An algorithm based on negative probabilities for a separability criterion

Abstract

Here, we demonstrate that entangled states can be written as separable states [$\rho _{1\ldots N}=\sum _{i}p_{i}\rho _{i}^{(1)}\otimes \cdots \otimes \rho _{i}^{(N)}$, 1 to N refering to the parts and $p_{i}$ to the nonnegative probabilities], although for some of the coefficients, $p_{i}$ assume negative values, while others are larger than 1 such to keep their sum equal to 1. We recognize this feature as a signature of non-separability or pseudoseparability. We systematize that kind of decomposition through an algorithm for the explicit separation of density matrices, and we apply it to illustrate the separation of some particular bipartite and tripartite states, including a multipartite ${\textstyle \bigotimes \nolimits ^{N}} 2$ one-parameter Werner-like state. We also work out an arbitrary bipartite $2\times 2$ state and show that in the particular case where this state reduces to an X-type density matrix, our algorithm leads to the separability conditions on the parameters, confirmed by the Peres-Horodecki partial transposition recipe. We finally propose a measure for quantifying the degree of entanglement based on these peculiar negative (and greater than one) probabilities.