论文标题
相对熵的近似张力，以进行不合规的条件期望
Approximate tensorization of the relative entropy for noncommuting conditional expectations
论文作者
论文摘要
在本文中，我们得出了熵的强度亚加性的新概括，以将一般条件期望设置为任意的有限维von Neumann代数。我们称之为相对熵的近似张力的后一种不平等可以表示为在给定密度及其各自的投影之间的相对熵的总和，以在两个相交的von Neumann代数上，以相同的密度与其对代数之间的相对熵的相对熵进行相对的距离，并在相互构造中添加型构造和繁殖。特别是，我们的不平等降低了通勤代数的熵的所谓准依然化，这是对数sobolev不平等现代证明古典晶格旋转系统的关键步骤。我们还根据量子晶格旋转系统中相关性的聚类条件提供了对常数的估计。一路上，我们显示了PETZ恢复图引起的条件期望与戴维斯将军的半群落之间的等价性。
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. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as 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.
