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Physical Review A

Computable multipartite multimode Gaussian quantum correlation measure and the monogamy relations for continuous-variable systems

Jinchuan Hou, Liang Liu, Xiaofei Qi

Published Mar 14, 2022• 4 citations
Publisher

Abstract

In this paper, definitions of the unification condition, the hierarchy condition, and three kinds of monogamy relations for multipartite quantum correlation measures are given and discussed. A computable multipartite multimode Gaussian quantum correlation measure ${\mathcal{M}}^{(k)}$ is proposed for any $k$-partite multimode continuous-variable systems with $k\ensuremath{\ge}2$. The value of ${\mathcal{M}}^{(k)}$ only depends on the covariance matrices of continuous-variable states, is invariant under any permutation of subsystems, has no ancilla problem, is nonincreasing under $k$-partite local Gaussian channels (particularly, invariant under $k$-partite locally Gaussian unitary operations), and vanishes on $k$-partite product states. For a $k$-partite Gaussian state $\ensuremath{\rho}, {\mathcal{M}}^{(k)}(\ensuremath{\rho})=0$ if and only if $\ensuremath{\rho}$ is a $k$-partite product state. Moreover, ${\mathcal{M}}^{(k)}$ satisfies the unification condition and the hierarchy condition that a multipartite quantum correlation measure should obey. We also show that ${\mathcal{M}}^{(k)}$ is not strongly monogamous but completely monogamous and tightly monogamous.

DOI
10.1103/physreva.105.032429

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In this paper, definitions of the unification condition, the hierarchy condition, and three kinds of monogamy relations for multipartite quantum correlation measures are given and discussed. A computable multipartite multimode Gaussian quantum correlation measure M(k){\mathcal{M}}^{(k)}M(k) is proposed for any kkk-partite multimode continuous-variable systems with k\ensuremath≥2k\ensuremath{\ge}2k\ensuremath≥2. The value of M(k){\mathcal{M}}^{(k)}M(k) only depends on the covariance matrices of continuous-variable states, is invariant under any permutation of subsystems, has no ancilla problem, is nonincreasing under kkk-partite local Gaussian channels (particularly, invariant under kkk-partite locally Gaussian unitary operations), and vanishes on kkk-partite product states. For a kkk-partite Gaussian state \ensuremathρ,M(k)(\ensuremathρ)=0\ensuremath{\rho}, {\mathcal{M}}^{(k)}(\ensuremath{\rho})=0\ensuremathρ,M(k)(\ensuremathρ)=0 if and only if \ensuremathρ\ensuremath{\rho}\ensuremathρ is a kkk-partite product state. Moreover, M(k){\mathcal{M}}^{(k)}M(k) satisfies the unification condition and the hierarchy condition that a multipartite quantum correlation measure should obey. We also show that M(k){\mathcal{M}}^{(k)}M(k) is not strongly monogamous but completely monogamous and tightly monogamous.
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.
We propose a quantification CνGn of n-mode Gaussian coherence. The value of CνGn only depends on the covariance matrices and displacement vectors of continuous-variable states without any optimization procedures, and thus is easily calculated. For n=1, CνG1 is a proper Gaussian coherence measure of single-mode CV system. For n≥2, CνGn is invariant under any permutation of submodes, nonincreasing under any n-mode local incoherent Gaussian channels, vanishes at incoherent Gaussian states, and, satisfies the unification condition and the hierarchy condition that a multi-partite quantum correlation measure should obey. Thus CνGn is a multi-partite Gaussian correlation measure, which reveals, though the quantum coherence lives in single-partite systems, that the multi-mode coherence for continuous-variable systems can be regarded as a multipartite Gaussian correlation between modes, and such multi-partite Gaussian correlation is also a quantum resource. Moreover, we show that CνGn is completely monogamous as a multipartite Gaussian correlation measure. This means that the n-mode Gaussian coherence subjects to the principles of resource allocation. In addition, CνGn is an upper bound of the geometric-based single-mode Gaussian coherence measure CBu by the Bures distance at pure Gaussian states of mode ≤2 and can be used to detect coherence in any n-mode Gaussian states more efficiently.
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 demonstrate the existence of general constraints on distributed quantum correlations, which impose a trade-off on bipartite and multipartite entanglement at once. For all N-mode Gaussian states under permutation invariance, we establish exactly a monogamy inequality, stronger than the traditional one, that by recursion defines a proper measure of genuine N-partite entanglement. Strong monogamy holds as well for subsystems of arbitrary size, and the emerging multipartite entanglement measure is found to be scale invariant. We unveil its operational connection with the optimal fidelity of continuous variable teleportation networks.
Quantum entanglement and quantum nonlocality are known to exhibit monogamy; that is, they obey strong constraints on how they can be distributed among multipartite systems. Quantum correlations that comprise and go beyond entanglement are quantified by, e.g., quantum discord. It was observed recently that for some states quantum discord is not monogamous. We prove, in general, that any measure of correlations that is monogamous for all states and satisfies reasonable basic properties must vanish for all separable states: only entanglement measures can be strictly monogamous. Monogamy of other than entanglement measures can still be satisfied for special, restricted cases: we prove that the geometric measure of discord satisfies the monogamy inequality on all pure states of three qubits.
We present a complete analysis of the multipartite entanglement of three-mode Gaussian states of continuous-variable systems. We derive standard forms which characterize the covariance matrix of pure and mixed three-mode Gaussian states up to local unitary operations, showing that the local entropies of pure Gaussian states are bound to fulfill a relationship which is stricter than the general Araki-Lieb inequality. Quantum correlations can be quantified by a proper convex roof extension of the squared logarithmic negativity, the continuous-variable tangle, or contangle. We review and elucidate in detail the proof that in multimode Gaussian states the contangle satisfies a monogamy inequality constraint [G. Adesso and F. Illuminati, New J. Phys8, 15 (2006)]. The residual contangle, emerging from the monogamy inequality, is an entanglement monotone under Gaussian local operations and classical communications and defines a measure of genuine tripartite entanglements. We determine the analytical expression of the residual contangle for arbitrary pure three-mode Gaussian states and study in detail the distribution of quantum correlations in such states. This analysis yields that pure, symmetric states allow for a promiscuous entanglement sharing, having both maximum tripartite entanglement and maximum couplewise entanglement between any pair of modes. We thus name these states GHZ∕W\mathrm{GHZ}∕WGHZ∕W states of continuous-variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the WWW states of three qubits. We finally consider the effect of decoherence on three-mode Gaussian states, studying the decay of the residual contangle. The GHZ∕W\mathrm{GHZ}∕WGHZ∕W states are shown to be maximally robust against losses and thermal noise.