Log-Euclidean R\'enyi Conditional Mutual Information

Abstract

Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum R\'enyi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum R\'enyi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the R\'enyi parameter, and converges to the QCMI in the limit $\alpha \to 1$. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive R\'enyi chain rules, which appear to be novel even in the classical setting.