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Entropy on a null surface for interacting quantum field theories and the Bousso bound

We study the vacuum-subtracted von Neumann entropy of a segment on a null plane. We argue that for interacting quantum field theories in more than two dimensions, this entropy has a simple expression in terms of the expectation value of the null components of the stress tensor on the null interval. More explicitly $ΔS = 2π\int d^{d-2}y \int_0^1 dx^+\, g(x^+)\, \langle T_{++}\rangle$, where $g(x^+)$ is a theory-dependent function. This function is constrained by general properties of quantum relative entropy. These constraints are enough to extend our recent free field proof of the quantum Bousso bound to the interacting case. This unusual expression for the entropy as the expectation value of an operator implies that the entropy is equal to the modular Hamiltonian, $ΔS = \langle ΔK \rangle $, where $K$ is the operator in the right hand side. We explain how this equality is compatible with a non-zero value for $ΔS$. Finally, we also compute explicitly the function $g(x^+)$ for theories that have a gravity dual.