Quantum systems usually suffer from unavoidable interactions with the environment. In most experiments, it is virtually impossible to isolate the system under study from the environment. It is thus imperative to study and understand how quantum systems interact with their surroundings. In the present study, I consider the evolution of quantum entanglement in a two-qubit system interacting with an environment in a regime where the non-Markovian effects are important. The present thesis is organized as follows: Chapter one is a general introduction to the thesis. In chapter two, some preliminary concepts that will be used in the subsequent chapters will be introduced and defined. These concepts include qubits, density matrix, Quantum entanglement and Entropic measures in quantum information. Ways to detect or quantify entanglement in a quantum system will also be discussed in the section on quantum entanglement. Chapter three will be about open quantum systems in general: concepts like master equations, Markovian quantum systems and non-Markovian quantum systems will be briefly reviewed. Chapter four will deal with time dependent entanglement features of two-qubit and multi-qubit systems interacting with an environment, basically all the results obtained in my study will be presented there. Some general conclusions will be drawn in chapter five. Copyright
Quantum systems usually suffer from unavoidable interactions with the environment. In most experiments, it is virtually impossible to isolate the system under study from the environment. It is thus imperative to study and understand how quantum systems interact with their surroundings. In the present study, I consider the evolution of quantum entanglement in a two-qubit system interacting with an environment in a regime where the non-Markovian effects are important. The present thesis is organized as follows: Chapter one is a general introduction to the thesis. In chapter two, some preliminary concepts that will be used in the subsequent chapters will be introduced and defined. These concepts include qubits, density matrix, Quantum entanglement and Entropic measures in quantum information. Ways to detect or quantify entanglement in a quantum system will also be discussed in the section on quantum entanglement. Chapter three will be about open quantum systems in general: concepts like master equations, Markovian quantum systems and non-Markovian quantum systems will be briefly reviewed. Chapter four will deal with time dependent entanglement features of two-qubit and multi-qubit systems interacting with an environment, basically all the results obtained in my study will be presented there. Some general conclusions will be drawn in chapter five. Copyright
In the first part of this thesis we extensively review the general theory of open quantum systems. We introduce new approaches to describe, in a hopefully more transparent way, some aspects which are usually misleading in a broad range of the literature. In addition, we revise new concepts and methods developed in recent years that complement the general theory mainly developed in the 1970s. In the second part, we focus our attention on the so-called Markovian processes which have been described in the previous part. We study the validity of typical equations used in simple systems such as harmonic oscillators, and propose and check different strategies to describe Markovian dynamics in more complicated interacting systems. Finally we propose a counter-intuitive situation where some level of noise can be actually useful for information tasks. In the third and last part, we approach the problem of the non-Markovian description of open quantum systems. To quantify how non-Markovian an open quantum system is we introduce a computable measure and apply it to some relevant situations. We also propose a way of probing the complexity of an environment by visualizing properties of a pair of qubits embedded in it. In addition, we develop a technique based on orthogonal polynomial theory to transform a wide range of environments into simple chains of particles which can be efficiently simulated with numerical techniques.
We explore various aspects of the quantum entanglement dynamics of systems of two, three and four qubits interacting with an environment at zero temperature in a non-Markovian regime, as described by the paradigmatic model recently studied by Bellomo et al (2007 Phys. Rev. Lett. 99 160502). We consider important families of initial states for the alluded systems. The average, typical entanglement evolution associated with each of these families is determined, and its relation with the evolution of the global degree of mixedness of the multi-qubit system is explored. For three and four qubits we consider the family of initial states equivalent under local unitary transformations to the |GHZ⟩ and | W ⟩ states, and compare their average behavior with the average behavior exhibited by initial maximally entangled two-qubit states. Furthermore, in the case of two qubits, the evolution of other manifestations of entanglement, related to measurable quantities, is also investigated. In particular, we consider the Mintert–Buchleitner concurrence lower bound and an entanglement indicator based upon the violation of local uncertainty relations.
