Dimensions, lengths and separability in finite-dimensional quantum systems

TL;DR

This paper investigates the geometric structure of separable states in finite-dimensional quantum systems, providing dimension calculations, existence proofs of certain states, and proposing conjectures for characterizing separability sets.

Contribution

It computes dimensions of key sets of quantum states, proves the existence of specific separable states with unique properties, and introduces conjectures for describing separability sets as semialgebraic.

Findings

Existence of separable states not expressible as d or fewer pure product states.

Separable states invariant under partial transpose are convex sums of real pure product states.

Reduction of the general multipartite separability problem to the real states case.

Abstract

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.