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Tractable measure of nonclassical correlation: Use of truncations of a density matrix

Akira Saitoh, Robabeh Rahimi, Mikio Nakahara

Published Jan 1, 2010• 0 citations

Abstract

AbstractFor the Oppenheim-Horodecki paradigm of nonclassical correlation, wepropose ameasure using truncations of a densitymatrix down to individual eigenspaces. It is computable within polynomial time in the dimension of the Hilbert space.Keywords: nonclassical correlation, computational tractability1. IntroductionClassical/nonclassical separation of correlations be-tween subsystems of a bipartite quantum system has beenan essential and insightful subject in the quantum infor-mation theory. The entanglement paradigm [1, 2] is basedon the state preparation stage: any quantum state that can-not be prepared by local operations and classical commu-nications (LOCC) [3] is entangled. There are paradigms[4, 5, 6, 7] based on post-preparation stages, which usediﬀerent deﬁnitions of classical and nonclassical correla-tions. On the basis of the Oppenheim-Horodecki deﬁni-tion [6, 8, 9], a quantum bipartite system consisting ofsubsystems A and B is (properly) classically correlatedif and only if it is described by a density matrix havinga product eigenbasis (PE), ρ

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AbstractFor the Oppenheim-Horodecki paradigm of nonclassical correlation, wepropose ameasure using truncations of a densitymatrix down to individual eigenspaces. It is computable within polynomial time in the dimension of the Hilbert space.Keywords: nonclassical correlation, computational tractability1. IntroductionClassical/nonclassical separation of correlations be-tween subsystems of a bipartite quantum system has beenan essential and insightful subject in the quantum infor-mation theory. The entanglement paradigm [1, 2] is basedon the state preparation stage: any quantum state that can-not be prepared by local operations and classical commu-nications (LOCC) [3] is entangled. There are paradigms[4, 5, 6, 7] based on post-preparation stages, which usediﬀerent deﬁnitions of classical and nonclassical correla-tions. On the basis of the Oppenheim-Horodecki deﬁni-tion [6, 8, 9], a quantum bipartite system consisting ofsubsystems A and B is (properly) classically correlatedif and only if it is described by a density matrix havinga product eigenbasis (PE), ρ

There is a commonly recognized paradigm in which a multipartite quantum system described by a density matrix having no product eigenbasis is considered to possess nonclassical correlation. Supporting this paradigm, we define two entropic measures of nonclassical correlation of a multipartite quantum system. One is defined as the minimum uncertainty about a joint system after we collect outcomes of particular local measurements. The other is defined by taking the maximum over all local systems about the minimum distance between a genuine set and a mimic set of eigenvalues of a reduced density matrix of a local system. The latter measure is based on an artificial game to create mimic eigenvalues of a reduced density matrix of a local system from eigenvalues of a density matrix of a global system. Numerical computation of these measures for several examples is performed.

Abstract Quantum nonlocality represents correlations between subsystems of a composite quantum system, usually including Bell nonlocality, steerability, and entanglement. According to the hypothesis of quantum mechanics, states of a quantum system Q described by a d -dimensional Hilbert space are denoted by density operators acting on . Under a basis e for the Hilbert space , every abstract density operator ρ of the system AB corresponds to a density matrix ρ e , which is a state of the complex Hilbert space . In this work, we discuss the consistency of quantum nonlocality of density operators ρ and their corresponding density matrices ρ e under the chosen basis e . It is proved that only when a basis e is a product one, a density operator ρ is entangled (respectively, Bell nonlocal, steerable) if and only if its density matrix ρ e is entangled (respectively, Bell nonlocal, steerable).

A measure of nonclassicality N in terms of local Gaussian unitary operations for bipartite Gaussian states is introduced. N is a faithful quantum correlation measure for Gaussian states as product states have no such correlation and every non product Gaussian state contains it. For any bipartite Gaussian state ρ A B , we always have 0 ≤ N ( ρ A B ) < 1 , where the upper bound 1 is sharp. An explicit formula of N for ( 1 + 1 ) -mode Gaussian states and an estimate of N for ( n + m ) -mode Gaussian states are presented. A criterion of entanglement is established in terms of this correlation. The quantum correlation N is also compared with entanglement, Gaussian discord and Gaussian geometric discord.