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arXiv:2001.01244

A computable multipartite multimode Gaussian correlation measure and the monogamy relation for continuous-variable systems

Jinchuan Hou, Liang Liu, Xiaofei Qi

Published Jan 5, 2020• 0 citations
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Abstract

In this paper, a computable multipartite multimode Gaussian quantum correlation measure ${\mathcal M}^{(k)}$ is proposed for any $k$-partite continuous-variable (CV) systems with $k\geq 2$. ${\mathcal M}^{(k)}$ depends only on the covariance matrix of CV states, is invariant under any permutation of subsystems, is a quantification without ancilla problem, nonincreasing under $k$-partite local Gaussian channels (particularly, invariant under $k$-partite local Gaussian unitary operations), vanishes on $k$-partite product states. For a $k$-partite Gaussian state $ρ$, ${\mathcal M}^{(k)}(ρ)=0$ if and only if $ρ$ is a $k$-partite product state. Thus, for the bipartite case, ${\mathcal M}={\mathcal M}^{(2)}$ is an accessible replacement of the Gaussian quantum discord and Gaussian geometric discord. Moreover, ${\mathcal M}^{(k)}$ satisfies the unification condition, hierarchy condition that a multipartite quantum correlation measure should obey. ${\mathcal M}^{(k)}$ is not bipartite like monogamous, but, ${\mathcal M}^{(k)}$ is complete monogamous and tight complete monogamous.

DOI
10.48550/arxiv.2001.01244

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In this paper, a computable multipartite multimode Gaussian quantum correlation measure M(k){\mathcal M}^{(k)}M(k) is proposed for any kkk-partite continuous-variable (CV) systems with k≥2k\geq 2k≥2. M(k){\mathcal M}^{(k)}M(k) depends only on the covariance matrix of CV states, is invariant under any permutation of subsystems, is a quantification without ancilla problem, nonincreasing under kkk-partite local Gaussian channels (particularly, invariant under kkk-partite local Gaussian unitary operations), vanishes on kkk-partite product states. For a kkk-partite Gaussian state ρρρ, M(k)(ρ)=0{\mathcal M}^{(k)}(ρ)=0M(k)(ρ)=0 if and only if ρρρ is a kkk-partite product state. Thus, for the bipartite case, M=M(2){\mathcal M}={\mathcal M}^{(2)}M=M(2) is an accessible replacement of the Gaussian quantum discord and Gaussian geometric discord. Moreover, M(k){\mathcal M}^{(k)}M(k) satisfies the unification condition, hierarchy condition that a multipartite quantum correlation measure should obey. M(k){\mathcal M}^{(k)}M(k) is not bipartite like monogamous, but, M(k){\mathcal M}^{(k)}M(k) is complete monogamous and tight complete monogamous.
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Generally speaking, it is difficult to compute the values of the Gaussian quantum discord and Gaussian geometric discord for Gaussian states, which limits their application. In the present paper, for any (n+m)-mode continuous-variable system, a computable Gaussian quantum correlation M is proposed. For any state ρAB of the system, M(ρAB) depends only on the covariant matrix of ρAB without any measurements performed on a subsystem or any optimization procedures, and thus is easily computed. Furthermore, M has the following attractive properties: (1) M is independent of the mean of states, is symmetric about the subsystems and has no ancilla problem; (2) M is locally Gaussian unitary invariant; (3) for a Gaussian state ρAB, M(ρAB)=0 if and only if ρAB is a product state; and (4) 0≤M((ΦA⊗ΦB)ρAB)≤M(ρAB) holds for any Gaussian state ρAB and any Gaussian channels ΦA and ΦB performed on the subsystem A and B, respectively. Therefore, M is a nice Gaussian correlation which describes the same Gaussian correlation as Gaussian quantum discord and Gaussian geometric discord when restricted on Gaussian states. As an application of M, a noninvasive quantum method for detecting intracellular temperature is proposed.
We propose a measure of nonclassical correlation [Formula: see text] in terms of local Gaussian unitary operations based on square of the fidelity [Formula: see text] for bipartite continuous-variable systems. This quantity is easier to be calculated or estimated and is a remedy for the local ancilla problem associated with the geometric measurement-induced nonlocality. A simple computation formula of [Formula: see text] for any [Formula: see text]-mode Gaussian states is presented and an estimation of [Formula: see text] for any [Formula: see text]-mode Gaussian states is given. For any [Formula: see text]-mode Gaussian states, [Formula: see text] does not increase after performing a local Gaussian channel on the unmeasured subsystem. Comparing [Formula: see text] in scale with other quantum correlations such as Gaussian geometric discord for two-mode symmetric squeezed thermal states reveals that [Formula: see text] is much better in detecting quantum correlations of Gaussian states.
A quantum correlation N F G , A for ( n + m ) -mode continuous-variable systems is introduced in terms of local Gaussian unitary operations performed on Subsystem A based on Uhlmann fidelity F. This quantity is a remedy for the local ancilla problem associated with the geometric measurement-induced correlations; is local Gaussian unitary invariant; is non-increasing under any Gaussian quantum channel performed on Subsystem B;and is an entanglement monotone when restricted to pure Gaussian states in the ( 1 + m ) -mode case. A concrete formula for ( 1 + 1 ) -mode symmetric squeezed thermal states (SSTSs) is presented. We also compare N F G , A with other quantum correlations in scale, such as Gaussian quantum discord and Gaussian geometric discord, for two-mode SSTSs, which reveals that N F G , A has some advantage in detecting quantum correlations of Gaussian states.
We compute the quantum maximal correlation for bipartite Gaussian states of continuous-variable systems. Quantum maximal correlation is a measure of correlation with the monotonicity and tensorization properties that can be used to study whether an arbitrary number of copies of a resource state can be locally transformed into a target state without classical communication, known as the local state transformation problem. We show that the required optimization for computing the quantum maximal correlation of Gaussian states can be restricted to local operators that are linear in terms of phase-space quadrature operators. This allows us to derive a closed-form expression for the quantum maximal correlation in terms of the covariance matrix of Gaussian states. Moreover, we define Gaussian maximal correlation based on considering the class of local Hermitian operators that are linear in terms of phase-space quadrature operators associated with local homodyne measurements. This measure satisfies the tensorization property and can be used for the Gaussian version of the local state transformation problem when both resource and target states are Gaussian. We also generalize these measures to the multipartite case. Specifically, we define the quantum maximal correlation ribbon and then characterize it for multipartite Gaussian states.