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arXiv:2312.04625 · Journal of High Energy Physics

A modified cosmic brane proposal for holographic Renyi entropy

Xi Dong, Jonah Kudler-Flam, Pratik Rath

Published Jun 19, 2024• 13 citations
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Abstract

A bstract We propose a new formula for computing holographic Renyi entropies in the presence of multiple extremal surfaces. Our proposal is based on computing the wave function in the basis of fixed-area states and assuming a diagonal approximation for the Renyi entropy. For Renyi index n ≥ 1, our proposal agrees with the existing cosmic brane proposal for holographic Renyi entropy. For n < 1, however, our proposal predicts a new phase with leading order (in Newton’s constant G ) corrections to the cosmic brane proposal, even far from entanglement phase transitions and when bulk quantum corrections are unimportant. Recast in terms of optimization over fixed-area states, the difference between the two proposals can be understood to come from the order of optimization: for n < 1, the cosmic brane proposal is a minimax prescription whereas our proposal is a maximin prescription. We demonstrate the presence of such leading order corrections using illustrative examples. In particular, our proposal reproduces existing results in the literature for the PSSY model and high-energy eigenstates, providing a universal explanation for previously found leading order corrections to the n < 1 Renyi entropies.

DOI
10.1007/jhep06(2024)120

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A bstract We propose a new formula for computing holographic Renyi entropies in the presence of multiple extremal surfaces. Our proposal is based on computing the wave function in the basis of fixed-area states and assuming a diagonal approximation for the Renyi entropy. For Renyi index n ≥ 1, our proposal agrees with the existing cosmic brane proposal for holographic Renyi entropy. For n < 1, however, our proposal predicts a new phase with leading order (in Newton’s constant G ) corrections to the cosmic brane proposal, even far from entanglement phase transitions and when bulk quantum corrections are unimportant. Recast in terms of optimization over fixed-area states, the difference between the two proposals can be understood to come from the order of optimization: for n < 1, the cosmic brane proposal is a minimax prescription whereas our proposal is a maximin prescription. We demonstrate the presence of such leading order corrections using illustrative examples. In particular, our proposal reproduces existing results in the literature for the PSSY model and high-energy eigenstates, providing a universal explanation for previously found leading order corrections to the n < 1 Renyi entropies.
Recently, Dong et al., [A modified cosmic brane proposal for holographic Renyi entropy, J. High Energy Phys. 06 (2024) 120JHEPFG1029-847910.1007/JHEP06(2024)120] proposed a modified cosmic brane prescription for computing the Rényi entropy S_{α} of a holographic system in the presence of multiple extremal surfaces. This prescription was found by assuming a diagonal approximation, where the Rényi entropy is computed after first measuring the areas of all extremal surfaces. We derive this diagonal approximation for the case of two extremal surfaces and show that it accurately computes Rényi entropies up to O(logG) corrections. For α<1, this allows us to derive the modified cosmic brane prescription, which differs from the original cosmic brane prescription at leading order in G. For α>1, it leads to the original cosmic brane prescription without needing to assume that replica symmetry is unbroken in the bulk.
A bstract We study Renyi entropies S n in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing S~n≡n2∂n(n−1nSn) {\tilde{S}}_n\equiv {n}^2{\partial}_n\left(\frac{n-1}{n}{S}_n\right) S~n​≡n2∂n​(nn−1​Sn​) S ˜ n ≡ n 2 ∂ n n − 1 n S n . We find that general operator algebra codes have a similar, more general prescription. Notably, for the AdS/CFT code to match the specific cosmic brane prescription, the code must have maximal entanglement within eigenspaces of the area operator. This gives us an improved definition of the area operator, and establishes a stronger connection between the Ryu-Takayanagi area term and the edge modes in lattice gauge theory. We also propose a new interpretation of existing holographic tensor networks as area eigenstates instead of smooth geometries. This interpretation would explain why tensor networks have historically had trouble modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a method to construct holographic networks with the correct spectrum.
We show that Renyi entropies satisfy an area law in quantum field theories with a holographic dual description in terms of a bulk gravitational theory. A derivative of the Renyi entropy SnS_nSn​ with respect to its index nnn is given by the area in Planck units of a bulk cosmic brane homologous to the entangling region. This geometric prescription for holographic Renyi entropies is a one-parameter generalization of the Ryu-Takayanagi prescription for entanglement entropy. As an example of applying this prescription, we calculate the mutual Renyi information InI_nIn​ between two disks to linear order in n−1n-1n−1 in holographic conformal field theories of arbitrary dimension.
Abstract In the context of the holographic duality, the entanglement entropy of ordinary QFT in a subregion in the boundary is given by a quarter of the area of an minimal surface embedded in the bulk spacetime. This rule has been also extended to a suitable one-parameter generalization of the von-Neuman entropy Ŝn that is related to the Rényi entropies Sn, as given by the area of a cosmic brane minimally coupled with gravity, with a tension related to n that vanishes as n → 1, and moreover, this parameter can be analytically extended to arbitrary real values. However, the brane action plays no role in the duality and cannot be considered a part of the theory of gravity, thus it is used as an auxiliary tool to find the correct background geometry. In this work we study the construction of the gravitational (reduced) density matrix from holographic states, whose wave-functionals are described as euclidean path integrals with arbitrary conditions on the asymptotic boundaries, and argue that in general, a non-trivial Hayward term must be haven into account. So we propose that the gravity model with a coupled Nambu-Goto action is not an artificial tool to account for the Rényi entropies, but it is present in the own gravity action through a Hayward term. As a result we show that the computations using replicas simplify considerably and we recover the holographic prescriptions for the measures of entanglement entropy; in particular, derive an area law for the original Rényi entropies (Sn) related to a minimal surface in the n replicated spacetime. Moreover, we show that the gravitational modular flow contains the area operator and can explain the Jafferis-Lewkowycz-Maldacena-Suh proposal.
In the context of the holographic duality, the entanglement entropy of ordinary QFT in a subregion in the boundary is given by a quarter of the area of an minimal surface embedded in the bulk spacetime. This rule has been also extended to a suitable one-parameter generalization of the von-Neuman entropy hatSn\\hat{S}_nhatSn​ that is related to the Renyi entropies SnS_nSn​, as given by the area of a \\emph{cosmic brane} minimally coupled with gravity, with a tension related to nnn that vanishes as nto1n\\to1nto1, and moreover, this parameter can be analytically extended to arbitrary real values. However, the brane action plays no role in the duality and cannot be considered a part of the theory of gravity, thus it is used as an auxiliary tool to find the correct background geometry. In this work we study the construction of the gravitational (reduced) density matrix from holographic states, whose wave-functionals are described as euclidean path integrals with arbitrary conditions on the asymptotic boundaries, and argue that in general, a non-trivial Hayward term must be haven into account. So we propose that the gravity model with a coupled Nambu-Goto action is not an artificial tool to account for the Renyi entropies, but it is present in the own gravity action through a Hayward term. As a result we show that the computations using replicas simplify considerably and we recover the holographic prescriptions for the measures of entanglement entropy; in particular, derive an area law for the original Renyi entropies (SnS_nSn​) related to a minimal surface in the nnn replicated spacetime. Moreover, we show that the gravitational modular flow contains the area operator and can explain the Jafferis-Lewkowycz-Maldacena-Suh proposal.
A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge-gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Rényi entropies. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Rényi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Rényi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.