Time-independence of gravitational R\'enyi entropies and unitarity in quantum gravity

Abstract

The Hubeny-Rangamani-Takayanagi surface $\gamma_{HRT}$ computing the entropy $S(D)$ of a domain of dependence $D$ on an asymptotically AdS boundary is known to be causally inaccessible from $D$. We generalize this gravitational result to higher replica numbers $n >1$ by considering the replica-invariant surfaces (aka `splitting surfaces') $\boldsymbol{\mathbf{\gamma}}$ of real-time replica-wormhole saddle-points computing R\'enyi entropies $S_n(D)$ and showing that there is a sense in which $D$ must again be causally inaccessible from $\boldsymbol{\mathbf{\gamma}}$ when the saddle preserves both replica and conjugation symmetry. This property turns out to imply the $S_n(D)$ to be independent of any choice of any Cauchy surface $\Sigma_D$ for $D$, and also that the $S_n(D)$ are independent of the choice of boundary sources within $D$. This is a key hallmark of unitary evolution in any dual field theory. Furthermore, from the bulk point of view it adds to the evidence that time evolution of asymptotic observables in quantum gravity is implemented by a unitary operator in each baby universe superselection sector. Though we focus here on pure Einstein-Hilbert gravity and its Kaluza-Klein reductions, we expect the argument to extend to any two-derivative theory who satisfies the null convergence condition. We consider both classical saddles and the effect of back-reaction from quantum corrections.