Quantum chaos and correlations in multipartite systems

In this thesis, we examine the dynamics, quantum correlations and the emergence of quantum chaos in multipartite systems, with emphasis on systems composed ofstrongly interacting discret

QUANTUM CORRELATIONS AND CHAOS IN MULTIPARTITE SYSTEMS

TEO YON SHIN

BSc (Hons), NTU

A THESIS SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY NUS GRADUATE SCHOOL FOR INTEGRATIVE SCIENCES AND

ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

AUGUST 2014

I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources

of information which have been used in the thesis.

This thesis has also not been submitted for any degree in any versity previously.

uni-TEO YON SHIN

20 August 2014

Dedicated to my parents, grandparents and great-aunt

is an inspiration to us all It has truly been a privilege to study under his guidancethroughout my PhD candidature.

I am also immensely grateful to my co-supervisor, Prof Kwek Leong Chuan, andthe chair of my thesis advisory committee, A/Prof Wang Zhisong This thesis wouldnot be possible without their persistent help and insightful inputs I truly appreciateeach and every discussion with them, which has significantly expanded the horizon

of my research I can never thank them enough, for always allocating time for themeetings as well as discussions despite their busy schedule

I would also like to thank all my fellow group members - Adam, Derek, Qifang,Hailong, Long Wen, Da Yang, Gaoyang, Shen Cheng and Naresh, for numerous en-lightening discussions and kind help I have received from them throughout all theseyears

My gratitude to NGS for giving me the opportunity to embark on this ible learning experience, and all the staffs at the NGS office for their support andassistance A special thanks to Joan who has expressed heartfelt concern on mywell-being, and helped me to cope with the difficult times during my candidature.

incred-To my high school physics teacher, Miss Yee Yoke Min She is the reason why Ibecome interested in physics She believes in me even when I doubted myself

To all my dear friends: Andy, Ben, Wee Shen, Pink, Sin Yin, Joyce, Gao Zheng,Yuan Jiun, Whelan, Xiao Wen, Wei Haw, Jess, Guan Pey, Vicky, Han Lin, Robert,Season, Lina, Yitjun, Wan Tung, Siew Huei, Shinyee, Jason, Sky, and forgive me fornot being able to include everyone here All the memories we have shared together

is so endearing to me; they kept me going when the road ahead seems daunting Aspecial thanks to Andy, who has given me valuable suggestions on my manuscript.And last but not least, I would like to thank all my family members for supporting

me and always make me feel home whenever I am with them No words can everexpress my gratitude to my mum and dad, my grandparents, and my great-aunt Iwould like to thank them for raising me up, for showering their love on me, and forallowing me to pursue my dreams and goals without any worries

In this thesis, we examine the dynamics, quantum correlations and the emergence

of quantum chaos in multipartite systems, with emphasis on systems composed ofstrongly interacting discrete and continuous variables subsystems Starting from an-alyzing the entanglement dynamics in triqubit and three coupled oscillators systems,

we comment on the role of environment on the sudden disappearances and rebirths ofentanglement, and demonstrate how local single mode squeezing can spuriously lead

to enhancement of multipartite entanglement Next we show how finite, chaotic tem may serve as an environment by considering a biqubit system weakly attached

sys-to two coupled anharmonic oscillasys-tors Due sys-to recent breakthough in engineeringstrong interaction in such discrete-continuous composite systems, it is important tounderstand the dynamics in chaotic regime where the rotating wave approximation(RWA) can no longer be applied With the construction of the double Rabi model,

we point out the importance of specifying the class of entanglement, in addressing therole of chaos in the steady state structure as well as dynamical generation of entan-glement We provide strong evidence that chaos in general suppresses entanglementthat does not fully involve all degrees of freedom in the chaotic system, while enhanc-ing global, multipartite entanglement Three different aspects of the entanglementdynamics in chaotic regime are presented, namely the failure of entanglement transfer

picture in chaos, evolution of noisy decoherence free state stabilized by chaos, andtwo competing effects of chaos in entanglement generation: chaos induced coherenceand decoherence Discord is regarded as a more fundamental quantum correlationmeasure than entanglement While discord has been studied in the quantum phasetransition of integrable systems, its behavior in the finite counterparts of these sys-tems which usually exhibit chaotic behavior has not been reported We show that,signatures of chaos can be identified with the fluctuations and production of quantumcorrelations in general, with discord showing more evident signatures By exploitingthe monogamy of quantum correlations which is a purely quantum mechanical effect,

we demonstrate that these signatures can be enhanced dramatically Finally we pose a new design of continuous quantum thermal machine based on Rabi model,which can be experimentally implemented in circuit QED Depending on the cou-pling between the superconducting qubit and the LC resonator, the dynamics of theworking components can be in quasiperiodic or highly chaotic regime We show that

pro-in the former, the engpro-ine produces coolpro-ing effect on an ordpro-inary qubit, while pro-in thelatter the engine functions as a heater instead

