Dimensions, lengths, and separability in finite-dimensional quantum systems
Lin Chen, Dragomir Ž. Djoković · Journal of Mathematical Physics · 2013
Many important sets of normalized states in a multipartite quantum system of finite dimension d, such as the set \documentclass[12pt]{minimal}\begin{document}${\cal S}$\end{document}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⊗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 ρ and a pure product state, whose mixture has smaller length than that of ρ. We show that any real \documentclass[12pt]{minimal}\begin{document}$\rho \in {\cal S}$\end{document}ρ∈S, which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2⊗N system, the number r of product states can be taken to be \documentclass[12pt]{minimal}\begin{document}$r=\mathop {\rm rank}\rho$\end{document}r= rank ρ. 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 \documentclass[12pt]{minimal}\begin{document}${\cal S}$\end{document}S as a semialgebraic set, which may eventually lead to an analytic solution in some low-dimensional systems such as 2⊗4, 3⊗3, and 2⊗2⊗2.