The convex sum of product states for a separable state

Hiroo Azuma

Shengjun Wu, J. Anandan

A quantum state which can be written as a convex sum of product states is a separable state; such a convex sum allows for a local description. There are many different local descriptions for a given separable state, and some new results about the minimal number of product states in the convex sum wil be given in this paper. Any quantum state is either entangled or separable. However it is very difficult to tell whether a given state is separable or entangled. This difficulty has not been solved yet although a lot of separability criteria have already be proposed. The various separability criteria will also be discussed.

Erik Alfsen, Fred Shultz

Let Sk be the set of separable states on B(Cm⊗Cn) admitting a representation as a convex combination of k pure product states, or fewer. If m>1, n>1, and k⩽max(m,n), we show that Sk admits a subset Vk such that Vk is dense and open in Sk, and such that each state in Vk has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains Vk. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of B(Cm⊗Cn) that preserve entanglement and separability.

Kil-Chan Ha, Seung-Hyeok Kye

Ting-Gui Zhang, Xiaofen Huang, Xianqing Li-Jost +2 more

Senlin Wu