  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 $H(AB|C) = H(A|BC)
+ H(B|C)$, 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 $H_{\alpha}(AB|C) \geqslant
H_{\beta}(A|BC) + H_{\gamma}(B|C)$, where the parameters $\alpha, \beta,
\gamma$ obey the relation $\frac{\alpha}{\alpha-1} = \frac{\beta}{\beta-1} +
\frac{\gamma}{\gamma-1}$ and $(\alpha-1)(\beta-1)(\gamma-1) > 1$; if
$(\alpha-1)(\beta-1)(\gamma-1) < 1$, the direction of the inequality is
reversed.

Quantum Physics