This paper reinterprets holographic duality as a geometric manifestation of information compression in the hierarchical structure of discrete constraint networks.The bulk constraint network forms a hierarchy through local constraint satisfaction,and the effective theory on the boundary emerges as the projection of constraintpropagation patterns onto the boundary subset. Tensor networks — particularlythe multi-scale entanglement renormalization ansatz — provide a computable andconcrete implementation of this correspondence. It is demonstrated that the intrinsic redundancy of the constraint network naturally yields an area-law bound forentropy, and its error-correcting code structure explains the microscopic origin ofsubregion duality and bulk locality. Furthermore, the dynamical evolution of bulkgeometry corresponds to an endogenous reconstruction process of the constraintnetwork, and the gravitational effects perceived by boundary observers are projection phenomena arising from the network’s pursuit of global consistency. Thisframework grounds the holographic principle in information theory and discretedynamics, and suggests possible generalizations to non-AdS geometries.
This paper reinterprets holographic duality as a geometric manifestation of information compression in the hierarchical structure of discrete constraint networks.The bulk constraint network forms a hierarchy through local constraint satisfaction,and the effective theory on the boundary emerges as the projection of constraintpropagation patterns onto the boundary subset. Tensor networks — particularlythe multi-scale entanglement renormalization ansatz — provide a computable andconcrete implementation of this correspondence. It is demonstrated that the intrinsic redundancy of the constraint network naturally yields an area-law bound forentropy, and its error-correcting code structure explains the microscopic origin ofsubregion duality and bulk locality. Furthermore, the dynamical evolution of bulkgeometry corresponds to an endogenous reconstruction process of the constraintnetwork, and the gravitational effects perceived by boundary observers are projection phenomena arising from the network’s pursuit of global consistency. Thisframework grounds the holographic principle in information theory and discretedynamics, and suggests possible generalizations to non-AdS geometries.
Dual constraint entropy, as a natural emergence of the information cost ofnode condensation in constraint networks, provides a fine information-theoreticmetric connecting boundary entanglement structure and bulk geometry for holographic duality. Starting from the two axioms of information conservation andcomputability, this work establishes constraint networks as the discrete skeletonof the bulk-boundary correspondence, and proves that the dual constraint entropySdual = −lnTr(ρ2 A) is the inevitable information cost incurred when the node condensation operator is triggered at the information density threshold. In the HaPPYholographic error-correcting code, the area law Sdual(A) = Amin/αH + γ is rigorously derived, where the effective holographic parameter αH = a/ln2 is uniquelydetermined by the lattice spacing and the fundamental unit of information, eliminating the free parameter nature of Newton’s constant. It is further proved thatwhen the subsystem dimension is no less than 3, dual constraint entropy can distinguish pure states that have identical von Neumann entropy but different entanglement distribution structures, revealing its sensitivity to the kurtosis of theSchmidt spectrum. The analysis of fluctuation scaling ∆Sdual ∼ √lndA in completely random tensor networks provides an information-theoretic interpretationof holographic quantum corrections and yields testable numerical predictions. Thecorrespondence between the information density gradient in the discrete-continuumlimit and bulk curvature is formulated as the constraint network coarse-grainingconjecture, charting the direction for a microscopic mechanism of gravity emergingfrom quantum information.
Tensor networks implementing quantum error correcting codes have recently been used to construct toy models of holographic duality explicitly realizing some of the more puzzling features of the AdS/CFT correspondence. These models reproduce the Ryu-Takayanagi entropy formula for boundary intervals, and allow bulk operators to be mapped to the boundary in a redundant fashion. These exactly solvable, explicit models have provided valuable insight but nonetheless suffer from many deficiencies, some of which we attempt to address in this article. We propose a new class of tensor network models that subsume the earlier advances and, in addition, incorporate additional features of holographic duality, including: (1) a holographic interpretation of all boundary states, not just those in a “code” subspace, (2) a set of bulk states playing the role of “classical geometries” which reproduce the Ryu-Takayanagi formula for boundary intervals, (3) a bulk gauge symmetry analogous to diffeomorphism invariance in gravitational theories, (4) emergent bulk locality for sufficiently sparse excitations, and (5) the ability to describe geometry at sub-AdS resolutions or even flat space.
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.
This project aims to construct a rigorous, computable, and testable discrete holographic duality model within the CDGM framework. Based on the discretization of the {5,4} tiling of the two-dimensional hyperbolic plane, the bulk network is exactly mapped to a multi-scale entanglement renormalization ansatz (MERA) tensor network on the boundary, with explicit constructions of isometry tensors and disentanglers. Anticipated results include: the entanglement entropy of a boundary region satisfies the discrete RT formula with a relative error less than 0.5% in the thermodynamic limit; the boundary two-point correlation function exhibits power-law decay with a scaling exponent consistent with a free scalar conformal field theory. This preregistration establishes a timestamped record of the research plan; full derivations and numerical details will be presented in subsequent publications.
The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand these bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1 + 1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices, obtaining highly accurate critical data. Within our framework, we also produce translation-invariant critical states by an efficiently contractible tensor network with the geometry of the multiscale entanglement renormalization ansatz. Furthermore, we establish a link between holographic quantum error-correcting codes and tensor networks. This work is expected to stimulate a more comprehensive study of tensor network models capturing bulk-boundary correspondences.
Tensor networks have previously been used to construct holographic quantum error correcting codes (HQECC), which are toy models of the AdS/CFT correspondence. HQECC have been able to exhibit many of the interesting features of the duality, such as complementary recovery and redundant encoding. However, in previous HQECC the boundary Hamiltonian which results from mapping a local bulk Hamiltonian to the boundary is a non-local Hamiltonian, with global terms that act over the entire boundary. In this work, we combine HQECC with recently developed Hamiltonian simulation theory to construct a bulk-boundary mapping in which local bulk Hamiltonians map to local boundary Hamiltonians. This allows us to construct a toy model of a complete holographic duality between models, not just states and observables. This mapping of local Hamiltonians extends the toy models to encompass the relationship between bulk and boundary energy scales, and in particular the mapping between bulk and boundary time dynamics. We discuss what insight this gives into toy models of black hole formation.