Generic Entanglement and Standard Form for N-mode Pure Gaussian States

We investigate the correlation structure of pure N-mode Gaussian resources which can be experimentally generated by means of squeezers and beam splitters, whose entanglement properties are generic. We show that those states are specified (up to local unitaries) by N(N-1)/2 parameters, corresponding to the two-point correlations between any pair of modes. Our construction yields a practical scheme to engineer such generic-entangled N-mode pure Gaussian states by linear optics. We discuss our findings in the framework of Gaussian matrix product states of harmonic lattices, raising connections with entanglement frustration and the entropic area law.Comment: 4 pages, 1 EPS figure. Revised, corrected and clarified. Final shortened version, published in PR

Bipartite and Multipartite Entanglement of Gaussian States

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.Comment: 21 pages, 4 figures, WS style. Published as Chapter 1 in the book "Quantum Information with Continuous Variables of Atoms and Light" (Imperial College Press, 2007), edited by N. Cerf, G. Leuchs, and E. Polzik. Details of the book available at http://www.icpress.co.uk/physics/p489.html . For recent follow-ups see quant-ph/070122

Equivalence between Entanglement and the Optimal Fidelity of Continuous Variable Teleportation

We devise the optimal form of Gaussian resource states enabling continuous variable teleportation with maximal fidelity. We show that a nonclassical optimal fidelity of $N$-user teleportation networks is {\it necessary and sufficient} for $N$-party entangled Gaussian resources, yielding an estimator of multipartite entanglement. This {\it entanglement of teleportation} is equivalent to entanglement of formation in the two-user protocol, and to localizable entanglement in the multi-user one. The continuous-variable tangle, quantifying entanglement sharing in three-mode Gaussian states, is operationally linked to the optimal fidelity of a tripartite teleportation network.Comment: 4 pages, 1 figure. Approved for publication in Phys. Rev. Let

Determination of continuous variable entanglement by purity measurements

We classify the entanglement of two--mode Gaussian states according to their degree of total and partial mixedness. We derive exact bounds that determine maximally and minimally entangled states for fixed global and marginal purities. This characterization allows for an experimentally reliable estimate of continuous variable entanglement based on measurements of purity.Comment: 4 pages, 3 EPS figures. Final versio

Quantum versus classical correlations in Gaussian states

Quantum discord, a measure of genuinely quantum correlations, is generalized to continuous variable systems. For all two-mode Gaussian states, we calculate analytically the quantum discord and a related measure of classical correlations, solving an optimization over all Gaussian measurements. Almost all two-mode Gaussian states are shown to have quantum correlations, while for separable states, the discord is smaller than unity. For a given amount of entanglement, it admits tight upper and lower bounds. Via a duality between entanglement and classical correlations, we derive a closed formula for the Gaussian entanglement of formation of all mixed three-mode Gaussian states whose normal mode decomposition includes two vacua.Comment: 4+2 pages, 1+1 figures. Close to published version including appendi

Entanglement sharing: from qubits to Gaussian states

It is a central trait of quantum information theory that there exist limitations to the free sharing of quantum correlations among multiple parties. Such 'monogamy constraints' have been introduced in a landmark paper by Coffman, Kundu and Wootters, who derived a quantitative inequality expressing a trade-off between the couplewise and the genuine tripartite entanglement for states of three qubits. Since then, a lot of efforts have been devoted to the investigation of distributed entanglement in multipartite quantum systems. In these proceedings we report, in a unifying framework, a bird's eye view of the most relevant results that have been established so far on entanglement sharing in quantum systems. We will take off from the domain of N qubits, graze qudits, and finally land in the almost unexplored territory of multimode Gaussian states of continuous variable systems.Comment: 11 pages. Proceedings of the workshop "Entanglement in physical and information sciences", Centro Ennio de Giorgi, Pisa, December 2004. (v2) References updated, final version published in Int. J. Quant. In