Entanglement Equilibrium and the Einstein Equation

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 the 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.

Figure 1

Causal diamond, in a maximally symmetric spacetime, for a geodesic ball $\mathrm{\Sigma }$ of radius $\ell$ with center $o$ and boundary $\partial \mathrm{\Sigma }$. The dashed curves are flow lines of $\zeta$, the conformal Killing vector field, whose flow preserves the diamond and which vanishes at the top and bottom vertices and on $\partial \mathrm{\Sigma }$. The vectors show $\zeta$ at four points of $\mathrm{\Sigma }$.