A Multiplicative Ergodic Theorem for Bistochastic Ergodic Quantum Processes with Applications to Entanglement

TL;DR

This paper establishes a multiplicative ergodic theorem for bistochastic quantum processes, providing insights into their long-term behavior and entanglement properties, with applications to quantum information theory.

Contribution

It introduces a new ergodic theorem for bistochastic quantum processes and classifies conditions under which these processes become entanglement breaking.

Findings

Classifies when compositions of random bcp maps are asymptotically entanglement breaking.

Shows occasionally PPT bcp maps are asymptotically entanglement breaking.

Demonstrates certain bcp cocycles become almost surely entanglement breaking in finite time.

Abstract

We prove a multiplicative ergodic theorem for bistochastic completely positive (bcp) linear cocycles acting on finite-dimensional matrix algebras, giving an invariant splitting described explicitly in terms of the multiplicative domains of the underlying bcp maps. As an application of our theorem, we classify when compositions of random bcp maps are asymptotically entanglement breaking, and use this classification to show that occasionally PPT bcp maps are asymptotically entanglement breaking. We conclude by demonstrating a certain class of bcp linear cocycles are almost surely entanglement breaking in finite time.