Dobrushin's ergodicity coefficient for Markov operators on cones

TL;DR

This paper generalizes Dobrushin's ergodicity coefficient to noncommutative settings, providing a unified framework for analyzing the contraction properties of both classical and quantum Markov operators.

Contribution

It introduces a new characterization of contraction ratios for operators on cones, extending Dobrushin's coefficient to quantum channels and noncommutative Markov chains.

Findings

Recovered classical Dobrushin's coefficient for stochastic matrices

Derived a noncommutative version for quantum channels

Provided algebraic criteria for convergence of noncommutative systems

Abstract

We give a characterization of the contraction ratio of bounded linear maps in Banach space with respect to Hopf's oscillation seminorm, which is the infinitesimal distance associated to Hilbert's projective metric, in terms of the extreme points of a certain abstract "simplex". The formula is then applied to abstract Markov operators defined on arbitrary cones, which extend the row stochastic matrices acting on the standard positive cone and the completely positive unital maps acting on the cone of positive semidefinite matrices. When applying our characterization to a stochastic matrix, we recover the formula of Dobrushin's ergodicity coefficient. When applying our result to a completely positive unital map, we therefore obtain a noncommutative version of Dobrushin's ergodicity coefficient, which gives the contraction ratio of the map (representing a quantum channel or a "noncommutative Markov chain") with respect to the diameter of the spectrum. The contraction ratio of the dual operator (Kraus map) with respect to the total variation distance will be shown to be given by the same coefficient. We derive from the noncommutative Dobrushin's ergodicity coefficient an algebraic characterization of the convergence of a noncommutative consensus system or equivalently the ergodicity of a noncommutative Markov chain.