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
