Pairwise completely positive matrices with applications in Quantum Information Theory

Abstract

This thesis introduces a generalization of the set of completely positive matrices that we call “pairwise completely positive” (PCP) matrices. PCP matrix pairs are defined so that one matrix in the pair is necessarily positive semidefinite while the other is necessarily entrywise non-negative. After PCP matrices are defined we explore their basic properties and establish numerous necessary and sufficient conditions that can help test whether or not a pair meets our definition of PCP. We then relate these matrix pairs to the separability of conjugate local diagonal unitary invariant (CLDUI) quantum states. In particular, we show that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a corresponding CLDUI state is separable.