Chain rules for quantum Rényi entropies

Abstract

We present chain rules for a new definition of the quantum Rényi conditional entropy sometimes called the "sandwiched" Rényi conditional entropy. More precisely, we prove analogues of the equation $H(AB|C)=H(A|BC)+H(B|C)$, which holds as an identity for the von Neumann conditional entropy. In the case of the Rényi entropy, this relation no longer holds as an equality, but survives as an inequality of the form $H_{\hat{I}}\pm (AB|C)⩾H_{\hat{I}}^2(A|BC)+H_{\hat{I}}^3(B|C)$, where the parameters $\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3$ obey the relation $\frac{\hat{I}}{\pm }\hat{I}\pm -1=\frac{\hat{I}}{²}{\hat{I}}^2-1+\frac{\hat{I}}{³}{\hat{I}}^3-1$ and $(\hat{I}\pm -1)({\hat{I}}^2-1)({\hat{I}}^3-1)>0$; if $(\hat{I}\pm -1)({\hat{I}}^2-1)({\hat{I}}^3-1)<0$, the direction of the inequality is reversed.

Typo in main statement fixed