# Relative entropy and curved spacetimes

July 14, 2021

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 stripe regions on
a globally hyperbolic spacetime, so to have non trivial modular inclusions of
von Neumann algebras, and make our analysis in this context.

Keywords:

*none*