Boundary Relative Entropy as Quasilocal Energy: Positive Energy Theorems and Tomography

Abstract

We argue that for a spherical region R on the boundary, relative entropy between the vacuum and an arbitrary holographic excited state can be computed in the bulk as a quasilocal energy associated to the volume between R and the minimal surface B ending on the boundary ∂R. Since relative entropy is monotonic and positive in any well-defined quantum theory, the associated quasilocal energy must also be positive and monotonic. This gives rise to an infinite number of constraints on the gravitational bulk, which must be satisfied in any theory of quantum gravity with a well-defined UV completion. For small regions $R$, these constraints translate into integrated positivity conditions of the bulk stress-energy tensor. When the bulk is Einstein gravity coupled to scalar fields, the boundary relative entropy can be related to an integral of the bulk action on the minimal surface B. Near the boundary, this expression can be inverted via the inverse Radon transform, to obtain the bulk stress energy tensor at a point in terms of the boundary relative entropy.