Entanglement Breaking Channels, Stochastic Matrices, and Primitivity

TL;DR

This paper explores the relationship between entanglement breaking quantum channels and stochastic matrices, establishing conditions for primitivity and providing bounds on their primitivity index, with implications for quantum information theory.

Contribution

It demonstrates that the primitivity of entanglement breaking channels is determined by their stochastic matrix representations, offering new bounds and insights.

Findings

Primitivity of channels depends on associated stochastic matrices.

Established bounds on the primitivity index of channels.

Provided examples and discussed open problems.

Abstract

We consider the important class of quantum operations (completely positive trace-preserving maps) called entanglement breaking channels. We show how every such channel induces stochastic matrix representations that have the same non-zero spectrum as the channel. We then use this to investigate when entanglement breaking channels are primitive, and prove this depends on primitivity of the matrix representations. This in turn leads to tight bounds on the primitivity index of entanglement breaking channels in terms of the primitivity index of the associated stochastic matrices. We also present examples and discuss open problems generated by the work.