标题： 施瓦茨希尔德黑洞的量子相对熵和面积定律 显示英文标题

标题： The quantum relative entropy of the Schwarzschild black-hole and the area law

摘要： 黑洞的热力学熵所遵循的面积定律是将引力与统计力学联系起来的基本结果之一。 在本工作中，我们提供了对任意施瓦茨希尔德半径的施瓦茨希尔德黑洞的量子相对熵的面积定律的推导。 在G. Bianconi《从熵产生的引力》（2024）中，流形的度规与几何和物质场诱导的度规之间的量子相对熵被提出作为熵量子引力的作用量，导致了修改后的爱因斯坦方程。 量子相对熵推广了Araki熵，并将零形式、一形式和二形式之间的度规视为量子算子。 尽管施瓦茨希尔德度规不是熵量子引力的修改爱因斯坦方程的精确解，但它在低耦合极限下是一个近似解。 在这里，我们证明与施瓦茨希尔德度规相关的量子相对熵在大的施瓦茨希尔德半径下遵循面积定律。 我们提供了对结果的完整统计力学解释。 显示英文摘要

摘要： The area law obeyed by the thermodynamic entropy of black holes is one of the fundamental results relating gravity to statistical mechanics. In this work we provide a derivation of the area law for the quantum relative entropy of the Schwarzschild black-hole for arbitrary Schwarzschild radius. The quantum relative entropy between the metric of the manifold and the metric induced by the geometry and the matter field has been proposed in G. Bianconi ``Gravity from entropy" (2024) as the action for entropic quantum gravity leading to modified Einstein equations. The quantum relative entropy generalizes Araki entropy and treats the metrics between zero-forms, one-forms, and two-forms as quantum operators. Although the Schwarzschild metric is not an exact solution of the modified Einstein equations of the entropic quantum gravity, it is an approximate solution valid in the low coupling limit. Here we show that the quantum relative entropy associated to the Schwarzschild metric obeys the area law for large Schwarzschild radius. We provide a full statistical mechanics interpretation of the results.