Complete entropic inequalities for quantum Markov chains

Understand

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.