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In this paper, we derive a new generalisation of the strong subadditivity ofthe entropy to the setting of general conditional expectations onto arbitraryfinite-dimensional von Neumann algebras. The latter inequality, which we callapproximate tensorization of the relative entropy, can be expressed as a lowerbound for the sum of relative entropies between a given density and itsrespective projections onto two intersecting von Neumann algebras in terms ofthe relative entropy between the same density and its projection onto analgebra in the intersection, up to multiplicative and additive constants. Inparticular, our inequality reduces to the so-called quasi-factorization of theentropy for commuting algebras, which is a key step in modern proofs of thelogarithmic Sobolev inequality for classical lattice spin systems. We alsoprovide estimates on the constants in terms of conditions of clustering ofcorrelations in the setting of quantum lattice spin systems. Along the way, weshow the equivalence between conditional expectations arising from Petzrecovery maps and those of general Davies semigroups.

Approximate tensorization of the relative entropy for noncommuting conditional expectations