1.1 A Brief Overview on Quantum Information Processing 1

1.2 Models Studied in This Thesis 4

1.3 Outline of the Thesis 5

2 Review on Quantum Entanglement and Discord 11 2.1 Detection of Entanglement 11

2.1.1 Definition of Entanglement 12

2.1.2 Separability Criteria 13

2.2 Classification of Entanglement 17

2.2.1 Entanglement Network 18

2.3 Several Important Entanglement Measures 20

2.3.1 Von Neumann Entropy and Entanglement of Formation 21

2.3.2 Distillable Entanglement 21

2.3.3 Entanglement Cost 22

2.3.4 Negativity 23

2.3.5 Concurrence and I-concurrence 24

2.4 Entanglement Measures for Tripartite Systems 29

2.4.1 Monogamy of Entanglement 29

2.4.2 3-tangle and E w 31

2.4.3 Extending Bipartite Measures to Multipartite Measures 33

2.5 Quantum Discord 35

3 Entanglement Dynamics of Multipartite Systems 42 3.1 Discrete Systems 42

3.1.1 Tripartite I-Concurrence and Negativity 44

3.1.2 Entanglement Dynamics of Discrete Systems 46

3.1.3 Analytical Results for Triqubit Systems 47

3.1.4 Monogamy of Entanglement in Triqubit Systems 50

3.1.5 ESD and ESB in Closed Mixed Triqubit Systems 51

3.2 Three Coupled Oscillator Systems 55

3.2.1 Entanglement Dynamics of Three Coupled Oscillators 56

3.2.2 Comparison of Entanglement Dynamics with quantum fluctuation 59 3.2.3 Effect of squeezing in the entanglement dynamics of TCO 65

3.3 Two Coupled Anharmonic Oscillators as a Finite Environment 68

4 Entanglement and Chaos in Double Rabi Model 73 4.1 Double Rabi Model 75

4.2 Identification of Quantum chaos in DRM 80

4.3 Entanglement Structure of Ground State 86

4.4 Entanglement Dynamics of DRM 89

4.4.1 Comparison of Analytical and Numerical Results 90

4.4.2 Failure of the Entanglement Transfer Picture 94

4.4.3 Steady Entanglement Dynamics with Mixed Initial States 99

4.4.4 Role of Chaos in Entanglement Generation 101

5 Signatures of Classical Bifurcation and Chaos in Quantum

5.1 Quantum Phase Transition in Dicke Model 108

5.2 Classical Bifurcation in Dicke Model 111

5.3 Signatures of Bifurcation in Tripartite Rabi Model 117

5.3.1 Quantum correlation structure of the ground state 119

5.4 Dynamics of Quantum Correlations in TRM 122

5.4.1 Werner State 122

5.4.2 Initially entangled qubit-cavity mode 127

6 Rabi Model in Circuit QED: A Quantum Thermal Machine with Dual Functionality 135 6.1 Introduction 135

6.1.1 Quantum Thermodynamics 136

6.1.2 Quantum Cyclic Engines 138

6.2 Two Qubits Quantum Refrigerator 140

6.3 Dynamics of the Two Qubits Quantum Refrigerator 145

6.4 Rabi Model as a Self-Contained Quantum Heat Engine 150

6.5 Dynamics of the Hybrid Circuit QED Quantum Refrigerator 154

6.5.1 Consistency of the Dynamical Description with Thermodynamics155 6.5.2 Dual Functionality of the Model 158

6.5.3 Understanding the Switching Effect 162

6.5.4 Semi-Continuous Quantum Thermal Machine: A Modified Setup166 6.6 Two-Step Quantum Thermal Machine 168

7 Conclusion and Outlook 172 7.1 Future Developments 174

LIST OF FIGURES

2.1 Schematic classification of three-qubit entangled states 172.2 Entanglement distribution between subsystems in a six-particle com-posite system 192.3 Graphical representation of three-way (GHZ type) and two-way (Wtype) tripartite entanglement 202.4 Entanglement Venn diagrams of triqubit systems for W state, GHZstate and a general state with hybrid tripartite entanglement 303.1 Entanglement of generalized Bell state and GHZ state 443.2 Entanglement of generalized W state 45

3.3 Entanglement dynamics of triqubit systems with S3 as the Hamiltonian

and |111i as the initial state 493.4 Demonstration of monogamy of entanglement in the dynamics of triqubitsystem 513.5 Sudden disappearance/revival of bipartite entanglement (concurrence)

in closed triqubit systems (S4) for GHZ type (left) and W type (right)Werner states 53

List of figures

3.6 Sudden disappearance/revival of tripartite entanglement (tripartite ativity) in closed triqubit systems (S4) for GHZ type (left) and W type(right) Werner states 533.7 Dynamics of tripartite I-concurrence (blue line) and negativity (redline) in three coupled oscillators system (Eq 3.2.1), with the initialstate |000i 58

neg-3.8 Dynamics of tripartite negativity N ABC (red line), bipartite negativity

NAB (cyan line) and the analytical result for ξ12 (star) 64

3.9 Dynamics of tripartite negativity N123 of initially three single-modesqueezed states (Eq 3.2.23) 67

3.10 Dynamics of tripartite negativity N123 of initially three mode squeezedstates (Eq 3.2.21) 673.11 Entanglement dynamics of a harmonic oscillator coupled bilinearly to

a two coupled anharmonic oscillators system 703.12 Dynamics of two qubits coupled to two coupled anharmonic oscillators

as a finite environment 724.1 Schematic diagram for the DRM 78

4.2 Level statistics for near-degenerate DRM with g = 0.4 (main) and

g = 0.001 (inset) in high resolution . 84

4.3 Ground state entanglement and delocalization (Inset: ln ξ) of DRM with respect to the qubit-cavity coupling g . 86

4.4 First excited state entanglement and delocalization (Inset: ln ξ) of DRM with respect to the qubit-cavity coupling g . 874.5 Spectrum of full DRM Hamiltonian (without applying RWA) 88

