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arXiv:1811.05171 · Journal of High Energy Physics

Holographic Renyi entropy from quantum error correction

Chris Akers, Pratik Rath

Published May 1, 2019• 132 citations
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Abstract

A bstract We study Renyi entropies S n in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing $$ {\tilde{S}}_n\equiv {n}^2{\partial}_n\left(\frac{n-1}{n}{S}_n\right) $$ 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.

DOI
10.1007/jhep05(2019)052

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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.
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.
We explore the role of sandwiched Renyi relative entropy in AdS/CFT and in finite-dimensional models of holographic quantum error correction. In particular, in the context of operator algebra quantum error correction, we discuss a suitable generalization of sandwiched Renyi relative entropy over finite-dimensional von Neumann algebras. It is then shown that the equality of bulk and boundary sandwiched relative Renyi entropies is equivalent to algebraic encoding of bulk and boundary states, the Ryu-Takayanagi formula, the equality of bulk and boundary relative entropy, and subregion duality. This adds another item to an equivalence theorem between the last four items established in arxiv:1607.03901. We then discuss the sandwiched Renyi relative entropy defined in terms of modular operators, and show that this becomes the definition naturally suited to the finite-dimensional models of holographic quantum error correction. Finally, we explore some numerical calculations of sandwiched Renyi relative entropies for a simple holographic random tensor network in order to obtain a better understanding of corrections to the exact equality of bulk and boundary sandwiched relative Renyi entropy.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
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 Tensor networks are useful toy models for understanding the structure of entanglement in holographic states and reconstruction of bulk operators within the entanglement wedge. They are, however, constrained to only prepare so-called “fixed-area states” with flat entanglement spectra, limiting their utility in understanding general features of holographic entanglement. Here, we overcome this limitation by constructing a variant of random tensor networks that enjoys bulk gauge symmetries. Our model includes a gauge theory on a general graph, whose gauge-invariant states are fed into a random tensor network. We show that the model satisfies the quantum-corrected Ryu-Takayanagi formula with a nontrivial area operator living in the center of a gauge-invariant algebra. We also demonstrate nontrivial, n -dependent contributions to the Rényi entropy and Rényi mutual information from this area operator, a feature shared by general holographic states.