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International Journal of Modern Physics B

Monogamy inequality and entanglement sharing in optomechanics

Jamila Hmouch, Mohamed Amazioug, Mostafa Nassik

Published Mar 25, 2023• 3 citations
Publisher

Abstract

In this paper, we quantitatively study the dynamics of the tripartite entanglement of the three-mode Gaussian state in a cavity-free optomechanical system. After deriving the explicit form of the evolved state matrix covariance, we elucidate in detail the evidence of entanglement monogamy in different mode partitions, based on Coofman–Kundu–Wotters inequalities, using the Gaussian Contangle quantifying bipartite entanglement. Next, we focus on the genuine tripartite entanglement quantified by means of the Residual Gaussian Contangle.

DOI
10.1142/s0217979224500784

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arXiv:1603.08173•Physical Review A•Jan 12, 2017•181 citations
In this paper, we quantitatively study the dynamics of the tripartite entanglement of the three-mode Gaussian state in a cavity-free optomechanical system. After deriving the explicit form of the evolved state matrix covariance, we elucidate in detail the evidence of entanglement monogamy in different mode partitions, based on Coofman–Kundu–Wotters inequalities, using the Gaussian Contangle quantifying bipartite entanglement. Next, we focus on the genuine tripartite entanglement quantified by means of the Residual Gaussian Contangle.
Abstract In this work, we discuss the monogamy property of quantum correlations (entanglement and steering) distributed in a cavity-free optomechanical system. In a tripartite pure Gaussian state, we verify the hierarchy of the studied quantum correlations. We investigate in detail the CKW-type monogamy inequalities, based on Gaussian Rényi-2 entropy entanglement, as well as, Gaussian steering. We quantify genuine tripartite entanglement by using the minimum residual Gaussian entanglement. In addition, we consider the minimum residual Gaussian steering as a quantifier of genuine tripartite steering. Our results indicate that, in the one-mode versus two-mode Gaussian state, the correlations hierarchy was respected; in the considered pure state, the monogamy of entanglement and steering distribution was proven and the studied genuine tripartite correlations have similar time evolution under the same conditions.
In this work, we study a theoretical scheme to investigate the monogamy of quantum steering in a hybrid exciton--cavity optomechanical (EOM) system. Within the Gaussian-state formalism, we quantify the steerability among the exciton, photon, and mechanical modes in both the steady-state and dynamical regimes. We establish and verify CKW-type monogamy inequalities for Gaussian steering and demonstrate that the distribution of bipartite steering among the three modes can be controlled through the system parameters. We further examine the influence of thermal noise and temporal evolution on the steering properties and the corresponding monogamy relations. Our results show that increasing the temperature generally suppresses the strength of Gaussian steering and modifies the degree of steering sharing imposed by the CKW-type monogamy relations. In contrast, in the dynamical regime, the steerability can increase with evolution time over suitable temporal windows, indicating that the transient dynamics can be exploited to enhance quantum correlations. These results highlight the potential of hybrid exciton--optomechanical platforms for controllable quantum correlations and provide a promising route toward the generation and manipulation of multipartite quantum resources.
In this paper, we analyze nonclassical correlations between bipartite states in two optomechanical systems. The first system (Sec. 2) consists of two nanoresonators spatially separated by broadband squeezed light, where each cavity has a fixed mirror and a movable one. The second system (Sec. 3) is an atom-optomechanical system consisting of an atomic ensemble placed inside an optical nanoresonator with a vibrating mirror. For both optomechanical systems, we give the Hamiltonian and the explicit expression of covariance matrix leading to the quantum equations describing the dynamic evolution of the system. Then, the nonclassical correlations are quantified using the logarithmic negativity and Gaussian quantum discord. We propose also a scheme for examining the evolution of Gaussian quantum steering and its asymmetry in each system. We show that the entanglement of the two mechanical modes is very strongly related to the parameters characterizing the environment where the movable mirrors evolve, in particular the squeeze parameter, the optomechanical cooperativity and thermal bath temperature.
