Entanglement Equilibrium and the Einstein Equation

Abstract

A link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein equation implies validity of the hypothesis. A more precise argument shows that, for first-order variations of the local vacuum state of conformal quantum fields, the vacuum entanglement is stationary if and only if the Einstein equation holds. For nonconformal fields, the same conclusion follows modulo a conjecture about the variation of entanglement entropy.

8 pages; v2: reorganized and rewritten; figure added; v3: reorganized; assumed conjecture weakened; noted that the derivation holds only for first order variations of the quantum field vacuum; v4: published version; introduction rewritten; results of arXiv:1601.00528 and arXiv:1602.01380 required that curvature of maximally symmetric local reference spacetime depend on the ball radius

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