In tensor networks, a geometric operation of pushing a bond cut surface toward a minimal surface corresponds to entanglement distillation. Cutting bonds defines a reduced transition matrix on the bond cut surface and the associated quantum state naturally emerges from it. We justify this picture quantitatively by evaluating the trace distance between the maximally entangled states and the states on bond cut surfaces in the multiscale entanglement renormalization ansatz (MERA) and matrix product states in a canonical form. Our numerical result for the random MERA is in a reasonable agreement with our proposal. The result sheds new light on a deeper understanding of the Ryu-Takayanagi formula for entanglement entropy in holography and the emergence of geometry from the entanglement structure.
In tensor networks, a geometric operation of pushing a bond cut surface toward a minimal surface corresponds to entanglement distillation. Cutting bonds defines a reduced transition matrix on the bond cut surface and the associated quantum state naturally emerges from it. We justify this picture quantitatively by evaluating the trace distance between the maximally entangled states and the states on bond cut surfaces in the multiscale entanglement renormalization ansatz (MERA) and matrix product states in a canonical form. Our numerical result for the random MERA is in a reasonable agreement with our proposal. The result sheds new light on a deeper understanding of the Ryu-Takayanagi formula for entanglement entropy in holography and the emergence of geometry from the entanglement structure.
Abstract A confining, non-Abelian gauge connection Uij ∈ SU(N) defined on the links of a dynamical graph has been proposed as the microscopic origin of classical spacetime geometry: the metric is reconstructed from the coset (soldering-form) sector of an enlarged connection, and black-hole and boundary-cut entropy are obtained as the logarithm of a combinatorial microstate count, matched, via the Freedman–Headrick theorem, to the bit-thread (max-flow/min-cut) formulation of holographic entanglement entropy. That matching is exact as a statement about counting, but it is not, by itself, a statement about the entanglement entropy of any specific quantum state: nothing in the combinatorial construction distinguishes a classical ensemble of flux occupation numbers from a genuine quantum wavefunction, so its connection to the entanglement/tensornetwork program of emergent geometry (Van Raamsdonk, Swingle, Ryu–Takayanagi, and the random-tensor-network derivation of Hayden, Qi, Kim, and Preskill) remains an analogy rather than a derivation. This paper closes that gap. We first restate the flux-gauge connection ensemble in full – its partition function, its block-link renormalization-group coarse-graining with closed-form Migdal–Kadanoff coefficients, its emergent metric and Newton’s-constant identification, and its combinatorial boundary-cut entropy – so that the paper is self-contained. We then show, by an exact character-expansion/Peter–Weyl derivation standard in lattice gauge theory but not previously applied to this construction, that the SU(N) link ensemble is a tensor network: each flux-tube segment (link) carries a bond dimension equal to the dimension of the SU(N) representation it transports, and each flux-tube junction (node) carries a gauge-invariant intertwiner tensor, fixed by representation theory rather than assumed. Only a single, narrowly scoped statistical assumption – on the distribution of intertwiners within the multiplicity space that gauge invariance already guarantees – is needed to complete this exact, deterministic tensor network into a genuine random tensor network in the technical sense of Hayden–Qi–Kim–Preskill (HQKP). We further identify the renormalization-group layers of Section 2’s coarse-graining as the isometric maps of a multi-scale entanglement renormalization ansatz (MERA), giving the emergent radial holographic direction a literal tensor-network origin rather than a scaling-dimension analogy. We then investigate, analytically and how the entanglement entropy of such a network reproduces the Ryu–Takayanagi/min-cut law. An exactly solvable limit built from classical (GHZ-type) perfect tensors reproduces the combinatorial law exactly for single-cut (boundary-anchored) regions but is shown to be degenerate for interior, two-sided regions, exposing a genuine gap in the combinatorial entropy argument: classical flux-occupation counting does not, by itself, certify multi-region (entanglement-wedgelevel) quantum entanglement structure. We then construct a genuinely quantum (non-classical, Gaussian-random) tensor-network realization of the link ensemble and confirm numerically, on chains of up to sixteen sites, (i) a clean area-law plateau in which the entanglement entropy of an interior region becomes independent of its size once the region is large enough, in place of continued volume-law growth, and (ii) a linear scaling of the plateau entropy with the logarithm of the bond dimension, S(A) = κNcut(A) logD, fit with R2 = 0.996 and an O(1) efficiency coefficient κ ≈ 0.68 for this finite, non-isometric network. We interpret κ < 1 as a genuine, quantifiable finite-network correction relative to the idealized D → ∞ isometric-projector limit in which HQKP prove κ → 1, and we list the construction of that idealized limit, directly on the SU(N) link ensemble, as the key open numerical task. We close by proposing an explicit correspondence between the tensor-network minimal surface of this paper and the Cartan soldering-form metric of the gauge-connection construction, framed as a falsifiable consistency conjecture rather than an established result.
