Sufficient separability criteria via quantum maps

Abstract

In this master’s thesis, we consider quantum linear maps as a tool to derive new sufficient conditions for separability in bipartite and multipartite systems. In specific, we focus on the so-called reduction map to strengthen the existing criteria for absolute separability in bipartite systems, i.e., for states that remain bi-separable under any global unitary transformation. To this aim, using powerful convex geometry techniques, we introduce tighter volumes and characterization of the set of absolutely separable states w.r.t. any bi-partition for arbitrary dimensions. Furthermore, we derive new conditions on the spectrum of bipartite entanglement witnesses. In addition, we address the multipartite scenario by presenting some non-optimal results. Finally, we provide some insights on the conjecture that having a positive partial transpose from spectrum is equivalent to being separable from spectrum for the symmetric subspace of N−qudits, as well as a new criterion for positive partial transpose from spectrum for arbitrary system sizes N.