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, \( {\gamma}_{A:B}^B \) and \( {\gamma}_{A:B}^{\overline{B}} \) lying respectively in the entanglement wedges of B and its complement \( \overline{B} \). Unlike the area \( \mathcal{A}\left({\gamma}_A\right) \) of the HRT surface γA, at least in the semiclassical limit, the area \( \mathcal{A}\left({\gamma}_{A:B}\right) \) of γA:B commutes with the area \( \mathcal{A}\left({\gamma}_B\right) \) of γB. To study the entropic interpretation of \( \mathcal{A}\left({\gamma}_{A:B}\right) \), 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 \( \mathcal{A}\left({\gamma}_A\right) \) over spacetimes defined by a boost angle conjugate to \( \mathcal{A}\left({\gamma}_B\right) \). In the case where the pieces \( {\gamma}_{A:B}^B \) and \( {\gamma}_{A:B}^{\overline{B}} \) intersect at a constant boost angle, a geometric argument shows that the n ≈ 1 Rényi entropy is then given by \( \frac{\mathcal{A}\left({\gamma}_{A:B}\right)}{4G} \). 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.
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Acknowledgments
We would like to thank Chris Akers, Luca Iliesiu, Geoff Penington and Arvin Shahbazi-Moghaddam for useful discussions. PR is supported in part by a grant from the Simons Foundation, and by funds from UCSB. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0360. This work was supported in part by the Berkeley Center for Theoretical Physics; by the Department of Energy, Office of Science, Office of High Energy Physics under QuantISED Award DESC0019380 and under contract DE-AC02-05CH11231. PR was supported by the National Science Foundation under Award Number 2112880.
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Dong, X., Marolf, D. & Rath, P. Constrained HRT Surfaces and their Entropic Interpretation. J. High Energ. Phys. 2024, 151 (2024). https://doi.org/10.1007/JHEP02(2024)151
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DOI: https://doi.org/10.1007/JHEP02(2024)151