Relative entropy and holography

Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation Delta S = Delta H for the first order variation of the entanglement entropy AS and the expectation value of the modular Hamiltonian Delta H. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation Delta S = Delta H for vacuum state tomography and obtain modified versions of the Bekenstein bound.