Log-Euclidean Rényi 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 Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→ 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 Rényi chain rules, which appear to be novel even in the classical setting.
Related
- Renormalization group and long-range conditional mutual information in hierarchical models
- Bosonic Quantum Conditional Mutual Information and Bose-Einstein Condensation
- Universal Scaling of Conditional Mutual Information in Classical Stochastic Dynamics
- Sharp Data-Processing Region for Quantum Conditional Rényi Entropies
- Free Mutual Information and Higher-Point Out-of-Time-Ordered Correlators
BibTeX
@misc{rubboli2026logeuclidean,
title = {{Log-Euclidean Rényi Conditional Mutual Information}},
author = {Roberto Rubboli and Amir Arqand and Mark M. Wilde},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34699},
eprint = {2609.34699},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34699},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Log-Euclidean Rényi Conditional Mutual Information
AU - Roberto Rubboli
AU - Amir Arqand
AU - Mark M. Wilde
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34699
UR - https://arxiv.org/abs/2609.34699
AB - 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 Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→ 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 Rényi chain rules, which appear to be novel even in the classical setting.
ER -