4.6 Analytical results for the dynamics of concurrence C12 under RWA 92

4.7 Numerical results for the dynamics of concurrence C12 under RWA 92

4.8 Analytical results for the dynamics of negativity N AB under RWA 93

List of figures

4.9 Entanglement dynamics of DRM characterized by N12 (dashed lines),

NAB (solid lines) and N 1A (dash-dotted lines), F = 1 and g = 0.8 . 95

4.10 Entanglement dynamics of DRM characterized by N12 (dashed lines)

and N AB (solid lines), F = 0.4 and g = 0.1 . 96

4.11 Entanglement dynamics of DRM characterized by N12 (dashed lines)

and N AB (solid lines), F = 0.4 and g = 0.8 974.12 Comparison of entanglement dynamics (top panel) to the entanglement

structure of twelfth excited state (bottom panel), characterized by N AB

and N 1A|2B 98

4.13 Entanglement Dynamics of DRM characterized by N12 (qubits) and

NAB (cavity), mixed input state (|Ψ−i) with F = 0.4 and g = 0.8 100

4.14 Schematic diagram for an asymmetrically coupled DRM 101

4.15 Entanglement Dynamics of DRM characterized by N 1A (solid lines)

and N 2A (dashed lines), g 1B = g 2A = g 2B = g = 0.01 and initial

state:|11i ⊗ |00i 102

4.16 Entanglement Dynamics of DRM characterized by N 1A(solid lines) and

N 2A (dashed lines), and purity dynamics of subsystem 2A γ 2A

(dashed-dotted lines), g 1B = g 2A = g 2B = g = 0.01 and initial state:|Φ+i ⊗ |00i 103

4.17 Dependence of averaged bipartite entanglement N12 (blue,

dashed-dotted line), N 1A (black, solid line) and N 2A (cyan, dashed line) on

g 1A for g = 0.01 105 5.1 Ground state entanglement C(ρ) (red, dashed line) and discord D(ρ)

(blue, solid line) for the TRM 119

5.2 Ground state quantum mutual information I(ρ) (cyan, solid line) and classical correlation J (ρ AB) (black, dashed line) for the TRM 1205.3 Spectrum of TRM (Eq 5.3.1) under rotating wave approximation 1215.4 Spectrum of TRM (Eq 5.3.1) 121

both panels) 1255.8 Dependence of mean amplitude (main) and fluctuations (inset) of con-

currence for F = 0.4 on the coupling g 126

5.9 Dependence of mean amplitude (main) and fluctuations (inset) of

dis-cord for F = 0.4 on the coupling g 126

5.10 Statistics of concurrence dynamics for maximally entangled qubit-cavitymode 1285.11 Statistics of discord dynamics for maximally entangled qubit-cavitymode Top panel: Fluctuations of discord Bottom panel: mean am-

plitude of discord Highly distinctive peaks located at g bp , g1

s and g2

s

can be seen clearly in both panels 129

5.12 Dynamics of N 1A (cyan, dashed line) and N 2A (black, solid line), forinitial state given by Eq 5.4.2 (left panel) and Eq 5.4.4 (right panel)

respectively with g = 0.5 In the left panel both measures oscillate at

∼ 0.3, while in the right panel both measures oscillate at ∼ 0.8 130 5.13 Statistics of qubit-cavity entanglement N 1A dynamics for maximallyentangled qubit-cavity mode 131

5.14 Statistics of biqubit-cavity entanglement S 12|Adynamics for maximallyentangled qubit-cavity mode 132

5.15 Statistics of qubit-cavity entanglement N 1A dynamics for maximallyentangled qubit-cavity mode with higher initial energy (Eq 5.4.6) 1336.1 Schematic diagram for two qubits quantum refrigerator 1436.2 The temperature of the cold qubit at the steady state 149

List of figures

6.3 Schematic diagram for circuit QED quantum refrigerator 1526.4 Sum of heat currents as given in Eq 6.2.2 for two different couplingparameters 1566.5 Total entropy production in the baths as given in Eq 6.2.4 for twodifferent coupling parameters 1576.6 The heat current between the working components and its bath, for

cir-limit, for g = 0.35 160 6.10 Comparison of the cooling for E h = 2, T h = 25, g = 0.42 (green dashed line) and E′

h = 4, T′

h = 50, g = 0.35 (cyan line), while the temperature

of the working bath is fixed (T r = 8) 160

6.11 Dependence of the cooling on the relaxation rate of the cold qubit p c 161

6.12 Energy for the transition |ii −→ |i−1i (solid lines) for i = 2, 4, 6, 8, 10, 12 (from top to bottom according to the arrows), E h = 4 162

6.13 Energy for the transition |ii −→ |i−1i (solid lines) for i = 2, 4, 6, 8, 10, 12 (from top to bottom according to the arrows), E h = 2 1636.14 Dependence of populations in the thermal state of the resonator on the

coupling strength g 165 6.15 Heat flowing from the bath at temperature T h = 50 to the super-

conducting qubit, for g = 0.35 (cooling mode, cyan dotted line) and

g = 0.8 (heating mode, red solid line) 166 6.16 Modified setup with p h, pc described by Eq 6.5.2 (k i = 0.3), p c = 0

and g = 0.35 (cooling mode) 167

6.17 Modified setup with p h, pc described by Eq 6.5.2 (k i = 0.3), p c = 0

and g = 1 (heating mode) 167

6.18 Cooling and heating of the cold qubit for the two-step cyclic enginebased on hybrid circuit QED component 170

CHAPTER 1

INTRODUCTION

At the heart of quantum information lies the intriguing phenomenon of quantumentanglement When two objects are entangled, the quantum state of one objectcannot be adequately described without the reference of the other The famous EPRpaper [1] studied the non-local realistic behavior of entangled systems and showedthat quantum mechanics is “incomplete”, leading to the question of whether thereexists a local hidden variable (LHV) theory It was John Bell who derived a theoreticalupper bound for any theory obeying “local realism” [2] This bound is known as the

