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arXiv:1904.08294 · Laser Physics

Entanglement production by statistical operators

Vyacheslav I. Yukalov, Elizaveta P. Yukalova, V A Yurovsky

Published Apr 16, 2019• 8 citations
arXivHTMLPDF

Abstract

Abstract In the problem of entanglement there exist two different notions. One is the entanglement of a quantum state, characterizing the state structure. The other is entanglement production by quantum operators, describing the action of operators in the given Hilbert space. Entanglement production by statistical operators, or density operators, is an important notion arising in quantum measurements and quantum information processing. The operational meaning of the entangling power of any operator, including statistical operators, is the property of the operators to entangle wave functions of the Hilbert space they are defined on. The measure of entanglement production by statistical operators is described and illustrated by entangled quantum states, equilibrium Gibbs states, as well as by the state of a complex multiparticle spinor system. It is shown that this measure is in intimate relation to other notions of quantum information theory, such as the purity of quantum states, linear entropy, or impurity, inverse participation ratio, quadratic Rényi entropy, the correlation function of composite measurements, and decoherence phenomenon. This measure can be introduced for a set of statistical operators characterizing a system after quantum measurements. The explicit value of the measure depends on the type of the Hilbert space partitioning. For a general multiparticle spinor system, it is possible to accomplish the particle-particle partitioning or spin-spatial partitioning. Conditions are defined showing when entanglement production is maximal and when it is zero. The study on entanglement production by statistical operators is important because, depending on whether such an operator is entangling or not, it generates qualitatively different probability measures, which is principal for quantum measurements and quantum information processing.

DOI
10.1088/1555-6611/ab0cf9

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Abstract In the problem of entanglement there exist two different notions. One is the entanglement of a quantum state, characterizing the state structure. The other is entanglement production by quantum operators, describing the action of operators in the given Hilbert space. Entanglement production by statistical operators, or density operators, is an important notion arising in quantum measurements and quantum information processing. The operational meaning of the entangling power of any operator, including statistical operators, is the property of the operators to entangle wave functions of the Hilbert space they are defined on. The measure of entanglement production by statistical operators is described and illustrated by entangled quantum states, equilibrium Gibbs states, as well as by the state of a complex multiparticle spinor system. It is shown that this measure is in intimate relation to other notions of quantum information theory, such as the purity of quantum states, linear entropy, or impurity, inverse participation ratio, quadratic Rényi entropy, the correlation function of composite measurements, and decoherence phenomenon. This measure can be introduced for a set of statistical operators characterizing a system after quantum measurements. The explicit value of the measure depends on the type of the Hilbert space partitioning. For a general multiparticle spinor system, it is possible to accomplish the particle-particle partitioning or spin-spatial partitioning. Conditions are defined showing when entanglement production is maximal and when it is zero. The study on entanglement production by statistical operators is important because, depending on whether such an operator is entangling or not, it generates qualitatively different probability measures, which is principal for quantum measurements and quantum information processing.
The review is devoted to two important quantities characterizing many-body systems, order indices and the measure of entanglement production. Order indices describe the type of order distinguishing statistical systems. Contrary to the order parameters characterizing systems in the thermodynamic limit and describing long-range order, the order indices are applicable to finite systems and classify all types of orders, including long-range, mid-range, and short-range orders. The measure of entanglement production quantifies the amount of entanglement produced in a many-partite system by a quantum operation. Despite that the notions of order indices and entanglement production seem to be quite different, there is an intimate relation between them, which is emphasized in the review.
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We define and analyse the concept of entanglement production during the evolution of a general quantum mechanical dissipative system. While it is important to minimise entropy production in order to achieve thermodynamical efficiency, maximising the rate of change of entanglement is important in quantum information processing. Quantitative relations are obtained between entropy and entanglement productions, under specific assumptions detailed in the text. We apply these to the processes of dephasing and decay of correlations between two initially entangled qubits. Both the Master equation treatment as well as the higher Hilbert space analysis are presented. Our formalism is very general and contains as special cases many reported individual instance of entanglement dynamics, such as, for example, the recently discovered notion of the sudden death of entanglement.
In early days of quantum theory it was believed that the results of measurements performed on two distant physical systems should be uncorrelated thus their quantum state should be separable it means described by a simple tensor product of the individual local state vectors or a tensor product of individual local density operators. It was shown many years ago by EPR that two systems which interacted in the past and separated afterwards had to be described in most cases by particular non-separable states which are called entangled. It was noticed by Zanardi et al. that a Hilbert space of possible state vectors of compound physical system could be partitioned in different way by introducing various tensor product structures induced by the experimentally accessible observables (interactions and measurements). Therefore a separable state in one partition could become entangled in different partition. In this sense the entanglement was relative to a particular set of experimental capabilities. Continuing this line of thought Torre et al. claimed to prove that for any separable state there existed strongly correlated physical observables therefore all quantum states were entangled. The same claims one may find in several recent papers. In this paper we will discuss in operational way the differences existing between separable, non-separable and entangled states and we will show that the conclusions of Torre et al. were unjustified. A sufficient condition for entanglement is the violation of BI-CHSH and/or steering inequalities. Since there exist experiments outside the quantum physics violating these inequalities therefore in the operational approach one cannot say that the entanglement is an exclusive quantum phenomenon.
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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.