Chain rules for quantum R\'enyi entropies

We present chain rules for a new definition of the quantum R'enyi conditional entropy sometimes called the “sandwiched” R'enyi conditional entropy. More precisely, we prove analogues of the equation $H(AB\mathrm{\mid }C)=H(A\mathrm{\mid }BC)+H(B\mathrm{\mid }C)$, which holds as an identity for the von Neumann conditional entropy. In the case of the R'enyi entropy, this relation no longer holds as an equality, but survives as an inequality of the form $H_{\alpha }(AB\mathrm{\mid }C)⩾H_{\beta }(A\mathrm{\mid }BC)+H_{\gamma }(B\mathrm{\mid }C)$, where the parameters $\alpha ,\beta ,\gamma$ obey the relation $\frac{\alpha }{\alpha -1}=\frac{\beta }{\beta -1}+\frac{\gamma }{\gamma -1}$ and $(\alpha -1)(\beta -1)(\gamma -1)>0$; if $(\alpha -1)(\beta -1)(\gamma -1)<0$, the direction of the inequality is reversed.