A Geometric Diagram of Separable States

Abstract

This paper present a geometric diagram of a separable state: If a mixed state $σ$ is separable, there are $2^{nS(σ)}$ linearly independant product vectors which span the same Hilbert space as the $2^{nS(σ)}$ ``likely'' strings of $σ^{\otimes n}$ do. This diagram results in a criterion for separability which is strictly stronger than the inorder criterion in [M.A. Nielsen and J. Kempe, Phys. Rev. Lett. 86, 5184 (2001)]. This means that the number of product bases of states of a system has close link to the nonlocality of the system.