From Quantum Relative Entropy to the Semiclassical Einstein Equations

Using modular theory, the authors show that the quantum relative entropy between the vacuum and coherent excitations of a scalar field on a bifurcate Killing horizon equals the energy flux across the horizon, which, assuming the Bekenstein-Hawking entropy-area relation, is proportional to the horizon’s area variation. Identifying this proportionality yields the semiclassical Einstein equations, providing a quantum-information-based derivation that generalizes Jacobson’s thermodynamic argument.