Due to its great importance, both from the fundamental and from the practical points of view, it is imperative that the concept of entanglement is explored. In this thesis I investigate the connection between information measures, entanglement and the “speed” of quantum evolution. In Chapter 1 a brief review of the different information and entanglement measures as well as of the concept of “speed” of quantum evolution is given. An illustration of the quantum no-cloning theorem in connection with closed timelike curves is also provided. The work leading up to this thesis has resulted in the following three publications and in one conference proceeding: (A) C. Zander and A.R. Plastino, “Composite systems with extensive Sq (power-law) entropies”, Physica A 364, (2006) pp. 145-156 (B) S. Curilef, C. Zander and A.R. Plastino, “Two particles in a double well: illustrating the connection between entanglement and the speed of quantum evolution”, Eur. J. Phys. 27, (2006) pp. 1193-1203 (C) C. Zander, A.R. Plastino, A. Plastino and M. Casas, “Entanglement and the speed of evolution of multi-partite quantum systems”, J. Phys. A: Math. Theor. 40 (11), (2007) pp. 2861-2872 (D) A.R. Plastino and C. Zander, “Would Closed Timelike Curves Help to Do Quantum Cloning?”, AIP Conference Proceedings: A century of relativity physics, ERE 841, (2005) pp. 570-573. Chapter 2 is based on (A) and is an application of the Sq (powerlaw) entropy. A family of models for the probability occupancy of phase space exhibiting an extensive behaviour of Sq and allowing for an explicit analysis of the thermodynamic limit is proposed. Chapter 3 is based on (B). The connection between entanglement and the speed of quantum evolution as measured by the time needed to reach an orthogonal state is discussed in the case of two quantum particles moving in a one-dimensional double well. This illustration is meant to be incorporated into the teaching of quantum entanglement. Chapter 4 is based on (C). The role of entanglement in time evolution is investigated in the cases of two-, three- and N-qubit systems. A clear correlation is seen between entanglement and the speed of evolution. States saturating the speed bound are explored in detail. Chapter 5 summarizes the conclusions drawn in the previous chapters.
The basic features of the dynamics of open quantum systems, such as the dissipation of energy, the decay of coherences, the relaxation to an equilibrium or non-equilibrium stationary state, and the transport of excitations in complex structures are of central importance in many applications of quantum mechanics. The theoretical description, analysis and control of non-Markovian quantum processes play an important role in this context. While in a Markovian process an open system irretrievably loses information to its surroundings, non-Markovian processes feature a flow of information from the environment back to the open system, which implies the presence of memory effects and represents the key property of non-Markovian quantum behavior. Here, we review recent ideas developing a general mathematical definition for non-Markoviantiy in the quantum regime and a measure for the degree of memory effects in the dynamics of open systems which are based on the exchange of information between system and environment. We further study the dynamical effects induced by the presence of system-environment correlations in the total initial state and design suitable methods to detect such correlations through local measurements on the open system.
Knowledge of the dynamical behavior of correlations with no classical counterpart, like entanglement, nonlocal correlations and quantum discord, in open quantum systems is of primary interest because of the possibility to exploit these correlations for quantum information tasks. Here we review some of the most recent results on the dynamics of correlations in bipartite systems embedded in non-Markovian environments that, with their memory effects, influence in a relevant way the system dynamics and appear to be more fundamental than the Markovian ones for practical purposes. Firstly, we review the phenomenon of entanglement revivals in a two-qubit system for both independent environments and a common environment. We then consider the dynamics of quantum discord in non-Markovian dephasing channel and briefly discuss the occurrence of revivals of quantum correlations in classical environments.
In this Thesis I discuss the exact dynamics of simple non-Markovian systems. I focus on fundamental questions at the core of non-Markovian theory and investigate the dynamics of quantum correlations under non-Markovian decoherence. \n \nIn the first context I present the connection between two different non-Markovian approaches, and compare two distinct definitions of non-Markovianity. The general aim is to characterize in exemplary cases which part of the environment is responsible for the feedback of information typical of non- Markovian dynamics. I also show how such a feedback of information is not always described by certain types of master equations commonly used to tackle non-Markovian dynamics. \n \nIn the second context I characterize the dynamics of two qubits in a common non-Markovian reservoir, and introduce a new dynamical effect in a wellknown model, i.e., two qubits under depolarizing channels. In the first model the exact solution of the dynamics is found, and the entanglement behavior is extensively studied. The non-Markovianity of the reservoir and reservoirmediated-interaction between the qubits cause non-trivial dynamical features. \n \nThe dynamical interplay between different types of correlations is also investigated. In the second model the study of quantum and classical correlations demonstrates the existence of a new effect: the sudden transition between classical and quantum decoherence. This phenomenon involves the complete preservation of the initial quantum correlations for long intervals of time of the order of the relaxation time of the system.