Quantum Counterpart of Gibbs Theorem on Entropy and Statistical Dependence

Abstract

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