“Bell’s Inequality” which is a special type of entanglement witness [3], as it is violated

by certain entangled states Subsequent experiments have shown that LHV theory isnot compatible with entangled quantum mechanical systems [4]

In addition to its counterintuitive features, quantum entanglement is also sidered as the key resource necessary for various algorithms in quantum informationprocessing that outperform their existing classical counterparts, by being able to solvecertain difficult tasks efficiently, i.e in polynomial time [5] Famous examples includeShor’s factorization [6], Grover’s search algorithm [7] and Deutsch-Jozsa algorithm

con-1.1 A Brief Overview on Quantum Information Processing Chapter 1

[8] Its appealing properties have also enabled exciting new prospects in quantumcommunication technologies such as quantum key distribution[9], superdense coding[10] and teleportation [11] Hence, quantum information processing has become one

of the fastest growing fields in Physics for practical reasons as well as the fact that itholds the key to address some of the most fundamental questions in modern quantummechanics

The realization that entanglement is not the one and only resource that givesquantum computing its edge came into light when it was demonstrated that a certainprotocol utilizing mixed input state with no entanglement (DQC1) leads to exponen-tial speed-ups over classical computation [12] Subsequently, a simplified version ofShor’s algorithm was performed by using a 7-qubit NMR system at room temperature[13], condition under which entanglement among qubits cannot be sustained [14] Thenotion that non-entangled states are useless for quantum computation was refuted

by the formulation of quantum discord [15], which was connected to the quantumcomputation speed-up provided by completely mixed separable states [16] Discordcharacterizes how much a composite quantum system is disturbed under the act oflocal measurements While there is no question that discord can be utilized to per-form many useful tasks, whether it is essential to quantum computation is still anopen question It could be the case that discord simply appears alongside a certainunknown quantum property which is responsible for the speed-ups

Practical implementation of quantum technology generally requires precise nipulation and characterization of multipartite systems Understanding multipartitequantum correlations is a huge step from the existing framework for bipartite systems

ma-It turns out that even for the simplest case of tripartite systems, introducing an tional particle adds new dimensions to the study of quantum correlations in bipartitesystems which is already very complex Issues such as entanglement sharing among

addi-subsystems and monogamy of entanglement [17, 18] must be addressed in order to

fully describe the entanglement properties of multipartite systems Foremost of all is

Chapter 1 1.1 A Brief Overview on Quantum Information Processing

the emergence of inequivalent classes of entangled states While a bipartite systemcan only be either entangled or separable, a tripartite system possesses six differentclasses of entangled states [19] The second being the monogamy of entanglement,that constraints the distribution of quantum correlations among different subsystems[17, 20, 21] Monogamy of entanglement is a truly quantum effect, as in classicalinformation theory there is no fundamental limit on the sharing of correlations One

of the major consequences of monogamy is that different partition of a set cannotattain maximum entanglement simultaneously It is found to be responsible for thesecurity of quantum key distribution [22] The concept of monogamy plays a majorrole in understanding how the entanglement structures and dynamical entanglementproduction vary in different dynamical regimes

The implementation of quantum information tasks is also constrained by the

ob-stacle imposed by decoherence [23], the non-unitary deterioration of quantum states

due to interactions between quantum system and its environment, which ultimatelyleads to the loss of entanglement Proposals to counter the detrimental effect ofsystem-environment interactions include quantum error correction [24, 25] and quan-tum Zeno effect [26, 27], and also preparing the initial state of the system in the

decoherence free subspace [28] which effectively decouples the system dynamics from

its environment Decoherence of a system often follows a half-life rule, characterized

by the exponential decay of the off-diagonal elements of the density matrix However,

the entanglement in the system may disappear in finite time, and this is known as

“en-tanglement sudden death” (ESD) [29–31] Intriguingly, the opposite effect coined by

the name “entanglement sudden birth” (ESB) [32–34] is also reported which signals

a backflow of information from environment into the system This backflow raisesthe question of whether there should be sudden birth of entanglement in reservoirstates while the system entanglement undergoes sudden death, a scenario commonly

observed and referred as the “entanglement transfer picture” (ETP) [35, 36].