We consider three modes AAA, BBB and CCC and derive continuous variable monogamy inequalities that constrain the distribution of bipartite entanglement amongst the three modes. The inequalities hold for all such tripartite states, without the assumption of Gaussian states, and are based on measurements of two conjugate quadrature phase amplitudes XiX_{i}Xi​ and PiP_{i}Pi​ at each mode i=A,Bi=A,Bi=A,B. The first monogamy inequality is DBA+DBC≥1D_{BA}+D_{BC}\geq1DBA​+DBC​≥1 where DBA<1D_{BA}<1DBA​<1 is the widely used symmetric entanglement criterion, for which DBAD_{BA}DBA​ is the sum of the variances of (XA−XB)/2(X_{A}-X_{B})/2(XA​−XB​)/2 and (PA+PB)/2(P_{A}+P_{B})/2(PA​+PB​)/2. A second monogamy inequality is EntBAEntBC≥1(1+(gBA(sym))2)(1+(gBC(sym))2)Ent_{BA}Ent_{BC}\geq\frac{1}{\left(1+(g_{BA}^{(sym)})^{2}\right)\left(1+(g_{BC}^{(sym)})^{2}\right)}EntBA​EntBC​≥(1+(gBA(sym)​)2)(1+(gBC(sym)​)2)1​ where EntBA<1Ent_{BA}<1EntBA​<1 is the EPR variance product criterion for entanglement. Here EntBAEnt_{BA}EntBA​ is a normalised product of variances of XB−gBA(sym)XAX_{B}-g_{BA}^{(sym)}X_{A}XB​−gBA(sym)​XA​ and PB+gBA(sym)PAP_{B}+g_{BA}^{(sym)}P_{A}PB​+gBA(sym)​PA​, and gBA(sym)g_{BA}^{(sym)}gBA(sym)​ is a parameter that gives a measure of the symmetry between the moments of AAA and BBB. We also show that the monogamy bounds are increased if a standard steering criterion for the steering of BBB is not satisfied. We illustrate the monogamy for continuous variable tripartite entangled states including the effects of losses and noise, and identify regimes of saturation of the inequalities. The monogamy relations explain the experimentally observed saturation at DAB=0.5D_{AB}=0.5DAB​=0.5 for the entanglement between AAA and BBB when both modes have 50\% losses, and may be useful to establish rigorous bounds of correlation for the purpose of quantum key distribution protocols.
We consider three modes AAA, BBB, and CCC and derive monogamy inequalities that constrain the distribution of bipartite continuous variable Einstein-Podolsky-Rosen entanglement amongst the three modes. The inequalities hold without the assumption of Gaussian states, and are based on measurements of the quadrature phase amplitudes Xi{X}_{i}Xi​ and Pi{P}_{i}Pi​ at each mode i=A,B,Ci=A,B,Ci=A,B,C. The first monogamy inequality involves the well-known quantity DIJ{D}_{IJ}DIJ​ defined by Duan-Giedke-Cirac-Zoller as the sum of the variances of $({X}_{I}\ensuremath{-}{X}_{J})/2$ and (PI+PJ)/2({P}_{I}+{P}_{J})/2(PI​+PJ​)/2 where $[{X}_{I},{P}_{J}]={\ensuremath{\delta}}_{IJ}$. Entanglement between III and JJJ is certified if DIJ<1{D}_{IJ}<1DIJ​<1. A second monogamy inequality involves the more general entanglement certifier EntIJ{\text{Ent}}_{IJ}EntIJ​ defined as the normalized product of the variances of ${X}_{I}\ensuremath{-}g{X}_{J}$ and PI+gPJ{P}_{I}+g{P}_{J}PI​+gPJ​, where ggg is a real constant. The monogamy inequalities give a lower bound on the values of DBC{D}_{BC}DBC​ and EntBC{\text{Ent}}_{BC}EntBC​ for one pair, given the values DBA{D}_{BA}DBA​ and EntBA{\text{Ent}}_{BA}EntBA​ for the first pair. This lower bound changes in the absence of two-mode Gaussian steering of BBB. We illustrate for a range of tripartite entangled states, identifying regimes of saturation of the inequalities. The monogamy relations explain without the assumption of Gaussianity the experimentally observed saturation at DAB=0.5{D}_{AB}=0.5DAB​=0.5 where there is symmetry between modes AAA and CCC.
We derive laws for the distribution of quantum steering among different parties in multipartite Gaussian states under Gaussian measurements. We prove that a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality holds quantitatively for a recently introduced measure of Gaussian steering. We then define the residual Gaussian steering, stemming from the monogamy inequality, as an indicator of collective steering-type correlations. For pure three-mode Gaussian states, the residual acts as a quantifier of genuine multipartite steering, and is interpreted operationally in terms of the guaranteed key rate in the task of secure quantum secret sharing. Optimal resource states for the latter protocol are identified, and their possible experimental implementation discussed. Our results pin down the role of multipartite steering for quantum communication.