Constrained HRT Surfaces and their Entropic Interpretation

A bstract 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, γA:BB γ A : B B and γA:BB̄ γ A : B B ¯ lying respectively in the entanglement wedges of B and its complement B̄ B ¯ . Unlike the area A(γA) A γ A of the HRT surface γ A , at least in the semiclassical limit, the area A(γA:B) A γ A : B of γ A : B commutes with the area A(γB) A γ B of γ B . To study the entropic interpretation of A(γA:B) 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 A(γA) A γ A over spacetimes defined by a boost angle conjugate to A(γB) A γ B . In the case where the pieces γA:BB γ A : B B and γA:BB̄ γ A : B B ¯ intersect at a constant boost angle, a geometric argument shows that the n ≈ 1 Rényi entropy is then given by A(γA:B)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.