  In this paper we argue that classical, asymptotically AdS spacetimes that
arise as states in consistent ultraviolet completions of Einstein gravity
coupled to matter must satisfy an infinite family of positive energy
conditions. To each ball-shaped spatial region $B$ of the boundary spacetime,
we can associate a bulk spatial region $\Sigma_B$ between $B$ and the bulk
extremal surface $\tilde{B}$ with the same boundary as $B$. We show that there
exists a natural notion of a gravitational energy for every such region that is
non-negative, and non-increasing as one makes the region smaller. The results
follow from identifying this gravitational energy with a quantum relative
entropy in the associated dual CFT state. The positivity and monotonicity
properties of the gravitational energy are implied by the positivity and
monotonicity of relative entropy, which holds universally in all quantum
systems.
