Quantum Counterpart of Gibbs Theorem on Entropy and Statistical Dependence

Abstract

In terms of a quantum statistical operator $\ensuremath{\rho}$ and associated reduced statistical operators ${\ensuremath{\rho}}^{(1)}$ and ${\ensuremath{\rho}}^{(2)}$, one defines corresponding entropies $S$, ${S}^{(1)}$, and ${S}^{(2)}$. It is shown that $S\ensuremath{\leqq}{S}^{(1)}+{S}^{(2)}$, with equality if $\ensuremath{\rho}$ equals the direct product ${\ensuremath{\rho}}^{(1)}\ensuremath{\bigotimes}{\ensuremath{\rho}}^{(2)}$. Thus the statistical dependence and greater information implicitly contained in $\ensuremath{\rho}$ yields an entropy smaller than that obtained for the case in which $\ensuremath{\rho}$ factors into the statistically independent form ${\ensuremath{\rho}}^{(1)}\ensuremath{\bigotimes}{\ensuremath{\rho}}^{(2)}$. This is the quantum counterpart of a classical theorem published by Gibbs in 1902.