Translation invariant state and its mean entropy-I

TL;DR

This paper proves the equivalence of various entropy measures for translation invariant states on infinite tensor product C*-algebras and characterizes pure states by zero entropy, extending classical entropy results to non-commutative settings.

Contribution

It establishes the equality of mean entropy, Connes-Størmer entropy, and Kolmogorov-Sinai entropy for translation invariant states, and characterizes pure states by zero entropy in a non-commutative context.

Findings

Mean entropy equals Connes-Størmer dynamical entropy.

Mean entropy equals Kolmogorov-Sinai entropy on a maximal abelian sub-algebra.

Pure states have zero mean entropy.

Abstract

Let $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ be the two sided infinite tensor product $C^*$-algebra of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$ and $\omega$ be a translation invariant state of $\IM$. In this paper, we have proved that the mean entropy $s(\omega)$ and Connes-St{\o}rmer dynamical entropy $h_{CS}(\IM,\theta,\omega)$ of $\omega$ are equal. Furthermore, the mean entropy $s(\omega)$ is equal to the Kolmogorov-Sinai dynamical entropy $h_{KS}(\ID_{\omega},\theta,\omega)$ of $\omega$ when the state $\omega$ is restricted to a suitable translation invariant maximal abelian $C^*$ sub-algebra $\ID_{\omega}$ of $\IM$. Futhermore, a translation invariant factor state of $\IM$ is pure if and only if its mean entropy is zero. The last statement can be regarded as a non commutative extension of Rokhlin-Sinai positive entropy theorem for non-pure factor states.