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Relations Between Classical and Quantum Properties Dynamics of quantum correlations in nonequilibrium systems: relations between classical and quantum properties Juan Camilo Castillo Hern´andez Advisor: Ferney Rodr´ıguez Departamento de F´ısica Universidad de los Andes A thesis submitted for the degree of F´ısica - Pregrado May 17, 2012 Advisor Ferney Rodr´ıguez Departamento de F´ısica Universidad de los Andes ii Abstract Quantum systems present unique behavior that cannot happen in classical systems. In order for such behavior to be seen, systems need to be coherent, which depends on the order of the medium in which the system evolves. How- ever, once the medium starts to show some level of disorder, the coherence is reduced, and the system starts to resemble its classical analogs. We show this type of behavior for two different systems: quantum walks and two-level systems. For quantum walks, we use the propagation rate as a sign of quan- tumness. For two-level systems, we use entanglement of formation, quantum discord, and violations to Leggett-Garg inequalities as measures of quantum- ness under both stochastic and thermal noise. In all cases we see clear evidence that an increase in disorder leads to a more classical behavior. For the case of thermal noise, we are also able to analyze the different behavior that emerges when the system is in thermal equilibrium and in nonequilibrium. We con- clude that at low temperatures moving away from equilibrium can lead to an increase in the quantumness of the system. iv Contents List of Figures v 1 Introduction 1 1.1 Quantum walks . .3 1.2 Two-qubit systems . .4 1.3 Outline . .5 2 Random Walks 7 2.1 The random walk model . .7 2.2 Classical random walks . .7 2.3 Quantum walks . 10 2.3.1 Ordered lattices . 11 2.3.1.1 Monoatomic lattice . 11 2.3.1.2 Diatomic lattices . 14 2.3.1.3 Propagation of the wavefunction . 16 2.3.2 Fibonacci diatomic lattices . 17 2.3.3 Random lattices . 20 2.3.3.1 Static disorder . 22 2.3.3.2 Dynamic random terms . 24 2.4 Summary . 26 i CONTENTS 3 Measures of quantumness in two-qubit systems 31 3.1 Wooters concurrence . 32 3.1.1 Entanglement of formation . 32 3.1.2 Difficulties of the EOF and the Wooters concurrence . 34 3.2 Quantum discord . 34 3.2.1 Classical mutual information . 35 3.2.2 Quantum mutual information and quantum discord . 37 3.3 Summary . 38 4 Leggett-Garg inequalities 41 4.1 Introduction . 41 4.2 The Einstein-Podolski-Rosen experiment and Bell's theorem . 43 4.3 Leggett-Garg inequalities . 45 4.3.1 Calculation of correlations in quantum systems . 47 4.3.2 Extension to an open system . 50 4.3.3 Leggett-Garg inequalities in the steady state . 54 4.4 Leggett-Garg inequalities as a measure of quantumness . 55 4.5 Summary . 56 5 Two qubits subject to stochastic noise 57 5.1 Description and evolution of the system . 58 5.1.1 Evolution in the Kraus representation . 60 5.1.2 Calculation of the entanglement of formation . 61 5.2 Loss of quantumness . 62 5.3 Summary . 64 6 Two qubits coupled with two baths at different temperatures 67 6.1 Description of the system . 68 6.2 General results . 70 6.3 Equilibrium between the baths . 73 6.4 Quantumness in non-equilibrium systems . 76 6.5 Approximations in symmetric systems . 79 ii CONTENTS 6.5.1 Maximum violation to the Leggett-Garg inequality . 82 6.6 Summary . 85 7 Conclusions 87 7.1 Random walks . 87 7.2 Two qubit systems . 88 A Results derived from this work 91 B The tight-binding model 93 C The mean of a complex exponential function 97 D Variance of a Gaussian colored noise 99 E Operator-sum representation and Kraus operators 103 F Open quantum systems and the Lindblad form 105 G Kraus representation from Lindbladian superoperators 109 G.1 Kraus represtentation for a 2-dimensional Hilbert space . 109 G.1.1 Evolution superoperator . 109 G.1.2 Kraus operators for the superoperator . 110 G.1.3 Kraus operators for a two level system in contact with two baths . 113 G.2 Kraus operators on a 4-dimensional Hilbert space . 115 G.2.1 Evolution superoperator . 115 G.2.2 Kraus operators for the superoperator . 118 G.2.3 Kraus operators for two interacting qubits in contact with two baths 121 References 125 iii CONTENTS iv List of Figures 2.1 Diagram of the system treated in a classical random walk . .7 2.2 Density plot, cross section, and dispersion for a classical random walk . .9 2.3 Diagram of the system treated in a quantum walk . 10 2.4 Density plot and cross section for a monoatomic lattice . 13 2.5 Density plot and cross section for diatomic lattices . 