  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 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 non unique. That means that if we start with some stress tensor, then
we can "improve" it adding a conserved term without modifying the Poincar\'e
charges. On the other hand, the presence of such improving term affects the
naive expectation for the relative entropy by a non vanishing 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 non uniqueness 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 coincides and give the
usual expression for the relative entropy calculated with the canonical stress
tensor.
