Relative entropy for coherent states from Araki formula

Abstract

We make a rigorous computation of the relative entropy between the vacuum state and a coherent state for a free scalar in the framework of algebraic description of quantum field theory (AQFT). We study the case of the Rindler wedge. Previous calculations including path integral methods and computations from the lattice give a result for such relative entropy which involves integrals of expectation values of the energy-momentum stress tensor along the considered region. However, the stress tensor is in general nonunique. That means that if we start with some stress tensor, then we can "improve" it adding a conserved term without modifying the Poincaré charges. On the other hand, the presence of such an improving term affects the naive expectation for the relative entropy by a nonvanishing boundary contribution along the entangling surface. In other words, this means that there is an ambiguity in the usual formula for the relative entropy coming from the nonuniqueness of the stress tensor. The main motivation of this work is to solve this puzzle. We first show that all choices of stress tensor except the canonical one are not allowed by positivity and monotonicity of the relative entropy. Then we fully compute the relative entropy between the vacuum and a coherent state in the framework of AQFT using the Araki formula and the techniques of modular theory. After all, both results coincide and give the usual expression for the relative entropy calculated with the canonical stress tensor.