As a quantum system is never truly isolated, understanding the realistic

perfor-1.2 Models Studied in This Thesis Chapter 1

mance of various quantum information protocols and the mechanism of degradation ofquantum states requires models that properly account for the system-environmentalinteractions There are many notable examples, and most are extremely difficult tosolve due to the formidable degree of complexity imposed by the many-body envi-ronment Recent advancement on controlling quantum systems with small degrees

of freedom has opened up a new paradigm of simulating a many-body environmentusing finite but chaotic systems In fact, it has been shown that a kicked rotor inchaotic regime is capable of reproducing the dynamical effect of a pure-dephasingenvironment [37] Hence, it is interesting to consider whether a closed, finite systemcan serve as an environment through generic bilinear coupling to the system

1.2 Models Studied in This Thesis

In this thesis, we study the dynamics of quantum entanglement and discord in

mul-tipartite systems, which includes triqubit systems, three coupled oscillators

systems and discrete-continuous variables systems A generic description of

two level systems coupled to the field modes of a cavity in quantum

electrodynam-ics is given by the well-known Rabi Hamiltonian [38, 39] The Hamiltonian can be

solved exactly under the rotating wave approximation (RWA), which reduces it into

the Jaynes-Cummings model [40] provided that the coupling is weak, as well as the

detuning and field amplitude are small [41] Experimental implementations of thesesystems have made significant breakthrough recently In circuit QED, for supercon-ducting qubits the role of cavity can be played by a LC resonator [42, 43], and the

coupling has been strengthened from strong coupling regime [44] (g/ω ∼ 10−3) inearlier experimental realization, to more recently ultra-strong coupling (USC) regime

[45–47] (g/ω ∼ 10−1), in which the evidence of RWA breaking down has been vided [47] As a result the counter-rotating terms in the Rabi Hamiltonian can no

pro-longer be neglected, and the corresponding classical system is chaotic in USC regime.

Without making the RWA, numerical [48, 49] and approximate analytical solutions

Chapter 1 1.3 Outline of the Thesis

[50–53] has been provided under idealized conditions There are two main schemesused to attack this problem, one is based on the photonic Fock states pioneered bySwain [54], the other so called “generalized rotating wave approximation” utilizespolaron-like transformations which basically transform the Hamiltonian in the newbasis such that the counter-rotating term is eliminated as a result [55] The full spec-trum of the Rabi model is derived by Braak in 2011 [56] The counter rotating termsneglected in RWA conserves the energy and parity of the system, but not the totalnumber of excitations [57] As the counter rotating terms are responsible for severalnovel quantum mechanical phenomena such as the onset of quantum chaos [58, 59],

we are motivated to investigate the entanglement dynamics without applying RWA

in our systems which will be further elaborated below It is worth mentioning that

the theoretical study of the so called deep-coupling regime, i.e g/ω ∼ 1, has been

reported as well [55] In this regime, the Rabi Hamiltonian transcends into a wholenew domain in which many new and exciting features become evident

1.3 Outline of the Thesis

The material presented in this thesis can be divided into five parts In chapter two

we give a brief introduction on concepts in quantum information theory relevant to ourstudies Despite continual efforts, questions regarding detection, quantification andclassification of entanglement for arbitrary quantum systems remain unsolved, exceptfor biqubits and qubit-qutrit systems Mixed entangled states exhibit a formidabledegree of complexity and even questions like whether a given mixed state is entan-gled is a N-P hard problem [60] Consequently, the problem of detecting separability

is addressed Various entanglement measures have been proposed but an operative,conclusive measure is yet to be found Concurrence [61] is by far the most impor-tant entanglement measure for biqubit systems The other two useful entanglementmeasures for arbitrary bipartite pure states are the I-concurrence [62] and negativity[63] We discuss how these two measures can be generalized to multipartite settings

1.3 Outline of the Thesis Chapter 1

[64, 65] Monogamy of entanglement, its connection to the entanglement Venn gram [66] and how inequivalent classes of entanglement can be quantified in general

dia-is also explained Thdia-is dia-is followed by a short review on quantum ddia-iscord, where wediscuss its significance, formal definition and how it can be quantified Althoughcalculation of discord for arbitrary biqubit systems is already a laborious task, theprocedure can be significantly simplified if the density matrix has the X-state struc-ture The X-state structure is preserved in the dynamics if the Hamiltonian possessesz-axis symmetry [67], for example the Rabi model and the double Rabi model (DRM)[68] consisting of two qubits and two cavity modes

The dynamics of quantum correlations in tripartite systems, as well as ite system with two coupled anharmonic oscillators in chaotic regime acting as the

compos-environment, are studied in chapter three By constructing analytically tractable

models in triqubit and three coupled harmonic oscillators systems, we verify the curacy of our approaches employed in numerical calculations For the case of triqubitsystems, a simple yet insightful example is constructed to point out the possible fallacy

ac-of the commonly accepted ETP On the other hand, squeezing in quantum optics iscommonly employed to demonstrate non-classical aspects of light and prepare entan-gled field modes Squeezing of light is typically achieved in single mode or two modes

By theoretically constructing a three-mode squeezed initial state [69], we show that,

as expected, the entanglement production is significantly enhanced Surprisingly, wehave also observed that local squeezing [70] on the initial state of three coupled oscil-lators systems may suppress or enhance the tripartite entanglement That is to say,

the experimentally more accessible local single-mode squeezing is actually capable of enhancement of dynamical production of global entanglement However the underly-

ing mechanism is not as straightforward as the multi-mode case; the effect of localsqueezing on entanglement dynamics is apparently two-fold Finally, we attach twocoupled anharmonic oscillators (TCAO) to simple harmonic oscillator and biqubitsystems By examining the dynamics of quantum correlations, purity and decay of

Chapter 1 1.3 Outline of the Thesis

coherence in these systems, we demonstrate that the TCAO is capable of functioning

as an environment, as it leads to saturation of entropy, dephasing effect and evenESD and ESB in the system This is a further indication that rather than the size,

it is the complexity of the environment that dictates the decoherence process.