We propose a holographic dual of the relative entropy of entanglement for a boundary tripartition A:B:C. Our proposal is that, at the leading order, \begin{align} E_R(ρ_{AB}) = \frac{1}{4G_N}\Big[\mathrm{Area}(Γ_{\min}) - \mathrm{Area}(γ_{AB})\Big] \notag \end{align} where Γmin is the minimal bulk tripartition surface and γAB is the minimal surface homologous to AB. For a general random tensor network, we prove the corresponding upper bound by constructing a separable state associated with Γmin and evaluating its relative entropy. We also establish the matching lower bound rigorously for networks of one, two, and three Haar random tensors by bounding the maximal tripartite product-state overlap. In appropriate geometries, the minimal tripartition surface can form a nontrivial tri-junction resembling the Mercedes logo. The proof proceeds by sequential optimization over candidate product states, which has a natural geometric interpretation as a local search for the minimal tripartition surface.
Pseudoentangled states are defined by their ability to hide their entanglement structure: they are indistinguishable from random states to any observer with polynomial resources, yet can have much less entanglement than random states. Existing constructions of pseudoentanglement based on phase and/or subset states are limited in the entanglement structures they can hide, e.g., the states may have low entanglement on a single cut, on all cuts at once, or on local cuts in one dimension. Here, we introduce new constructions of pseudoentangled states based on (pseudo)random tensor networks that afford much more flexibility in the achievable entanglement structures. We illustrate our construction with the simplest example of a matrix product state, realizable as a staircase circuit of pseudorandom unitary gates, which exhibits pseudo-area-law scaling of entanglement in one dimension. We then generalize our construction to arbitrary tensor network structures that admit an isometric realization. A notable application of this result is the construction of pseudoentangled "holographic" states whose entanglement entropy obeys a Ryu-Takayanagi "minimum-cut" formula, answering a question posed in Aaronson et al. [arXiv:2211.00747].
Extending the holographic principle to de Sitter (dS) spacetimes remains one of the most vital open frontiers in quantum gravity, where a microscopic, bottom-up tensor-network framework that relates boundary quantum data to emergent de Sitter spacetime is still lacking. In this work, we first show the emergence of de Sitter spacetime from boundary entanglement by formulating a non-unitary continuous multi-scale entanglement renormalization ansatz (cMERA) for a concrete non-Hermitian critical fermion chain. Within this emergent spacetime, we analyze the associated geodesics and show that they act as extremal Ryu-Takayanagi (RT) surfaces undergoing a smooth timelike-to-null transition. Remarkably, we demonstrate that this continuum trajectory suggests a distinct tensor-network architecture in which the bond-counting contribution naturally truncates at the discrete timelike-to-null transition toward the deep infrared. In the resulting architecture, the null ray along the horizon is represented by zero-cost links, since the associated cut severs no tensor legs. This network structure successfully reproduces the logarithmic scaling of non-unitary critical entanglement entropy, offering a bond-counting picture for the de Sitter RT formula. Our results provide the long-sought dS/(c)MERA correspondence at the level of both emergent spacetime and discrete holographic entanglement.
We study the evaluation of Rényi multi-entropies Sn(q) in Random Tensor Network (RTN) states in the large bond-dimension limit. For the case of Rényi index n=2 and arbitrary number of parties q, we prove that that multi-entropies are determined by minimal multiway cuts through the network. When the minimal multiway cut is degenerate, we characterize the full minimizer set via compatible families of minimal cuts and give a criterion for all minimizers to come from ordinary cut partitions. For n=2, this gives a natural generalization of the minimal cut description of bipartite entanglement to multipartite systems with arbitrarily many parties. For the case of integer n>2, we show that the minimal multiway cut conjecture is in general \emph{not true} by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. We discuss the implication for our results on the multipartite entanglement structures in RTN and holography.
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.