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arXiv:1312.7359 · Physical Review A

Fraction of isospectral states exhibiting quantum correlations

Michał Oszmaniec, Marek Kuś

Published Jul 8, 2014• 22 citations
arXivHTMLPDF

Abstract

For several types of correlations (mixed-state entanglement in systems of distinguishable particles, particle entanglement in systems of indistinguishable bosons and fermions, and non-Gaussian correlations in fermionic systems) we estimate the fraction of noncorrelated states among the density matrices with the same spectra. We prove that for the purity exceeding some critical value (depending on the considered problem) fraction of noncorrelated states tends to zero exponentially fast with the dimension of the relevant Hilbert space. As a consequence, a state randomly chosen from the set of density matrices possessing the same spectra is asymptotically a correlated one. To prove this we developed a systematic framework for detection of correlations via nonlinear witnesses.

DOI
10.1103/physreva.90.010302

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For several types of correlations (mixed-state entanglement in systems of distinguishable particles, particle entanglement in systems of indistinguishable bosons and fermions, and non-Gaussian correlations in fermionic systems) we estimate the fraction of noncorrelated states among the density matrices with the same spectra. We prove that for the purity exceeding some critical value (depending on the considered problem) fraction of noncorrelated states tends to zero exponentially fast with the dimension of the relevant Hilbert space. As a consequence, a state randomly chosen from the set of density matrices possessing the same spectra is asymptotically a correlated one. To prove this we developed a systematic framework for detection of correlations via nonlinear witnesses.
For several types of correlations: mixed-state entanglement in systems of distinguishable particles, particle entanglement in systems of indistinguishable bosons and fermions and non-Gaussian correlations in fermionic systems we estimate the fraction of non-correlated states among the density matrices with the same spectra. We prove that for the purity exceeding some critical value (depending on the considered problem) fraction of non-correlated states tends to zero exponentially fast with the dimension of the relevant Hilbert space. As a consequence a state randomly chosen from the set of density matrices possessing the same spectra is asymptotically a correlated one. To prove this we developed a systematic framework for detection of correlations via nonlinear witnesses.
The problem of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices $\ensuremath{\varrho}$ describing N-dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N> 8N>~8N> 8, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for N=8N=8N=8 and to estimate the volume of the bound entangled states in this case.
One of the most important questions in quantum information theory is the so-called separability problem. It involves characterizing the set of separable (or, equivalently entangled) states among mixed states of a multipartite quantum system. In this thesis we study the generalization of this problem to types of quantum correlations that are defined in a manner analogous to entanglement. We start with the subset of set of pure states of a given quantum system and call states belonging to the convex hull of this subset "non-correlated" states. Consequently, the states laying outside the convex hull are referred to as "correlated". In this work we focus on cases when there exist a symmetry group that preserves the class of "non-correlated" pure states. The presence of symmetries allows to obtain a unified treatment of many types of seemingly unrelated types of correlations. We apply our general results to particular types of correlations: (i) entanglement of distinguishable particles, (ii) particle entanglement of bosons, (iii) "entanglement" of fermions, (iv) non-convex-Gaussian correlations in fermionic systems, (v) genuine multiparty entanglement, and finally (vi) refined notions of bipartite entanglement based on the concept of the Schmidt number. We investigate the natural problems and questions concerning the correlations defined above: (I) We provide explicit polynomial characterization of various types of correlations for pure states. (II) We examine cases in which it is possible to give a complete analytical characterization of correlated mixed states. (III) We derive a variety of polynomial criteria for detection of correlations in mixed states.(IV) We use the above criteria and the technique of measure concentration to study typical properties of correlations on sets of isospectral density matrices.
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.
We introduce the notion of hidden quantum correlations. We present the mean values of observables depending on one classical random variable described by the probability distribution in the form of correlation functions of two (three, etc.) random variables described by the corresponding joint probability distributions. We develop analogous constructions for the density matrices of quantum states and quantum observables. We consider examples of four-dimensional Hilbert space corresponding to the "quantum roulette" and "quantum compass."