1206.3775 (Lin Chen et al.)

Some semialgebraic sets of states and separability conjectures    [PDF]

Lin Chen, Dragomir Z. Djokovic

We show that many important sets of normalized states, such as the set, D, of all states or the set, S of all separable states, in a multipartite quantum system of finite dimension, d, are real semialgebraic sets, and so possess a well defined dimension. We compute many of these dimensions for several low-dimensional systems. By using dimension arguments, we show that there exist separable states which are not convex combinations of d or less pure product states. For instance this is the case for the bipartite 3x4 system. We also put forward two conjectures describing the set S as a semialgebraic set which may eventually lead to an analytic solution of the separability problem in some low-dimensional systems such as 2x4, 3x3 and 2x2x2.

View original: http://arxiv.org/abs/1206.3775