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 γB for B. In that context, the constrained HRT surface γA: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 γB. As a result, γA:B is the union of two pieces, $$ {\gamma}_{A:B}^B $$ γ A : B B and $$ {\gamma}_{A:B}^{\overline{B}} $$ γ A : B B ¯ lying respectively in the entanglement wedges of B and its complement $$ \overline{B} $$ B ¯ . Unlike the area $$ \mathcal{A}\left({\gamma}_A\right) $$ A γ A of the HRT surface γA, at least in the semiclassical limit, the area $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ A γ A : B of γA:B commutes with the area $$ \mathcal{A}\left({\gamma}_B\right) $$ A γ B of γB. To study the entropic interpretation of $$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ A γ A : B , 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 $$ \mathcal{A}\left({\gamma}_A\right) $$ A γ A over spacetimes defined by a boost angle conjugate to $$ \mathcal{A}\left({\gamma}_B\right) $$ A γ B . In the case where the pieces $$ {\gamma}_{A:B}^B $$ γ A : B B and $$ {\gamma}_{A:B}^{\overline{B}} $$ γ A : B B ¯ intersect at a constant boost angle, a geometric argument shows that the n ≈ 1 Rényi entropy is then given by $$ \frac{\mathcal{A}\left({\gamma}_{A:B}\right)}{4G} $$ A γ A : B 4 G . 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.