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

在本文中，我们将熵的强可加性推导出到任意有限维冯诺依曼代数上的一般条件期望的设置。这种概括称为相对熵的近似张量化，包括给定密度与其各自投影到两个相交冯诺依曼代数之间的相对熵总和的下限，根据相同密度与其密度之间的相对熵投影到交集的代数上，直到乘法和加法常数。特别是，我们的不等式简化为所谓的交换代数熵的准因式分解，这是经典晶格自旋系统对数 Sobolev 不等式现代证明的关键步骤。我们还根据量子晶格自旋系统设置中的相关性聚类条件提供了对常数的估计。在此过程中，我们展示了由 Petz 恢复图产生的条件期望与一般 Davies 半群的条件期望之间的等价性。