Separability Criterion for Density Matrices

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Abstract

A quantum system consisting of two subsystems is separable if its density matrix can be written as \(\rho=\sum_A w_A\,\rho_A'\otimes\rho_A''\), where \(\rho_A'\) and \(\rho_A''\) are density matrices for the two subsytems. In this Letter, it is shown that a necessary condition for separability is that a matrix, obtained by partial transposition of \(\rho\), has only non-negative eigenvalues. This criterion is stronger than Bell's inequality.

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Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels

John Smolin, Benjamin Schumacher, Sandu Popescu … (1995)

Two separated observers, by applying local operations to a supply of not-too-impure entangled states ({\em e.g.} singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ({\em e.g.} near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transfrom one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield \(D(M)\) of pure singlets (\(\ket{\Psi^-}\)) distillable from mixed states \(M\), showing \(D(M)>0\) if \(\bra{\Psi^-}M\ket{\Psi^-}>\half\).