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Information measures, entanglement and quantum evolution

Claudia Zander

Published Sep 6, 2007• 0 citations
OpenAlex

Abstract

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.

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•May 8, 2012•0 citations
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.
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
The concept of quantum speed limit-time (QSL) was initially introduced as a lower bound to the time interval that a given initial state ${\ensuremath{\psi}}_{I}$ may need so as to evolve into a state orthogonal to itself. Recently [V. Giovanetti, S. Lloyd, and L. Maccone, Phys. Rev. A 67, 052109 (2003)] this bound has been generalized to the case where ${\ensuremath{\psi}}_{I}$ does not necessarily evolve into an orthogonal state, but into any other ${\ensuremath{\psi}}_{F}$. It was pointed out that, for certain classes of states, quantum entanglement enhances the evolution ``speed'' of composite quantum systems. In this work we provide an exhaustive and systematic QSL study for pure and mixed states belonging to the whole 15-dimensional space of two qubits, with ${\ensuremath{\psi}}_{F}$ a not necessarily orthogonal state to ${\ensuremath{\psi}}_{I}$. We display convincing evidence for a clear correlation between concurrence, on the one hand, and the speed of quantum evolution determined by the action of a rather general local Hamiltonian, on the other one.
The time needed by a quantum system to reach a state fully distinguishable from the original one provides a natural way of determining how fast the corresponding dynamical evolution is. This orthogonality time admits a lower bound, expressible in terms of the energy's expectation value and the energy's standard deviation, that yields a 'quantum speed limit'. Composite quantum systems need entanglement in order to achieve this limit. So far, most studies on the connection between entanglement and the quantum speed limit have focused on the case of non-interacting systems. The connection between quantum speed and entanglement is systematically investigated here for the case of a system of two interacting qubits, taking into consideration all possible initial states that evolve into an orthogonal one.
Entanglement is closely related to some fundamental features of the dynamics of composite quantum systems: quantum entanglement enhances the "speed" of evolution of certain quantum states, as measured by the time required to reach an orthogonal state. The concept of "speed" of quantum evolution constitutes an important ingredient in any attempt to determine the fundamental limits that basic physical laws impose on how fast a physical system can process or transmit information. Here we explore the relationship between entanglement and the speed of quantum evolution in the context of the quantum brachistochrone problem. Given an initial and a final state of a composite system we consider the amount of entanglement associated with the brachistochrone evolution between those states, showing that entanglement is an essential resource to achieve the alluded time-optimal quantum evolution.
Due to their great importance, both from the fundamental and from the practical points of view, it is imperative that the various facets of the concepts of information and entanglement are explored systematically in connection with diverse physical systems and processes. These concepts are at the core of the emerging field of the Physics of Information. In this Thesis I investigate some aspects of the dynamics of information in both classical and quantum mechanical systems and then move on to explore entanglement in fermion systems by searching for novel ways to classify and quantify entanglement in fermionic systems. In Chapter 1 a brief review of the different information and entropic measures as well as of the main evolution equations of classical dynamical and quantum mechanical systems is given. The conservation of information as a fundamental principle both at the classical and quantum levels, and the implications of Landauer's theorem are discussed in brief. An alternative and more intuitive proof of the no-broadcasting theorem is also provided. Chapter 2 is a background chapter on quantum entanglement, where the differences between the concept of entanglement in systems consisting of distinguishable subsystems and the corresponding concept in systems of identical fermions are emphasized. Different measures of entanglement and relevant techniques such as majorization, are introduced. To illustrate some of the concepts reviewed here I discuss the entanglement properties of an exactly soluble many-body model which was studied in paper (E) of the publication list corresponding to the present Thesis. An alternative approach to the characterization of quantum correlations, based on perturbations under local measurements, is also briefly reviewed. The use of uncertainty relations as entanglement indicators in composite systems having distinguishable subsystems is then examined in some detail. Chapter 3 is based on papers (A) and (B) of the list of publications. Extended Landauer-like principles are developed, based amongst others on the conservation of information of divergenceless dynamical systems. Conservation of information within the framework of general probabilistic theories, which include the classical and quantum mechanical probabilities as particular instances, is explored. Furthermore, Zurek's information transfer theorem and the no-deleting theorem are generalized. Chapter 4 is based on articles (C) and (D) mentioned in the publication list, and investigates several separability criteria for fermions. Criteria for the detection of entanglement are developed based either on the violation of appropriate uncertainty relations or on inequalities involving entropic measures. Chapter 5 introduces an approach for the characterization of quantum correlations (going beyond entanglement) in fermion systems based upon the state disturbances generated by the measurement of local observables. Chapter 6 summarizes the conclusions drawn in the previous chapters. The work leading up to this Thesis has resulted in five publications in peer reviewed science research journals.