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 ${\hat{I}}_B^3$ for $B$. In that context, the constrained HRT surface ${\hat{I}}_{A:B}^3$ can be defined as the codimension-2 bulk surface anchored to $A$ that is obtained by a maximin construction restricted to Cauchy slices containing ${\hat{I}}_B^3$. As a result, ${\hat{I}}_{A:B}^3$ is the union of two pieces, $γ^B_{A:B}$ and $γ^{\bar B}_{A:B}$ lying respectively in the entanglement wedges of $B$ and its complement $\bar{B}$. Unlike the area $\mathcal{A}\left({\hat{I}}_A^3\right)$ of the HRT surface ${\hat{I}}_A^3$, at least in the semiclassical limit, the area $\mathcal{A}\left({\hat{I}}_{A:B}^3\right)$ of ${\hat{I}}_{A:B}^3$ commutes with the area $\mathcal{A}\left({\hat{I}}_B^3\right)$ of ${\hat{I}}_B^3$. To study the entropic interpretation of $\mathcal{A}\left({\hat{I}}_{A:B}^3\right)$, we analyze the Rényi entropies of subregion $A$ in a fixed-area state of subregion $B$. We use the gravitational path integral to show that the $n\approx 1$ Rényi entropies are then computed by minimizing $\mathcal{A}\left({\hat{I}}_A^3\right)$ over spacetimes defined by a boost angle conjugate to $\mathcal{A}\left({\hat{I}}_B^3\right)$. In the case where the pieces $γ^B_{A:B}$ and $γ^{\bar B}_{A:B}$ intersect at a constant boost angle, a geometric argument shows that the $n\approx 1$ Rényi entropy is then given by $\frac{\mathcal{A}({\hat{I}}_{A:B}^3)}{4G}$. We discuss how the $n\approx 1$ Rényi entropy differs from the von Neumann entropy due to a lack of commutativity of the $n\to 1$ and $G\to 0$ 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.

16 pages, 3 figures, minor edits in v2