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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 x N systems when (M-1)(N-1)>1. This solves an open problem proposed in [J. Mod. Opt. 47 (2000), 377-385]. 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 in S, which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2 x 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 x 4, 3 x 3 and 2 x 2 x 2.