.01m, and δ = 0.001, and Bxy = 9/16. This gives:

∆ = 79.9 and Ω = 9.99x10−4m. The remaining parameters differ for the two beams

due to the difference in the values of δxy. For one beam, δxy = δ = 0.001m which gives

∆xy = 79.9 and Ωxy = 9.99x10−4m. For the second beam, δxy = 43δ = 4

3(0.001)m which

gives ∆xy = 59.8 and Ωxy = 1.33x10−3m.

When these parameters are substituted into (5.26) and plotted for two diametrically

opposite points (one at r and the other at−r) as a function of r (half the distance between

the points), the correlation profile in Fig. 5.3 is obtained. As in Fig. 5.2, the difference

in the degree of cross-polarization between the two beams results in a difference in the

correlation in intensity fluctuations at two different points. However, this difference is

greater when the beams are allowed to propagate in space away from the source, as can

be seen in Fig. 5.3.


2

4

6

8

10

-0.1 -0.05 0 0.05 0.1

〈∆I(r

)∆

(−r)〉

×10−

8

r

Figure 5.3: Correlations in Intensity Fluctuations of Two Propagated Beams.

Here we have two beams with the same degree of coherence and the same degree of po-

larization, but with different degrees of cross-polarization that are allowed to propagate.

This is a plot of the correlations in intensity fluctuations of the beams, 10km away from

the source, of two points at r and -r (diametrically opposite) versus r. Notice how the

profile of the correlations are different. This is due to the difference in the degree of

cross-polarization of the two beams. As in Fig. 5.2, the solid curve corresponds to Q = 0

at the source, while for the dashed curve Q = 716

at the source.


5.6 Conclusion

In conclusion, we have shown by using an example of Gaussian Schell-model beams that

to describe the correlations in intensity fluctuations of a beam at two different points,

the degree of coherence and the degree of polarization are not sufficient. In general, these

correlations will also depend on the newly discovered statistical parameter: the degree

of cross-polarization. We illustrate our result with examples and show that the effect of

the degree of cross-polarization can be greater on propagated beams than at the source.

Chapter 6

Conclusions

In this thesis, I presented a study in quantum and classical correlations. I investigated

the dynamics of entanglement, the most famous form of quantum correlations, in two-

level bipartite systems placed in an environment with a nonzero temperature. I proved

that in these systems, entanglement always decays in a finite time. I also looked at

bipartite harmonic oscillator systems, whose dynamics involved dephasing only; i.e., I

assumed that there is no loss of population. In this case, I showed that entanglement is

always long-lived. However, I also illustrated, using NOON states as an example, that

whatever is left of this entanglement is not necessarily useful as a resource in experiments.

In more recent years, quantum discord has gained attention as a different measure for

quantum correlations. To help understand its relation with entanglement, I calculated

both quantities for density matrices that describe the most general two-qubit systems

and plotted the result. I also identified that states that bound the relation. Moreover,

I produced similar results about the relationship between discord and linear entropy.

For my classical optics project, I studied the Hanbury-Brown Twiss effect in Gaussian

Schell-model beams. I showed that when the vector nature of the field is taken into

consideration, the effect depends on the newly discovered statistical parameter, known

as the degree of cross-polarization.

85

Chapter 6. Conclusions 86

One of the first lessons I learned while completing these projects is that entanglement

is not very useful. When I say it is not useful, I mean on the practical level. It is

true that in an ideal world where environmental interactions are omitted, many exciting

operations can be performed using entanglement as a resource, which is something that

was shown theoretically. However, as soon as we look at realistic systems, we see that it

decays too quickly for us to make impressive progress with quantum computing. In the

case of two-qubit systems, with a non-zero temperature environment, entanglement not

only decays, but does so in a short finite time. In other words, after this period of time,

it disappears completely; no remnants of it is left for us to try to make any use of it.

Moreover, although in the 2-mode-N-photon states undergoing dephasing, entanglement

is long-lived, whatever is left of it is of little practical use. In terms of practicality this

shows that whether entanglement decays in a finite time or not, its initial decay rate is

too high for any remnants of entanglement, if they exist, to be useful.

