Entanglement Breaking Quantum Channels, Stochastic Matrices and Primitivity

Abstract

Quantum entanglement is a fundamental phenomenon in quantum information where states of different quantum systems are connected in a way that they cannot be described independently of each other irrespective of the spatial distance between them. We study a class of quantum channels called entanglement breaking channels. These channels break the presence of entanglement when acting on bipartite states. We also study an underlying structure describing such channels known as their Holevo form and generate stochastic matrices from them. Upon examination, we find that the nonzero spectrum and the Jordan canonical forms of such channels and their associated stochastic matrices are closely related. Focusing on conditions for primitivity of our channels, we further investigate the relationship between their primitivity indices and those of their matrices and eventually provide a quantum bound for the primitivity index of these channels. Finally, we also introduce the notion of the Holevo rank and provide a bound on these entanglement breaking channels in terms of this rank.