Theoretical Statistics and Mathematics Unit, ISI Delhi

On the equivalence of separability and extendability of quantum states

by B. V. Rajarama Bhat, K. R. Parthasarathy and Ritabrata Sengupta

Motivated by the notions of $k$-extendability and complete
extendability of the state of a finite level quantum system as
described by Doherty et al (Phys. Rev. A, 69:022308),
we introduce parallel definitions in the context of
Gaussian states and using only properties of their
covariance matrices derive necessary and sufficient
conditions for their complete extendability. It turns out
that the complete extendability property is equivalent to
the separability property of a bipartite Gaussian state.
Following the proof of quantum de Finetti
theorem as outlined in Hudson and Moody (Z.
Wahrscheinlichkeitstheorie und Verw. Gebiete,
33(4):343--351), we show that separability is equivalent to
complete extendability for a state in a bipartite Hilbert
space where at least one of which is of dimension greater
than 2. This, in particular, extends the result of Fannes,
Lewis, and Verbeure (Lett. Math. Phys. 15(3): 255--260) to
the case of an infinite dimensional Hilbert space whose C*
algebra of all bounded operators is not separable.

isid/ms/2016/06 [fulltext]

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