Given any half-sided modular inclusion of standard subspaces, we show that the entropy function associated with the decreasing one-parameter family of translated standard subspaces is convex for any given (not necessarily smooth) vector in the underlying Hilbert space. In second quantisation, this infers the convexity of the vacuum relative entropy with respect to the translation parameter of the modular tunnel of von Neumann algebras. This result allows us to study the QNEC inequality for coherent states in a free Quantum Field Theory on a stationary curved spacetime, given a KMS state. To this end, we define wedge regions and appropriate (deformed) subregions. Examples are given by the Schwarzschild spacetime and null translated subregions with respect to the time translation Killing flow. More generally, we define wedge and strip regions on a globally hyperbolic spacetime, so to have non trivial modular inclusions of von Neumann algebras, and make our analysis in this context.

给定标准子空间的任何半边模包含，我们表明，对于底层希尔伯特空间中的任何给定（不一定是平滑的）向量，与减少的一个参数族平移标准子空间相关的熵函数是凸的。在第二次量化中，这推断真空相对熵相对于冯诺依曼代数模隧道的平移参数的凸性。这个结果使我们能够在给定 KMS 状态的情况下，研究自由量子场论中固定弯曲时空上相干态的 QNEC 不等式。为此，我们定义了楔形区域和适当的（变形的）子区域。Schwarzschild 时空和空平移子区域给出了关于时间平移杀死流的例子。更普遍，