Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence

Gerardo Adesso, Alessio Serafini, and Fabrizio Illuminati

Phys. Rev. A 73, 032345 – Published 30 March 2006

Abstract

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 $GHZ ∕ W$ states of continuous-variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the $W$ states of three qubits. We finally consider the effect of decoherence on three-mode Gaussian states, studying the decay of the residual contangle. The $GHZ ∕ W$ states are shown to be maximally robust against losses and thermal noise.

Received 16 December 2005

DOI:https://doi.org/10.1103/PhysRevA.73.032345

Authors & Affiliations

Gerardo Adesso^1,2, Alessio Serafini^3,4, and Fabrizio Illuminati¹

¹Dipartimento di Fisica “E. R. Caianiello,” Università degli Studi di Salerno, CNR-Coherentia, Gruppo di Salerno, and INFN Sezione di Napoli-Gruppo Collegato di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy
²Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
³Institute for Mathematical Sciences, Imperial College, London, London SW7 2PE, United Kingdom and QOLS, The Blackett Laboratory, Imperial College, London, Prince Consort Road, London SW7 2BW, United Kingdom
⁴Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom

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Vol. 73, Iss. 3 — March 2006

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Figure 1
(Color online) Range of the entropic quantities $a_{l}^{'} = μ_{l}^{- 1} - 1$ for pure three-mode Gaussian states. The three parameters $a_{l}^{'}$ , with $l = 1, 2, 3$ , have to vary inside the pyramid represented in plot (a) or, equivalently, for fixed values of one of them, say $a_{1}^{'}$ , inside the shaded slice represented in plot (b), in order to determine the CM of a physical state, Eq. (42). The expression of the boundary surfaces and/or curves comes from the saturation of the triangular inequality (40) for all possible mode permutations. In particular, for the projected two-dimensional plot (b), the equations of the three boundaries are I. $a_{3}^{'} = a_{1}^{'} - a_{2}^{'}$ ; II. $a_{3}^{'} = a_{1}^{'} + a_{2}^{'}$ ; III. $a_{3}^{'} = a_{2}^{'} - a_{1}^{'}$ . All quantities plotted are dimensionless.Reuse & Permissions
Figure 2
(Color online) Pictorial representation of Eq. (73), defining the residual Gaussian contangle $G_{τ}^{r e s}$ of generic (nonsymmetric) three-mode Gaussian states. $G_{τ}^{r e s}$ quantifies the genuine tripartite entanglement shared among mode 1 $(●)$ , mode 2 $(∎)$ , and mode 3 $(▴)$ . The optimal decomposition that realizes the minimum in Eq. (73) is always the one for which the CM of the reduced state of the reference mode has the smallest determinant.Reuse & Permissions
Figure 3
(Color online) Three-dimensional plot of the residual Gaussian contangle $G_{τ}^{r e s} (σ^{p})$ in pure three-mode Gaussian states $σ^{p}$ , determined by the three local mixedness $a_{l}$ , $l = 1, 2, 3$ . One of the local mixedness is kept fixed $(a_{1} = 2)$ . The remaining ones vary constrained by the triangle inequality (55), as depicted in 1b. The explicit expression of $G_{τ}^{r e s}$ is given by Eq. (27). See text for further details. All quantities plotted are dimensionless.Reuse & Permissions
Figure 4
(Color online) Plot, as a function of the single-mode mixedness $a$ , of the tripartite residual Gaussian contangle $G_{τ}^{r e s}$ Eq. (79) in the CV $GHZ ∕ W$ states (dashed line); in the $T$ states Eq. (83) (solid line); and in $50 000$ randomly generated mixed symmetric three-mode Gaussian states of the form Eq. (46) (dots). The $GHZ ∕ W$ states, that maximize any bipartite entanglement, also achieve maximal genuine tripartite quantum correlations, showing that CV entanglement distributes in a promiscuous way in symmetric Gaussian states. Note also how all random mixed states have a non-negative residual Gaussian contangle. This confirms the results presented in Ref. 12, and discussed in detail and extended in Sec. 4C, on the strict validity of the CKW monogamy inequality for CV entanglement in three-mode Gaussian states. All quantities plotted are dimensionless.Reuse & Permissions
Figure 5
(Color online) Evolution of the residual Gaussian contangle $G_{τ}^{r e s}$ for $GHZ ∕ W$ states with local mixedness $a = 2$ (solid curves) and $T$ states with local mixedness $a = 2.8014$ (dashed curves). Such states have an equal initial residual contangle. The uppermost curves refer to a homogeneous bath with $N = 0$ (pure losses), while the lowermost curves refer to a homogeneous bath with $N = 1$ . As apparent, thermal photons are responsible for the vanishing of entanglement at finite times.Reuse & Permissions

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