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arXiv:2508.14877

Proof of a Generalized Ryu-Takayanagi Conjecture

Artem Averin

Published Aug 20, 2025• 0 citations
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Abstract

We derive a generalized version of the Ryu-Takayanagi formula for the entanglement entropy in arbitrary diffeomorphism invariant field theories. We use a recent framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory's phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface $Σ,$ the choice of a codimension-2 surface $B$ without boundary contained in $Σ$ specifies a submanifold in the phase space. For diffeomorphism invariant field theories, a functional integral expression for their density matrices was recently given and then used to derive bounds on phase space volumes in the considered submanifold associated to $B.$ These bounds formalize the gravitational entropy bound. Here, we present an implication of this derivation in that we show the obtained functional integral expression for density matrices to be naturally suited for the replica trick. Correspondingly, we prove a functional integral expression for the associated entanglement entropies and derive a practical prescription for their evaluation to leading order and beyond. An important novelty of our approach is the contact to phase space. This allows us both to obtain a prescription for entanglement entropies in arbitrary diffeomorphism invariant field theories not necessarily possessing a holographic dual as well as to use entanglement entropies to study their phase space structure. In the case of the bulk-boundary correspondence, our prescription consistently reproduces and hence provides a natural and independent proof of the Ryu-Takayanagi formula as well as its various generalizations. These include the covariant holographic entanglement entropy proposal, Dong's proposal for higher-derivative gravity as well as the quantum extremal surface prescription.

DOI
10.48550/arxiv.2508.14877

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We derive a generalized version of the Ryu-Takayanagi formula for the entanglement entropy in arbitrary diffeomorphism invariant field theories. We use a recent framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory's phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface Σ,Σ,Σ, the choice of a codimension-2 surface BBB without boundary contained in ΣΣΣ specifies a submanifold in the phase space. For diffeomorphism invariant field theories, a functional integral expression for their density matrices was recently given and then used to derive bounds on phase space volumes in the considered submanifold associated to B.B.B. These bounds formalize the gravitational entropy bound. Here, we present an implication of this derivation in that we show the obtained functional integral expression for density matrices to be naturally suited for the replica trick. Correspondingly, we prove a functional integral expression for the associated entanglement entropies and derive a practical prescription for their evaluation to leading order and beyond. An important novelty of our approach is the contact to phase space. This allows us both to obtain a prescription for entanglement entropies in arbitrary diffeomorphism invariant field theories not necessarily possessing a holographic dual as well as to use entanglement entropies to study their phase space structure. In the case of the bulk-boundary correspondence, our prescription consistently reproduces and hence provides a natural and independent proof of the Ryu-Takayanagi formula as well as its various generalizations. These include the covariant holographic entanglement entropy proposal, Dong's proposal for higher-derivative gravity as well as the quantum extremal surface prescription.
A bstract The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the information cost in Bell pairs of restoring correlations inaccessible to observers in the strip. A general realization of this formalism in higher dimensions is unknown. We first prove a no-go theorem eliminating candidate expressions for higher dimensional differential entropy based on entropic c-theorems. Then we propose a new formula in terms of an integral of shape derivatives of the entanglement entropy of ball shaped regions. Our proposal stems from the physical requirement that differential entropy must be locally finite and conformally invariant. Demanding cancelation of the well-known UV divergences of entanglement entropy in field theory guides us to our conjecture, which we test for surfaces in AdS 4 . Our results suggest a candidate c-function for field theories in arbitrary dimensions.
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Abstract The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the information cost in Bell pairs of restoring correlations inaccessible to observers in the strip. A general realization of this formalism in higher dimensions is unknown. We first prove a no-go theorem eliminating candidate expressions for higher dimensional differential entropy based on entropic c-theorems. Then we propose a new formula in terms of an integral of shape derivatives of the entanglement entropy of ball shaped regions. Our proposal stems from the physical requirement that differential entropy must be locally finite and conformally invariant. Demanding cancelation of the well-known UV divergences of entanglement entropy in field theory guides us to our conjecture, which we test for surfaces in AdS4. Our results suggest a candidate c-function for field theories in arbitrary dimensions.
We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}AdS3​/CFT2​. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are codimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A\mathcal{A}A and the points on the Ryu-Takayanagi (RT) surface EA{\mathcal{E}}_{\mathcal{A}}EA​. In the same sense an arbitrary subinterval A2{\mathcal{A}}_{2}A2​ of A\mathcal{A}A will correspond to a subinterval E2{\mathcal{E}}_{2}E2​ of EA{\mathcal{E}}_{\mathcal{A}}EA​. This fine correspondence indicates that the length of E2{\mathcal{E}}_{2}E2​ captures the contribution sA(A2){s}_{\mathcal{A}}({\mathcal{A}}_{2})sA​(A2​) from A2{\mathcal{A}}_{2}A2​ to the entanglement entropy SA{S}_{\mathcal{A}}SA​, hence gives the contour function for entanglement entropy. Furthermore we propose that sA(A2){s}_{\mathcal{A}}({\mathcal{A}}_{2})sA​(A2​) in general can be written as a simple linear combination of entanglement entropies of single intervals inside A\mathcal{A}A. This proposal passes several nontrivial tests.
We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Rényi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.
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