Constrained HRT Surfaces and their Entropic Interpretation

Abstract

Consider two boundary subregions $A$ and $B$ that lie in a common boundary Cauchy surface, and consider also the associated HRT surface $\gamma_B$ for $B$. In that context, the constrained HRT surface $\gamma_{A:B}$ can be defined as the codimension-2 bulk surface anchored to $A$ that is obtained by a maximin construction restricted to Cauchy slices containing $\gamma_B$. As a result, $\gamma_{A:B}$ is the union of two pieces, $\gamma^B_{A:B}$ and $\gamma^{\bar B}_{A:B}$ lying respectively in the entanglement wedges of $B$ and its complement $\bar B$. Unlike the area $\mathcal{A}\left(\gamma_A\right)$ of the HRT surface $\gamma_A$, at least in the semiclassical limit, the area $\mathcal{A}\left(\gamma_{A:B}\right)$ of $\gamma_{A:B}$ commutes with the area $\mathcal{A}\left(\gamma_B\right)$ of $\gamma_B$. To study the entropic interpretation of $\mathcal{A}\left(\gamma_{A:B}\right)$, we analyze the R\'enyi entropies of subregion $A$ in a fixed-area state of subregion $B$. We use the gravitational path integral to show that the $n\approx1$ R\'enyi entropies are then computed by minimizing $\mathcal{A}\left(\gamma_A\right)$ over spacetimes defined by a boost angle conjugate to $\mathcal{A}\left(\gamma_B\right)$. In the case where the pieces $\gamma^B_{A:B}$ and $\gamma^{\bar B}_{A:B}$ intersect at a constant boost angle, a geometric argument shows that the $n\approx1$ R\'enyi entropy is then given by $\frac{\mathcal{A}(\gamma_{A:B})}{4G}$. We discuss how the $n\approx1$ R\'enyi entropy differs from the von Neumann entropy due to a lack of commutativity of the $n\to1$ and $G\to0$ limits. We also discuss how the behaviour changes as a function of the width of the fixed-area state. Our results are relevant to some of the issues associated with attempts to use standard random tensor networks to describe time dependent geometries.