Dimensions, lengths, and separability in finite-dimensional quantum systems

Many important sets of normalized states in a multipartite quantum system of finite dimension d, such as the set S of all separable states, are real semialgebraic sets. We compute dimensions of many such sets in 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, such states exist in bipartite M circle times N systems when (M - 2)(N - 2) > 1. This solves an open problem proposed by DiVincenzo, Terhal and Thapliyal about 12 years ago. We prove that there exist a separable state rho and a pure product state, whose mixture has smaller length than that of rho. We show that any real rho is an element of S, which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2 circle times N system, the number r of product states can be taken to be r = rank rho. We also show that the general multipartite separability problem can be reduced to the case of real states. Regarding the separability problem, we propose two conjectures describing S as a semialgebraic set, which may eventually lead to an analytic solution in some low-dimensional systems such as 2 circle times 4, 3 circle times 3, and 2 circle times 2 circle times 2. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4790405]