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arXiv:quant-ph/0309037 · Modern Physics Letters B

EVOLUTIONAL ENTANGLEMENT IN NONEQUILIBRIUM PROCESSES

Vyacheslav I. Yukalov

Published Feb 10, 2003• 24 citations
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Abstract

Entanglement in nonequilibrium systems is considered. A general definition for entanglement measure is introduced, which can be applied for characterizing the level of entanglement produced by arbitrary operators. Applying this definition to reduced density matrices makes it possible to measure the entanglement in nonequilibrium as well as in equilibrium statistical systems. An example of a multimode Bose–Einstein condensate is discussed.

DOI
10.1142/s021798490300497x

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Entanglement in nonequilibrium systems is considered. A general definition for entanglement measure is introduced, which can be applied for characterizing the level of entanglement produced by arbitrary operators. Applying this definition to reduced density matrices makes it possible to measure the entanglement in nonequilibrium as well as in equilibrium statistical systems. An example of a multimode Bose–Einstein condensate is discussed.
The problem of entanglement produced by an arbitrary operator is formulated and a related measure of entanglement production is introduced. This measure of entanglement production satisfies all properties natural for such a characteristic. A particular case is the entanglement produced by a density operator or a density matrix. The suggested measure is valid for operations over pure states as well as over mixed states, for equilibrium as well as nonequilibrium processes. Systems of arbitrary nature can be treated, described by field operators, spin operators, or any other kind of operator that is realized by constructing generalized density matrices. The interplay between entanglement production and phase transitions in statistical systems is analyzed using the examples of Bose-Einstein condensation, the superconducting transition, and magnetic transitions. The relation between the measure of entanglement production and order indices is analyzed.
A general description of entanglement is suggested as an action realized by an arbitrary operator over given disentangled states. The related entanglement measure is defined. Because of its generality, this definition can be employed for any physical systems, pure or mixed, equilibrium or nonequilibrium, and characterized by any type of operators, whether these are statistical operators, field operators, spin operators, or anything else. Entanglement of any number of parts from their total ensemble forming a multiparticle composite system can be determined. Interplay between entanglement and ordering, occurring under phase transitions, is analyzed by invoking the concept of operator order indices.
Abstract A measure of entanglement production by quantum operations is suggested. This measure is general, being valid for operations over pure states as well as over mixed states, for equilibrium as well as for nonequilibrium processes. The measure of entanglement production satisfies all properties typical of such a characteristic. Systems of arbitrary nature can be treated, described by field operators, spin operators, or any other operators, which is realized by defining generalized correlation matrices. Particular cases of entanglement production are considered.
A measure of correlations for statistical systems is advanced, called correlation index. The measure is general, being applicable to quantum and classical, equilibrium and nonequilibrium statistical systems. After formulating the general notion of correlation indices for arbitrary trace-class operators, the calculation of the correlation indices are illustrated by several examples of density operators and reduced density matrices. The computation of the correlation indices for equilibrium systems is exemplified for bosonic systems with the Bose-Einstein condensate, superconducting systems, and for spin systems. Correlation indices for nonequilibrium systems are considered for nonlinear dynamics of spin and pseudospin systems.
The survey discusses the problem of measuring correlations in composite quantum and statistical systems. After briefly recalling the standard methods of describing correlations, the emphasis is on the method of correlation indices. The latter method for quantifying the strength of correlations in composite physical systems is sufficiently general allowing for measuring in a unique way the correlation strength in both quantum and statistical systems, for static as well as for dynamic processes. The correlation indices, can be defined for correlation operators, reduced density matrices, and other operators containing information on correlations in the studied composite systems. The correlation indices measure all types of correlations, quantum correlations, such as entanglement, as well as classical correlations. They are applicable for characterizing static as well as dynamical processes. Examples are given of correlation indices for several quantum states and for spin and pseudospin systems. The quantification of correlations in a nonequilibrium system is exemplified by calculating the correlation index for a trapped Bose-Einstein condensate subject to the action of an alternating field.
A measure of entanglement production by quantum operations is suggested. This measure is general, being valid for operations over pure states as well as over mixed states, for equilibrium as well as for nonequilibrium processes. The measure of entanglement production satisfies all properties typical of such a characteristic. Systems of arbitrary nature can be treated, described by field operators, spin operators, or any other operators, which is realized by defining generalized correlation matrices. Particular cases of entanglement production are considered.