Relative entropy and curved spacetimes

TL;DR

This paper proves the convexity of entropy functions in curved spacetimes using modular theory, enabling analysis of quantum energy inequalities in quantum field theory on stationary curved backgrounds.

Contribution

It establishes the convexity of vacuum relative entropy in curved spacetimes and applies this to study the quantum null energy condition in free quantum field theories.

Findings

Convexity of entropy functions in curved spacetimes.

Application to quantum null energy condition (QNEC) for coherent states.

Examples include Schwarzschild spacetime and null translated regions.

Abstract

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.