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The positivity of relative entropy for spatial subsystems in a holographicCFT implies the positivity of certain quantities in the dual gravitationaltheory. In this note, we consider CFT subsystems whose boundaries lie on thelightcone of a point $p$. We show that the positive gravitational quantitywhich corresponds to the relative entropy for such a subsystem $A$ is a novelnotion of energy associated with a gravitational subsystem bounded by theminimal area extremal surface $\tilde{A}$ associated with $A$ and by the AdSboundary region $\hat{A}$ corresponding to the part of the lightcone from $p$bounded by $\partial A$. This generalizes the results of arXiv:1605.01075 forball-shaped regions by making use of the recent results in arXiv:1703.10656 forthe vacuum modular Hamiltonian of regions bounded on lightcones. As part of ouranalysis, we give an analytic expression for the extremal surface in pure AdSassociated with any such region $A$. We note that its form immediately impliesthe Markov property of the CFT vacuum (saturation of strong subadditivity) forregions bounded on the same lightcone. This gives a holographic proof of theresult proven for general CFTs in arXiv:1703.10656. A similar holographic proofshows the Markov property for regions bounded on a lightsheet for non-conformalholographic theories defined by relevant perturbations of a CFT.

Positive gravitational subsystem energies from CFT cone relative entropies