Entanglement is a purely quantum mechanical concept with no classical part On the contrary, chaos signaled by sensitive dependence on initial conditions ofaperiodic deterministic systems is incompatible with quantum mechanics Sensitivity

counter-on initial ccounter-onditicounter-ons implies that nearby trajectories separate at an expcounter-onential rate,but the notion of trajectory is not permitted in the quantum domain Intriguingly,studies on quantum systems that display chaotic behavior in the classical limit haverevealed the manifestation of chaos in entanglement Dynamics of entanglement in

DRM is investigated in chapter four, centralized on two seemingly contradicting

concepts, namely: chaos induced coherence and chaos induced decoherence We study

the quantum chaos in DRM, whose level statistics obey Wigner-Dyson distribution[71, 72] at sufficiently large coupling between the qubits and cavity modes In thefirst half of chapter four, we investigate how quantum chaos affects the entanglementstructure of the steady states We have found that chaos in general enhances themore “global” class of entanglement that involves more degrees of freedom, whilesuppresses the pairwise entanglement This is an aspect of the “two competing fac-tors” mentioned above Transformation of entanglement structure is closely related

to the delocalization of eigenstates, and can be regarded as signature of quantumchaos Pairwise entanglement between the continuous variables systems (i.e the cav-ity modes) is also found to be much more dominant than the biqubit systems, since

the former is not constrained by monogamy of entanglement Next we turn to the

dynamical settings, and examine the effect of chaos on entanglement production fromthree different perspectives First we demonstrate that the ETP is non-existent in thechaotic regime, followed by an example of how chaos leads to stabilized entanglementdynamics of noisy decoherence-free initial states Finally, we construct a variant of

1.3 Outline of the Thesis Chapter 1

our model that clearly demonstrates the manifestation of two competing factors of

chaos on entanglement We conclude that, the “chaos induced coherence” becomes

dominant over “chaos induced decoherence” in the deep chaotic regime, where theentanglement production within the chaotic subsystems is enhanced

Studies on the ground states of quantum systems undergoing quantum phase

tran-sition [73–75] at zero temperature have revealed that the quantum correlations display

scaling behavior in the vicinity of critical point [76–79] Although these systems areintegrable in the thermodynamical limit, chaotic behavior are usually exhibited when

N is finite [59] The behavior of quantum discord in chaotic systems has not been

studied, as the evaluation of discord is generally very difficult However, as mentionedearlier, the Rabi model (and its variant) that describes generic spin-boson interactionspreserves the X-state structure of the biqubit state, which simplifies the numerical cal-culation of discord significantly In the previous chapter, we have also demonstratedhow signature of chaos can be observed, albeit indirectly, via transformation in theentanglement distribution Previous studies have shown that signature of chaos canalso be observed in the entanglement production rate of Dicke model [80], near thecoupling that results to onset of chaos from the perspective of dynamical symmetry

breaking [81] This coupling is referred as the “bifurcation point” By employing the

concurrence as the entanglement measure for the biqubit system and also calculating

the discord, in chapter five we show that signatures of chaos can be observed in the

production as well as fluctuations of these two measures, by showing a well-definedpeak around the bifurcation point We take a step forward and consider how uniquecharacteristics of quantum correlations can be exploited, in order to make these sig-natures even more evident Inspired by the concept of monogamy and the insights ofhow different types of entanglement are generated dynamically, we have constructed

a simple initial state that leads to highly distinguished amplification of the ture observed previously near the bifurcation point Two additional peaks manifest,

signa-in the vicsigna-inity of couplsigna-ing parameters that roughly correspond to the clustersigna-ing of

Chapter 1 1.3 Outline of the Thesis

level repulsions in the spectrum This suggests the signatures we observed are notexclusive to the onset of bifurcation, but are in general sensitive to the details of theHamiltonian

In the most part of this thesis, we consider principally qubit-oscillator systems inwhich ultra-strong coupling between discrete and continuous variables subsystems can

be achieved owing to recent breakthrough in circuit QED engineering As a result,how these systems function as certain quantum devices in different dynamical regimescan be explored experimentally The construction of miniature quantum thermal de-vices [82] has received much attention Besides providing a platform to understandthe quantum limit of classical thermodynamical concepts, studies on these models alsostems from obvious practical motivations The understanding on quantum analogues

of four-stroke cyclic engines, such as the Carnot and Otto cycles are well developed inthe literature [83–85] On the other hand, quantum engines that operate in a contin-uous fashion by extracting work from a supply of heat flowing between two workingbaths at different temperatures have also gained considerable interests, as these en-gines do not require any external controls and input of work So far the most notable

example of self-contained thermal device is given by the two qubits refrigerator model

introduced in [86] As a purely theoretical construction the two qubits refrigerator haslittle to offer in practical applications, although understanding its working principleshas provided valuable insights In light of an emerging field known as “hybrid circuit

QED” (HCQED), in chapter six we propose an engine containing two working parts,

namely a superconducting qubit and a LC resonator We show that, when coupled

to an ordinary atom via interactions that can be engineered readily in HCQED, the

engine can function as a refrigerator when the coupling between the working circuit QED components is intermediate, and a heater when the coupling is large Besides

a higher feasibility in experimental implementations, the dual functionality of ourquantum thermal machine achieved by controlling a single parameter is also absent

in the two qubit model This is the first evidence that a quantum engine operating in

1.3 Outline of the Thesis Chapter 1

different dynamical regimes can deliver opposite outputs.

