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arXiv:quant-ph/0401056 · Physical Review A

P -representable subset of all bipartite Gaussian separable states

M. C. de Oliveira

Published Sep 17, 2004• 22 citations
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Abstract

$P$-representability is a necessary and sufficient condition for separability of bipartite Gaussian states only for the special subset of states whose covariance matrix are $\mathrm{Sp}(2,R)\ensuremath{\bigotimes}\mathrm{Sp}(2,R)$ locally invariant. Although this special class of states can be reached by a convenient $\mathrm{Sp}(2,R)\ensuremath{\bigotimes}\mathrm{Sp}(2,R)$ transformation over an arbitrary covariance matrix, it represents a loss of generality, avoiding inference of many general aspects of separability of bipartite Gaussian states.

DOI
10.1103/physreva.70.034303

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PPP-representability is a necessary and sufficient condition for separability of bipartite Gaussian states only for the special subset of states whose covariance matrix are Sp(2,R)\ensuremath⨂Sp(2,R)\mathrm{Sp}(2,R)\ensuremath{\bigotimes}\mathrm{Sp}(2,R)Sp(2,R)\ensuremath⨂Sp(2,R) locally invariant. Although this special class of states can be reached by a convenient Sp(2,R)\ensuremath⨂Sp(2,R)\mathrm{Sp}(2,R)\ensuremath{\bigotimes}\mathrm{Sp}(2,R)Sp(2,R)\ensuremath⨂Sp(2,R) transformation over an arbitrary covariance matrix, it represents a loss of generality, avoiding inference of many general aspects of separability of bipartite Gaussian states.
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A necessary and sufficient condition for characterization and quantification of entanglement of any bipartite Gaussian state belonging to a special symmetry class is given in terms of classicality measures of one-party states. For Gaussian states whose local covariance matrices have equal determinants it is shown that separability of a two-party state and classicality of a one-party state are completely equivalent to each other under a nonlocal operation, allowing entanglement features to be understood in terms of any available classicality measure.