Constrained HRT Surfaces and their Entropic Interpretation

Abstract

Abstract Consider two boundary subregions A and B that lie in a common boundary Cauchy surface, and consider also the associated HRT surface γ B for B. In that context, the constrained HRT surface γ 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 γ B . As a result, γ A:B is the union of two pieces, γ A : B B $$ {\gamma}_{A:B}^B $$ and γ A : B B ¯ $$ {\gamma}_{A:B}^{\overline{B}} $$ lying respectively in the entanglement wedges of B and its complement B ¯ $$ \overline{B} $$ . Unlike the area A γ A $$ \mathcal{A}\left({\gamma}_A\right) $$ of the HRT surface γ A , at least in the semiclassical limit, the area A γ A : B $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ of γ A:B commutes with the area A γ B $$ \mathcal{A}\left({\gamma}_B\right) $$ of γ B . To study the entropic interpretation of A γ A : B $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ , 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 ≈ 1 Rényi entropies are then computed by minimizing A γ A $$ \mathcal{A}\left({\gamma}_A\right) $$ over spacetimes defined by a boost angle conjugate to A γ B $$ \mathcal{A}\left({\gamma}_B\right) $$ . In the case where the pieces γ A : B B $$ {\gamma}_{A:B}^B $$ and γ A : B B ¯ $$ {\gamma}_{A:B}^{\overline{B}} $$ intersect at a constant boost angle, a geometric argument shows that the n ≈ 1 Rényi entropy is then given by A γ A : B 4 G $$ \frac{\mathcal{A}\left({\gamma}_{A:B}\right)}{4G} $$ . We discuss how the n ≈ 1 Rényi entropy differs from the von Neumann entropy due to a lack of commutativity of the n → 1 and G → 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.

Keywords

• AdS-CFT Correspondence
• Gauge-Gravity Correspondence