A Cutoff Phenomenon for Quantum Markov Chains

TL;DR

This paper introduces the concept of the cutoff phenomenon in quantum information theory, analyzing how certain quantum processes converge and providing bounds and scaling behaviors for quantum state preparation.

Contribution

It establishes the cutoff phenomenon for quantum Markov chains and derives bounds on their convergence, revealing that convergence is not solely determined by spectral gaps.

Findings

Quantum Markov chains exhibit cutoff phenomena.

Graph states can be prepared efficiently via dissipation.

Exact scaling of convergence time to stationarity is provided.

Abstract

We derive upper and lower bounds on the convergence behavior of certain classes of one-parameter quantum dynamical semigroups. The classes we consider consist of tensor product channels and of channels with commuting Liouvillians. We introduce the notion of Cutoff Phenomenon in the setting of quantum information theory, and show how it exemplifies the fact that the convergence of (quantum) stochastic processes is not solely governed by the spectral gap of the transition map. We apply the new methods to show that graph states can be prepared efficiently, albeit not in constant time, by dissipation, and give the exact scaling behavior of the time to stationarity.