Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

In this paper, we derive a new generalisation of the strong subadditivity entropy to setting general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, which call approximate tensorization relative entropy, can be expressed as lower bound for sum entropies between given density and its respective projections two intersecting algebras in terms same projection an algebra intersection, up multiplicative additive constants. particular, our inequality reduces so-called quasi-factorization commuting algebras, is key step modern proofs logarithmic Sobolev classical lattice spin systems. We also provide estimates on constants conditions clustering correlations quantum Along way, show equivalence arising from Petz recovery maps those Davies semigroups.