Chain rules for quantum R\'enyi entropies

TL;DR

This paper establishes chain rule inequalities for the quantum Re9nyi conditional entropy, generalizing the von Neumann entropy identity to a broader class of entropies with parameter-dependent inequalities.

Contribution

It introduces and proves chain rule inequalities for the sandwiched quantum Re9nyi conditional entropy, extending classical entropy relations to the quantum Re9nyi setting.

Findings

Derived inequalities relating quantum Re9nyi entropies with different parameters.

Identified conditions under which the inequalities hold with reversed direction.

Extended classical entropy chain rules to quantum Re9nyi entropies.

Abstract

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 $(\alpha-1)(\beta-1)(\gamma-1) > 0$; if $(\alpha-1)(\beta-1)(\gamma-1) < 0$, the direction of the inequality is reversed.