Quantum discord, although it differs from entanglement by depending on the con-

cept of measurement in quantum systems, is defined using similar methods as used in

defining entanglement in the general case of mixed states. For the pure state case, from

fundamental concepts, discord and entanglement are identical. However, the way they

are both extended to the mixed state case is by taking the worst case scenario. In the

case of entanglement, as with Wootters’ Concurrence for example, a minimization is done

over all pure state decomposition of a mixed state. With discord, a maximization over

all possible measurements on one of the subsystems is made. These are all methods to

make safe claims about the existence of quantum correlations without requiring the users

to use their brains; it gives them a recipe that they can use all the time. My work has

shown me that these methods are very much state-dependent. What this means is that

for some density matrices they can give a very accurate indication of how much quantum

correlations exist. In others, they can be very pessimistic, underestimating how much

correlations exist in the system. Also this pessimism depends on the quantity used; i.e.


discord or entanglement. It is like someone who owns a hat shop and to make sure that

all his customers’ heads will fit into the hats he sells, he makes them all super large in

size. This makes it safer because the chances that a customer’s head will not fit into

the head because almost zero. However, this does not mean that these hats accurately

represent the head sizes of the different customers. It would be better to make the hats

as customers order them by making the necessary measurements and analysis to see what

design and size would suit them best!

An approach to eliminate this one size fits all concept of quantifying quantum cor-

relations was made by Luo when he introduced MID [24]. Based on the measurement

concept of discord, Luo symmetrizes the quantification by performing the measurements

on both subsystems in the two-qubit system. He also eliminates the maximization by

selecting a set of projectors to perform these measurements. His choice of projectors is

very much dependent on the system under study; it works splendidly in some, but not

in others. The moral lesson is that we can still use the concept of discord, but choose

the projectors that are appropriate for our system. Just like the owner of the hat shop

does not need to produce super large hats into which everybody’s head can safely fit,

but rather tailor make them as his customers place the order, we can also design the

set of projectors that will be able to detect quantum correlations in our system. This

will depend very much on the system under study and the experimental setup. Here,

a collaboration between experimentalists and theorists will be needed to discuss these

concepts to come up with the optimal way to capture the correlations.

Measures of quantumness must stem from experiments. This is very much the ap-

proach taken in classical optics, where the derivation of an expression for correlations is

done by taking an operational approach. For example, the formalism, which is credited

to Professor Emil Wolf, used to describe the classical correlations in Chapter 5 of this

thesis, is based on the correlation experiment that studies the Hanbury-Brown Twiss

Effect. Using a similar approach, methods to capture quantum correlations should be


constructed, which should be based on the experiment of interest.

One of the most impactful results in my study of quantum discord is the pimple re-

gion in the discord-entropy relation. This is a region where an increase in linear entropy,

which corresponds to an increase of noise in the system, increases discord. Clearly, states

that display such a behaviour are not susceptible to the effects of decoherence caused

by environmental interactions, which is an uncommon behaviour in quantum systems.

However, from the quantum technology point of view, this result also implies that these

states could be the ideal ones to use in computations where environmental interactions

usually destroy the correlations and make the computation inefficient. Therefore, experi-

mentalists, who suffer directly from environmental noise in their labs, can benefit directly

from considering using these states.

As for the results about the importance of the degree of cross-polarization in the

Hanbury-Brown Twiss effect for Gaussian Schell model beams, the main lesson to learn

is that to describe the correlations in intensity fluctuations of the beam at two different

points, the degree of coherence and the degree of polarization alone are not enough

to describe these correlations. In general, it is very important to consider the degree of

cross-polarization as well. In cases where the field is completely polarized, the correlation

will not depend on this newly discovered statistical parameter. However, in the case of

partially polarized light, it will. That is why one way for experimentalists to be on the

safe side, if they want to ignore the degree of cross-polarization, is to use a polarizer in

their experimental set up.

I think that at this stage, we are far from understanding what quantum correlations

are and how to quantify them. Instead of focusing the research in this field on developing

quantum technologies and building powerful quantum computers, I believe that it is very

important to dedicate more efforts to understand the nature of quantum correlations.

The physicist’s job of describing the principles behind how quantum systems function

and what special properties they possess needs to be addressed first. When all the


mysteries are solved, the task of building great technologies can be handed to engineers

who can apply the concepts. This is what happened with classical computers. First,

the physical laws that describe how transistors and all the other structures that were

eventually used to build computers work were made clear by physicists. Following that,

the job of building these structures rested on the shoulders of engineers. That is why

at this stage, physicists have the big burden of finding clearer explanations to quantum

theory and the nature of quantum correlations.

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