Constrained HRT Surfaces and their Entropic Interpretation

Abstract

Abstract Consider two boundary subregionsAandBthat lie in a common boundary Cauchy surface, and consider also the associated HRT surfaceγ B forB. In that context, the constrained HRT surfaceγ A:B can be defined as the codimension-2 bulk surface anchored toAthat is obtained by a maximin construction restricted to Cauchy slices containingγ B . As a result,γ A: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 ofBand its complement$$ \overline{B} $$ B ¯ . Unlike the area$$ \mathcal{A}\left({\gamma}_A\right) $$ A γ A of the HRT surfaceγ A , at least in the semiclassical limit, the area$$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ A γ A : B ofγ A:B commutes with the area$$ \mathcal{A}\left({\gamma}_B\right) $$ A γ B ofγ B . To study the entropic interpretation of$$ \mathcal{A}\left({\gamma}_{A:B}\right) $$ A γ A : B , we analyze the Rényi entropies of subregionAin a fixed-area state of subregionB. We use the gravitational path integral to show that then ≈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 then ≈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 then ≈1 Rényi entropy differs from the von Neumann entropy due to a lack of commutativity of then→ 1 andG →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.