CHAPTER 2

REVIEW ON QUANTUM ENTANGLEMENT AND DISCORD

The objective of this chapter is to provide an overview for two important conceptscharacterizing quantum correlations, i.e entanglement and discord

In the first part of this chapter we shall focus on the detection, classification andquantification of quantum entanglement, with special emphasis on the concepts ofmonogamy and multipartite entanglement measures As we shall see later, these con-cepts are of paramount significance in understanding the distribution and generation

of entanglement in different dynamical regimes In the second part we discuss theconcept of quantum discord, which captures the quantum correlations from a morefundamental perspective than entanglement, namely how quantum systems behavedifferently from their classical counterparts under local measurements At the end ofthis chapter we shall examine how discord may be calculated for biqubit systems

The detection and measurement of entanglement is closely related and the former isusually the starting point to answer many questions regarding entanglement As amatter of fact, many entanglement measures are born out of or borrowed the ideafrom certain entanglement criteria Understanding various approaches of entangle-

2.1 Detection of Entanglement Chapter 2

ment detection also exposes the major problem encountered in quantum informationtheory Currently entanglement is well understood in pure bipartite systems of arbi-trary dimensions, but the obscurity of entanglement in mixed states and multipartitesystems poses a potentially severe problem for practical applications [87] In thissection the definition of separability is given for the cases of pure and mixed statesrespectively, followed by a brief overview on the detection of entanglement Moreemphasis will be placed on concepts of more relevance such as positive partial trans-pose criterion [88, 89] and entanglement witnesses [3, 90] The idea of bound andfree entanglement will be described too as it is related to the operational meaning ofnegativity which is one of the most important entanglement measures applied in ourstudy Introduction on several entanglement measures follows in the next section

2.1.1 Definition of Entanglement

Quantum entanglement is a subtle nonlocal correlation that has no classical analog,which manifests itself in the event of measurement performed on the entangled sub-systems Thus, entanglement is best characterized as a feature of the system thatcannot be created through local operations and classical communications (LOCC) [5].LOCC implies that local operations are performed independently on different parts ofthe joint system, and the results are communicated via a classical channel Although

we can describe a pure entangled composite system precisely, it is not possible toassign a self-sufficient quantum mechanical description on each individual subsystem

In other words, the whole is in a definite state, while the parts taken separately are

not The global states of an entangled composite system cannot be written as product

states, which is equivalent to having at least two coefficients in the Schmidt form [91–

93] For the case of a pure state of N constituents A1, A2, A3, , AN, separabilityimplies that the wavefunction can be expressed in terms of product state:

ρ = ρ A1 ⊗ ρ A2 ⊗ ρ A3 ⊗ · · · ⊗ ρ AN = ⊗N

Chapter 2 2.1 Detection of Entanglement

where ρ Ai is the reduced density matrix [94] given by ρ Ai = Tr{A j}(ρ), ∀A j 6= A i,otherwise the state is entangled Mathematical structures of entangled states arisefrom the tensor product formalism of the Hilbert space of composite quantum systems

On the other hand, since the quantum systems are hardly isolated from the ronment, the state of the system given by taking partial trace over the environment isoften mixed Description in terms of mixed states is also required if the preparationhistory of the system is uncertain A mixed state does not contain any entanglement

envi-if it is a convex sum of separable states [95] The proper definition of separability for

a mixed state of N subsystems A1, A2, A3, , AN is therefore:

j=1 pj = 1 Hence a mixed state is separable if and only if all the

M pure density matrices ρ j are separable (factorizable), otherwise it is entangled

2.1.2 Separability Criteria

Given an arbitrary quantum state, we wish to know whether it is entangled In otherwords, we have to check the separability based on the definition given previously.This problem seems straightforward but is actually still an open question In fact,proving separability for a mixed quantum state is “Non-deterministic Polynomialtime” (N-P) hard [60] in computational complexity theory There are no generalprocedures that allow us to find the optimum decomposition of any given state Forpure states, there are criteria that allow one to differentiate entangled and separablestates unequivocally, but for mixed states similar conclusions can be obtained onlyfor low-dimensional systems [96] There are two types of criteria on the separability

of quantum states, namely operational and non-operational criteria Operationalcriteria [97] refer to specific methods that will yield affirmative, negative or non-decisive answers Non-operational criterion gives us conclusive requirements which

2.1 Detection of Entanglement Chapter 2

can be verified but no general operational procedure is provided

Operational Criteria

The operational criteria exploit the mathematical structure of density matrices fromdifferent perspectives In particular we shall focus on PPT criterion [88] as it leads

to negativity [63] which is a very important entanglement measure in our study

1 Positive partial transpose (PPT) criterion [98]: First we shall introduce theconcept of partial transpose The partial transpose of a composite density matrix

AB is given by transposing with respect to one of its subsystems:

(ρ AB)TA aα,bβ = (ρ AB)bα,aβ (2.1.3)

The PPT criterion states that, bipartite states which are negative under partialtranspose are entangled That is if

!TA

i piρ i A T ⊗ ρ i B ≥ 0 (2.1.5)

Since ρ i

A≥ 0, from the fact that transposition preserves the signs of eigenvalues,

we can conclude that ρ i

A T

must be also a positive matrix This implies that

if we obtain negative eigenvalues in ρ AB after partial transposition, ρ AB mustrepresent an entangled state [88] The PPT criterion is only necessary andsufficient for 2 × 2 and 2 × 3, and only necessary for higher dimensional systems[98] For CV systems PPT is necessary and sufficient for separability of Gaussianstates [99], while some bound entangled non-Gaussian states can be detected via

Chapter 2 2.1 Detection of Entanglement

condition derived in [89] We shall make use of range criterion in next section

to introduce the bound entanglement which cannot be detected by PPT

2 Range criterion [100]: The range of an operator A acting on the Hilbert space H

is given by R(A) = {A(ψ) : ψ ∈ H}, i.e the range is spanned by the eigenvectors

of A with nonzero eigenvalues The range criterion states that if ρ is separable,

then there exist a collection of product states n

ψ i

A ⊗ φ j B

osuch that they span

the range of ρ and nψ i

A⊗φ j B∗o span the range of ρ TB, where the complex

conjugation and the partial transposition on ρ are taken in the same subspace.

