State space structure of tripartite quantum systems

Abstract

State space structure of tripartite quantum systems is analyzed. In particular, it has been shown that the set of states separable across all the three bipartitions [say, ${\mathcal{B}}^{int}(ABC)$] is a strict subset of the set of states having positive partial transposition across the three bipartite cuts [say, ${\mathcal{P}}^{int}(ABC)$] for all the tripartite Hilbert spaces ${\mathbb{C}}_{A}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}_{B}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}_{C}^{{d}_{3}}$ with $min{{d}_{1},{d}_{2},{d}_{3}}\ensuremath{\ge}2$. The claim is proved by constructing a state belonging to the set ${\mathcal{P}}^{int}(ABC)$ but not to ${\mathcal{B}}^{int}(ABC)$. For ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}3}$ with $d\ensuremath{\ge}3$, the construction follows from a specific type of multipartite unextendible product bases. However, such a construction is not possible for ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}3}$ since bipartite systems ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ do not allow any unextendible product bases for arbitrary $n$ [Phys. Rev. Lett. 82, 5385 (1999)]. For the three-qubit system, we therefore come up with a different construction.