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].
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].
We construct a family of 2D-local constant-depth quantum circuits that output states whose entanglement entropy across a specified cut cannot be estimated in quantum polynomial time. As constant-depth quantum circuits can be learned from polynomially many quantum samples, our resulting pseudoentangled states are implicitly public-key and not pseudorandom. This separates pseudoentanglement from pseudorandomness in the shallow-circuit regime: the former is possible, while the latter is not. The construction is based on the quantum intractability of the Dense-Sparse Learning Parity with Noise problem introduced in [DJ25] and uses a bounded-fan-in, bounded-fan-out classical randomized encoding for linear maps x↦Mx, which could be of independent interest. As applications, we obtain quantum hardness for the problem of learning the entanglement structure (across a fixed cut) of the ground-state of 1D and 2D local Hamiltonians. The 1D Hamiltonian has an inverse polynomial gap, whereas the 2D one has a constant gap. This complements the result of [BZZ24] that showed only factoring-based hardness for the 1D case, though achieving a volume versus area entanglement difference.
Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement'', a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states -- first defined by Ji, Liu and Song -- which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to logn across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously.
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.
Holographic tensor networks serve as toy models for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, capturing many of its essential features in a concrete manner. However, existing holographic tensor network models remain far from a complete theory of quantum gravity. A key obstacle is their discrete structure, which only approximates the semi-classical geometry of gravity in a qualitative sense. In \cite{Lin:2024dho}, it was shown that a network of partial-entanglement-entropy (PEE) threads, which are bulk geodesics with a specific density distribution, generates a perfect tessellation of AdS space. Moreover, such PEE-network tessellations can be constructed for more highly symmetric geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop several holographic tensor network models: (1) the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs; (2) the HaPPY-like PEE tensor network constructed from perfect tensors; and (3) the random PEE tensor network. In all these models, we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly equals the area of that surface.
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.
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.