The range criterion is stronger than the PPT in some cases An example for theviolation of range criterion by a PPT 2 × 4 system is shown below:

where 0 < b < 1 It is straightforward to check that ρ TB ≥ 0, but it is impossible

to find any product states which span the range of ρ and after partial conjugation

is performed, also spans the range of ρ TB Hence, according to the range criterion

Eq 2.1.6 is an entangled state

Non-Operational Criteria

Non-operational criteria is stronger than the operational criteria but there are nogeneral instructions to obtain conclusive results The entanglement witness is partic-ularly insightful as it arises from the geometrical nature of quantum states [101, 102]

2.1 Detection of Entanglement Chapter 2

A pure state ρ is entangled iff there exist a Hermitian operator W such that:

We can conclude that any state ρ with Tr(ρW ) < 0 is entangled, hence W is the

“entanglement witness” [3] This is necessary and sufficient for any bipartite system,

but no conclusion can be drawn if Tr(ρW) ≥ 0 As an example, Bell’s inequalities

are actually a special type of entanglement witness [103] All separable states mustsatisfy Bell’s inequalities, but satisfying Bell’s inequalities doesn’t mean the statesare separable Interestingly, entanglement witness can be constructed from the ther-modynamics point of view, and the link between entanglement and thermodynamicalquantities such as energy [104] and heat capacity [105] has been discussed Distilla-bility of entangled states can also be characterized by entanglement witness [106]

We have illustrated the difficulty of determining whether a general mixed state isseparable, and discussed some useful concepts related to the detection of entangle-ment In particular we talked about the strengths of different operational criteria andhow they can complement each other, and presented the geometrical interpretation ofentanglement witness We emphasize that the PPT criterion leads to the negativity,which is one of the most important entanglement measure applied in our study

Chapter 2 2.2 Classification of Entanglement

2.2 Classification of Entanglement

Entanglement in bipartite systems is relatively straightforward, but the combinations

of various entangled subsystems increases rapidly with the number of constituents[107] Consequently there are various types and classes of entanglement in multi-partite systems For example, in a tripartite system there are already six differentclasses of entanglement [19, 108] Since our studies are carried out on multipartitesystems, the class of entanglement in question must be specified clearly Here we shall

make use of the simplest multipartite system, namely a triqubit system to elaborate

the inequivalent classes of entanglement in tripartite systems In Fig.2.1 (adapted

Figure 2.1: (adapted from [108]) Schematic classification of three-qubit entangled states From the

outermost set towards the center : The GHZ class (describes the three-way tripartite entanglement),

W class (describes the two-way tripartite entanglement), three biseparable classes (describes the

bipartite entanglement between three different partitions of the system A|BC, B|AC, C|AB) and

the separable class States in the GHZ class can be converted into the W class via LOCC, but the reverse cannot be achieved Similarly for the W class and the biseparable class B.

from [108]) a hierarchy of six inequivalent classes of triqubit entanglement is shown,

according to convertibility via LOCC Two states are said to have same class of

entan-glement if they can be transformed into each other via LOCC with nonzero successrate There exists two incompatible tripartite entangled classes, namely the GHZ andthe W class [109] The GHZ class entangled state is represented by the famous GHZ

state [110, 111], which is three-way entangled and can be regarded as the three-qubit

2.2 Classification of Entanglement Chapter 2

analogue of Bell states It is the most “nonlocal” state for three qubit systems, butfragile as the remaining particles become disentangled upon losing any one of the par-

ticles [112] The maximally entangled state in the W class is the two-way entangled W

state [109], which in contrast is more robust as the remaining particles are still gled even if any one of them is traced off Within the W class there are three different

entan-biseparable classes B as there exists three different ways (i.e A|BC, B|AC, C|AB)

to bipartition a tripartite system, and finally at the center is the fully separable class

S All these sets are convex, compact, and satisfies the following:

The GHZ class contains all types of entanglement Any state belongs to the subsetGHZ/W can be converted to any physical states contained in the W class, but thereverse process is not possible no matter how many copies of input states in the Wclass one possess [113] Similarly, states that belong to the B class cannot be convertedinto tripartite entangled states, as entanglement cannot increase under LOCC

2.2.1 Entanglement Network

Understanding and controlling multipartite entanglement is crucial to practical cations of entanglement in quantum information processing Addressing problems likethe distribution and quantification of entanglement among the subsystems is highlynontrivial, as there are several aspects of multipartite entanglement that contribute

appli-to the complexity of the problem In consideration of this we have developed a scheme

to describe entanglement in a multipartite system, and we shall address it as

“entan-glement network” [114] The entan“entan-glement network helps us to visualize the possible

entanglement sharing structures in multipartite systems, which is extremely useful

in our study Multiple n-way entanglement may co-exist in a N-partite entangled system, where 2 ≤ n ≤ N and partitions of subsystems can also be entangled to each