Pure inductive limit state and Kolmogorov's property-II

TL;DR

This paper investigates the conditions under which a translation invariant quantum state on an infinite tensor product algebra is pure and exhibits Kolmogorov's property, linking purity to the type of von Neumann algebra factors involved.

Contribution

It provides an asymptotic criterion for the purity of quantum Markov chain states and characterizes when such states have Kolmogorov's property based on the algebraic type.

Findings

Purity of the state is characterized by an asymptotic criterion on the Markov map.

Pure states restrict to either type-I or type-III factors on one side of the chain.

Kolmogorov's property holds when the restricted state is of type-I.

Abstract

A translation invariant state $\omega$ on $C^*$-algebra $\clb=\otimes_{k \in \IZ}M^{(k)}$, where $M^{(k)}=M_d(\IC)$ is the $d-$dimensional matrices over field of complex numbers, give rises a stationary quantum Markov chain and associates canonically a unital completely positive normal map $\tau$ on a von-Neumann algebra $\clm$ with a faithful normal invariant state $\phi$. We give an asymptotic criteria on the Markov map $(\clm,\tau,\phi)$ for purity of $\omega$. Such a pure $\omega$ gives only type-I or type-III factor $\omega_R$ once restricted to one side of the chain $\clb_R=\otimes_{\IZ_+}M^{(k)}$. In case $\omega_R$ is type-I, $\omega$ admits Kolmogorov's property.