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Non-Unitary Tensor Networks from Holographic Delamination

Schwabe Henning

Published Aug 8, 2026• 0 citations
Publisher

Abstract

Standard Ryu-Takayanagi (RT) bounds and Multiscale Entanglement Renormalization Ansatz (MERA) networks assume a costless, unitary mapping between continuous bulk geometry and discrete boundary information. In this Letter, we show that projecting continuous bulk fields into discrete boundary states introduces a mandatory topological phase-space friction governed by the Euler-Mascheroni constant $\gamma$. Normalizing this harmonic discretization residue across the geometric capacity of a two-dimensional boundary yields the exact holographic delamination invariant, $\Lambda_2 = \exp(\gamma/\pi^2) \approx 1.060228$. By demonstrating that discrete MERA layer scaling is the exact mathematical dual to the logarithmic AdS radial measure, we prove this additive topological penalty exponentiates into a strict multiplicative scaling of thermodynamic work. We construct non-unitary dissipative MERA disentangler operators ($\hat{\Upsilon}$) that enforce a factorization bound on accessible boundary entropy $S_{\text{acc}}$. Acting as a covariant topological scalar, this invariant preserves the Equivalence Principle while establishing a fundamental thermodynamic and semantic limit on holographic state projection.

DOI
10.5281/zenodo.21851187

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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.
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.
A central insight of holography is that spacetime geometry emerges from the quantum entanglement of a boundary theory. This work proposes and validates a precise, local manifestation of this principle: in a 1+1D boundary conformal field theory (CFT) dual to an asymptotically bulk, the Ricci scalar curvature is proportional to the negative second derivative of the entanglement entropy density. This hypothesis, motivated by the Ryu-Takayanagi formula and the geodesic deviation equation, posits that local fluctuations in boundary entanglement directly encode local bulk curvature. The relation is tested numerically using a tensor network toy model inspired by the Multi-scale Entanglement Renormalization Ansatz (MERA). Localized suppressions of boundary entanglement, analogous to removing disentanglers in MERA, are shown to induce localized positive curvature spikes in the emergent bulk geometry. The results provide direct computational evidence for the entanglement-curvature correspondence, reinforcing the paradigm that spacetime geometry is built from quantum entanglement.
The multi-scale entanglement renormalization ansatz (MERA) is a tensor network that can efficiently parameterize critical ground states on a 1D lattice, and also suggestively implement some aspects of the holographic correspondence of string theory on a lattice. Extending our recent work [S. Singh, Physical Review D 97, 026012 (2018); S. Singh, N. A. McMahon, and G. K. Brennen, Phys. Rev. D 97, 026013 (2018)], we show how the MERA representation of a 1D critical ground state---which has long range entanglement---can be viewed as a strange correlator: the overlap of a 2D state with short range entanglement and a 2D product state. Strange correlators were recently introduced to map 2D symmetry protected or topologically ordered quantum states to critical systems in one lower dimension. The 2D quantum state dual to the input 1D critical state is obtained by lifting the MERA, a procedure which introduces bulk quantum degrees of freedom by inserting intertwiner tensors on each bond of the MERA tensor network. We show how this dual 2D bulk state exhibits several features of holography, for example, appearance of horizon-like holographic screens and bulk gauging of global on-site symmetries at the boundary. We also derive a quantum corrected Ryu-Takayanagi formula relating boundary entanglement entropy to bulk geodesic lengths---as measured by bulk entropy---and numerically test it for ground states of a set of unitary minimal model CFTs, as realized by 1D anyonic Heisenberg models.
We propose a discrete holographic framework for analyzing the thermodynamic origins of classical chaos in the three-body problem. By representing interacting point masses as localized Bekenstein horizons on a unitary MERA tensor network, we propose that classical Lyapunov divergence is an emergent kinematic projection of epistemic coarse-graining. At the discrete zero-dissipation limit, Node Symmetry Invariance (NSI) dictates strictly unitary evolution and exact Poincaré recurrence. The mapping of this network into a continuous macroscopic phase space requires the continuous truncation of Schmidt coefficients due to the Coffman-Kundu-Wootters (CKW) monogamy bound of tripartite entanglement. By invoking Pesin's Identity, we mathematically map the observed continuous Lyapunov exponent directly to the thermodynamic integration of this Landauer boundary erasure over time. Finally, we demonstrate that the fractal geometries of macroscopic strange attractors correspond to generalized Ryu-Takayanagi minimal surfaces, indicating that chaotic orbits are thermodynamic geodesics minimizing the continuous projection of discrete quantum noise.
This paper encodes the subsystem entanglement structure of boundary conformal field theory into a weighted graph and defines the spectral entanglemententropy via the spectral theory of the Dirichlet-to-Neumann map. Within the constraint network framework, a continuum limit and a mechanism for dimensionalemergence are introduced to establish a rigorous transition from discrete spectrato continuous holographic geometry. It is proved that, in the rigid limit, the spectral entanglement entropy strictly decomposes into the sum of a geometric areaterm and a topological correction term: the area term arises from a strictly positive lower bound on the spectral gap of the Dirichlet-to-Neumann map and ispinned by the curvature of the extremal surface; the topological correction termoriginates from the nullity of the coarse-grained rigid kernel, whose value is proportional to the absolute value of the Euler characteristic. It is further proved thatthe stability of a quantum extremal surface is equivalent to the algebraic connectivity of the quotient graph being strictly positive, a condition characterized bythe Fiedler value. This yields three testable predictions, encompassing a universal difference in the topological correction, multi-surface transitions for non-convexregions, and the finite-temperature spectral-gap scaling, and reveals a deep isomorphism with holographic quantum error correction. This framework unifies thegeometric origin, quantum corrections, and the recoverability of error correction ofthe Ryu-Takayanagi formula within a single spectral flow structure.