We derive the Ryu–Takayanagi (RT) formula for holographic entanglement entropy from a resource-allocation principle. We model bulk reconstruction as a budget-constrained distributed optimization problem: spacelike-separated boundary observers must decide which bulk operators to reconstruct under a shared budget, without communicating their local inputs. We prove three results. (1) Any local (non-communicating) protocol for this game achieves expected utility at most 1/2, while a centralized protocol achieves 2/3; the gap Δ = 1/6 is tight. (2) The information cost of closing this gap is O(1) bits per degree of freedom, with residual loss decaying as 4−k in the number of communicated bits k. (3) In holographic tensor networks (HaPPY codes and random tensor networks), the allocation decisions across distinct bonds on the minimal surface are independent, so the total information cost sums linearly: S(A) = |γA| log2 χ, reproducing the RT formula. The derivation is non-circular: the RT formula emerges as the output, not an input. We identify the bipartite inefficiency ratio η(2) = 1/4 and discuss its possible connection to the Bekenstein–Hawking prefactor. We also characterize the entanglement wedge phase transition as a discrete reallocation event in the bond-level optimization landscape.Extended proofs available at doi:10.5281/zenodo.19213268.
We derive the Ryu–Takayanagi (RT) formula for holographic entanglement entropy from a resource-allocation principle. We model bulk reconstruction as a budget-constrained distributed optimization problem: spacelike-separated boundary observers must decide which bulk operators to reconstruct under a shared budget, without communicating their local inputs. We prove three results. (1) Any local (non-communicating) protocol for this game achieves expected utility at most 1/2, while a centralized protocol achieves 2/3; the gap Δ = 1/6 is tight. (2) The information cost of closing this gap is O(1) bits per degree of freedom, with residual loss decaying as 4−k in the number of communicated bits k. (3) In holographic tensor networks (HaPPY codes and random tensor networks), the allocation decisions across distinct bonds on the minimal surface are independent, so the total information cost sums linearly: S(A) = |γA| log2 χ, reproducing the RT formula. The derivation is non-circular: the RT formula emerges as the output, not an input. We identify the bipartite inefficiency ratio η(2) = 1/4 and discuss its possible connection to the Bekenstein–Hawking prefactor. We also characterize the entanglement wedge phase transition as a discrete reallocation event in the bond-level optimization landscape.Extended proofs available at doi:10.5281/zenodo.19213268.
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
We propose a framework for preparing quantum states with a holographic entanglement structure, in the sense that the entanglement entropies are governed by minimal surfaces in a chosen bulk geometry. We refer to such entropies as holographic because they obey a relation between entropies and bulk minimal surfaces, known as the Ryu-Takayanagi (RT) formula, that is a key feature of holographic models of quantum gravity. Typically in such models, the bulk geometry is determined by solving Einstein's equations. Here, we simply choose a bulk geometry, then discretize the geometry into a coupling graph comprising bulk and boundary nodes. Evolving under this graph of interactions and measuring the bulk nodes leaves behind the desired pure state on the boundary. We numerically demonstrate that the resulting entanglement properties approximately reproduce the predictions of the Ryu-Takayanagi formula in the chosen bulk geometry. We consider graphs associated with hyperbolic disk and wormhole geometries, but the approach is general. We also introduce a procedure for decorating the graph which, when applied to the hyperbolic disk, yields a genuine CFT ground state with power-law correlations but which only partially obeys RT. The minimal ingredients in our proposal involve only Gaussian operations and measurements and are readily implementable in photonic and cold-atom platforms.
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.
We consider the special case of random tensor networks (RTNs) endowed with gauge symmetry constraints on each tensor. We compute the R\'enyi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large-bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background-independent quantum gravity, and for importing quantum gravity tools into tensor network research.
We establish a quantitative equivalence between the holographic duality of theAdS/CFT correspondence and the theory of quantum error-correcting codes (QECC).By identifying the code subspace with the low-energy bulk effective field theoryand the physical Hilbert space with the boundary CFT, we show that the RyuTakayanagi formula for entanglement entropy directly determines the code distanced —the minimal weight of an undetectable error. We prove a holographic Hammingbound: d ≤ Area(γmin)/(4Gℏln2), with equality for perfect tensor networks. Thegravitational constant G thus quantifies the error-correcting capacity of spacetime.A testable prediction is derived: the code distance for a black hole horizon saturates the bound, explaining the unitarity of Hawking evaporation as a property of amaximum-distance separable code. We further show that the entanglement wedgereconstruction theorem is equivalent to the Knill–Laflamme conditions for QECC
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.