Complete entropic inequalities for quantum Markov chains

TL;DR

This paper establishes fundamental entropic inequalities for quantum Markov chains, including modified log-Sobolev inequalities, strong data processing inequalities, and approximate tensorization of relative entropy, with broad applications in quantum information theory.

Contribution

It introduces new entropic inequalities for quantum Markov semigroups and channels, extending quantum entropy subadditivity and providing bounds relevant for quantum designs and Markovian evolutions.

Findings

Proves modified log-Sobolev inequalities for GNS-symmetric quantum Markov semigroups.

Establishes strong data processing inequalities for finite-dimensional quantum channels.

Demonstrates approximate tensorization of quantum relative entropy, extending SSA.

Abstract

We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All the three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or $L_2$-estimates. As applications, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive bounds for examples of both theoretical and practical relevance, including representation of sub-Laplacians on $\operatorname{SU}(2)$ and various classes of local quantum Markov semigroups such as quantum Kac generators and continuous time approximate unitary designs. For the latter, our bounds imply the existence of local continuous time Markovian evolutions on $nk$ qudits forming $\epsilon$-approximate $k$-designs in relative entropy for times scaling as $\widetilde{\mathcal{O}}(n^2 \operatorname{poly}(k))$.