Multipartite entanglement sharing in continuous variable systems

Peter van Loock, Samuel L. Braunstein

Gerardo Adesso, Fabrizio Illuminati

In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global and local amounts of mixedness, and efficiently estimated by direct measurements of the associated purities. For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pair of modes by local, unitary operations. We then analyze the distribution of entanglement among multiple parties in multimode Gaussian states. We introduce the continuous-variable tangle to quantify entanglement sharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure, symmetric three-mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. Finally, we investigate the connection between multipartite entanglement and the optimal fidelity in a continuous-variable quantum teleportation network. We show how the fidelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimal fidelity provides a clearcut operational interpretation of several measures of bipartite and multipartite entanglement, including the entanglement of formation, the localizable entanglement, and the continuous-variable tangle.

M. K. Olsen, Joel Frederick Corney

Gerardo Adesso, Fabrizio Illuminati

For continuous-variable (CV) systems, we introduce a measure of entanglement, the CV tangle ( contangle ), with the purpose of quantifying the distributed (shared) entanglement in multimode, multipartite Gaussian states. This is achieved by a proper convex-roof extension of the squared logarithmic negativity. We prove that the contangle satisfies the Coffman–Kundu–Wootters monogamy inequality in all three-mode Gaussian states, and in all fully symmetric N -mode Gaussian states, for arbitrary N . For three-mode pure states, we prove that the residual entanglement is a genuine tripartite entanglement monotone under Gaussian local operations and classical communication. We show that pure, symmetric three-mode Gaussian states allow a promiscuous entanglement sharing, having both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. These states are thus simultaneous CV analogues of both the GHZ and the W states of three qubits: in CV systems monogamy does not prevent promiscuity, and the inequivalence between different classes of maximally entangled states, holding for systems of three or more qubits, is removed.

E Wu, Jia Liu, Luomeng Chao +5 more

Abstract Macroscopic multiple continuous-variable (CV) entanglement plays a central role in recent CV-based approaches to quantum multipartite network, quantum communication and quantum secret sharing. We propose a novel scheme to generate stationary quadripartite CV entanglement at the macroscopic level in two coupling Fabry–Perot cavities with a flux qutrit. We treat two cavity fields and two movable mirrors as intrinsically quadripartite and study quadripartite CV entanglement among them. We demonstrate that genuine quadripartite CV entanglement can be generated in this system.