Relative entropy and curved spacetimes

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