Pairwise completely false matrices with applications in Quantum Information Theory

Kai-yan Lee

Ichiro Fujimoto

洋 長谷川, 利也 高見

Olivia R. MacLean

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.

Hiroshi Hasegawa, Toshiya Takami

Andrzej Jamiołkowski, Grzegorz Pastuszak

Assume that are complex matrices. We give a computable criterion for existence of a common eigenvector of which generalize the result of D. Shemesh established for two matrices. We use this criterion to prove some necessary and sufficient condition for to have a common invariant subspace of dimension , , if every has pairwise different eigenvalues. Since the set of all matrices having multiple eigevalues has Lebesgue measure , the condition is sufficient in practical applications. Being motivated by quantum information theory, we give a flavour of such applications for irreducible completely positive superoperators.