15 2.6 Propagation of the wavefunction as a function of time for diatomic ordered lattices . 16 2.7 Density plots for Fibonacci lattices . 19 2.8 Propagation of the wavefunction as a function of time for Fibonacci lattices 20 2.9 Density plots for some diatomic random lattices . 21 2.10 Density plots for some random lattices whose energies follow a normal dis- tribution . 23 2.11 Propagation of the wavefunction as a function of time for random lattices . 25 2.12 Density plots for lattices with dynamic disorder. 27 2.13 Propagation of the wavefunction as a function of time for lattices with random disorder, and different values of σE................... 28 2.14 Cross section of the wavefunction with random disorder for a large time. 29 3.1 Venn diagram of the amounts of uncertainty that add up to the mutual information. 36 v LIST OF FIGURES 4.1 Diagram that summarizes all of the probabilities involved in the calculation of the correlations based on (4.9) . 47 4.2 LGI for a two-level system . 51 5.1 Evolution of Wooters concurrence and quantum discord under a white noise. 63 5.2 Evolution of Wooters concurrence and quantum discord under correlated Gaussian noise. 65 6.1 Behavior of the LGI in time . 70 6.2 Detail of the behavior of the LGI in time . 71 6.3 Entanglement of formation, quantum discord and maximum violation to LGI as a function of mean temperature and heat current . 72 6.4 Cross section of maximum violations in equilibrium. 73 6.5 Behavior of EOF, quantum discord, and maximum violation to LGI in equilibrium . 74 6.6 Cross section of maximum violations out of equilibrium at a fixed mean temperature. 75 6.7 Measures of quantumness for a fixed mean temperature when moving away from equilibrium . 76 6.8 Regions with different behaviors for EOF and quantum discord . 77 6.9 Phase diagram for the violations to LGI . 78 6.10 Phase diagram for the violations to LGI with logarithmic axes . 79 6.11 Second derivatives of Vmax, G, and H with respect to ∆T for = 3, Γ = 0:01. 84 vi 1 Introduction Coherence is the fundamental element which gives quantum systems their particular prop- erties, and which makes them different from classical systems. The most striking differ- ences between quantum and classical behavior are related to systems in pure states that evolve unitarily. For instance, let us talk about Young's double slit experiment with electrons[1]. Each electron starts in a coherent state from a source at one side of the two slits, crosses to the other side and reaches a screen, creating a pattern of maxima and minima, the maxima being the places where the greatest quantity of electrons arrive. In a given point in the screen, the number of electrons detected depends on the total prob- ability amplitude of the wavefunction, which is the sum of the wavefunction coming from each of the two slits. If they arrive in phase, the probability density is at a maximum, but if they arrive with a phase of π, the wavefunctions cancel each other and the probability is at a minimum. Therefore, the precise pattern is very sensitive to the relative phases of each of the two paths for the electron. It also depends on the two electrons coming in with a well-defined relative phase. If the phase difference were somehow random, which is the case of decoherence, the pattern described before would not be present, instead showing two sharp maxima, one behind each of the slits as predicted by classical mechanics. This quantum mechanical example of coherence presents behavior very similar to double slit diffraction with light. For coherent light, such as a laser, a clear pattern of maxima and minima can be seen. However, if the light is not coherent, such as that from a simple light 1 1. INTRODUCTION bulb, the two maxima can be seen behind each of the slits. As interesting as the behavior of a coherent quantum system is, its coherence depends on perfect media, which allow us to exploit the quantum characteristics of systems. How- ever, the assumption of a perfect medium is only an approximation of reality. Any medium at room temperature ceases to be static due to the random motion caused by contact with the surroundings, which leads to random fluctuations. Such random fluctuations change the relative phases of the wavefunctions that are summed to find the final observed prob- abilities. This kind of decoherence is thus inevitable in any real system which can be experimentally reproduced. Thus it is very valuable to explore what happens when the assumption of a perfect medium is relaxed, allowing the presence of some type of disor- der, inevitably leading to some decoherence.
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