Quantum Relative Entropy implies the Semiclassical Einstein Equations

Abstract: We prove that the semiclassical Einstein equations emerge directly from quantum information theory. Using modular theory, we establish that the relative entropy between the vacuum state and coherent excitations of a scalar quantum field on a bifurcate Killing horizon is given by the energy flux across the horizon. Under the assumption of the Bekenstein-Hawking entropy-area formula, this energy flux is proportional to a variation in the surface area of the horizon cross section. The semiclassical Einstein equations follow automatically from this identification. Our approach provides a rigorous quantum field theoretic generalization of Jacobson's thermodynamic derivation of Einstein's equations, replacing classical thermodynamic entropy with the well-defined quantum relative (Araki-Uhlmann) entropy. This suggests that quantum information plays a fundamental role in what is seen as a zeroth order approximation of a theory of quantum gravity, namely quantum field theory in curved spacetimes.