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Graph restricted tensors: building blocks for holographic networks

Rafaĺ Bistroń, Márton Mestyán, Balázs Pozsgay, Karol Życzkowski

TL;DR

This work introduces graph-constrained tensors, a graph-theoretic framework to encode maximal entanglement constraints on selected bipartitions, unifying 1-uniform, dual-unitary, and AME-type states and enabling solvable holographic tensor networks. The authors develop two concrete solvable families—planar pentagonal and planar hexagonal tensors—and derive exact analytic solutions, including AME(5,2) and Type I hexagonal families, with explicit coefficient parameterizations. They then show how these tensors yield nontrivial, holography-inspired correlation functions on AdS/CFT-like disk tilings, deriving scaling dimensions Δ from reduced-path eigenvalues and demonstrating power-law decays along geodesic paths. The results provide non-stabilizer building blocks for holographic lattice models, offer analytic tools for computing correlation functions, and open avenues toward broader hypergraph/constrained tensor constructions and tighter bounds on scaling exponents in holographic codes.

Abstract

We analyze few-body quantum states with particular correlation properties imposed by the requirement of maximal bipartite entanglement for selected partitions of the system into two complementary parts. A novel framework to treat this problem by encoding these constraints in a graph is advocated; the resulting objects are called ``graph-restricted tensors''. This framework encompasses several examples previously treated in the literature, such as 1-uniform multipartite states, quantum states related to dual unitary operators and absolutely maximally entangled states (AME) corresponding to 2-unitary matrices. Original examples of presented graph-restricted tensors are motivated by tensor network models for the holographic principle. In concrete cases we find exact analytic solutions, demonstrating thereby that there exists a vast landscape of non-stabilizer tensors useful for the lattice models of holography.

Graph restricted tensors: building blocks for holographic networks

TL;DR

This work introduces graph-constrained tensors, a graph-theoretic framework to encode maximal entanglement constraints on selected bipartitions, unifying 1-uniform, dual-unitary, and AME-type states and enabling solvable holographic tensor networks. The authors develop two concrete solvable families—planar pentagonal and planar hexagonal tensors—and derive exact analytic solutions, including AME(5,2) and Type I hexagonal families, with explicit coefficient parameterizations. They then show how these tensors yield nontrivial, holography-inspired correlation functions on AdS/CFT-like disk tilings, deriving scaling dimensions Δ from reduced-path eigenvalues and demonstrating power-law decays along geodesic paths. The results provide non-stabilizer building blocks for holographic lattice models, offer analytic tools for computing correlation functions, and open avenues toward broader hypergraph/constrained tensor constructions and tighter bounds on scaling exponents in holographic codes.

Abstract

We analyze few-body quantum states with particular correlation properties imposed by the requirement of maximal bipartite entanglement for selected partitions of the system into two complementary parts. A novel framework to treat this problem by encoding these constraints in a graph is advocated; the resulting objects are called ``graph-restricted tensors''. This framework encompasses several examples previously treated in the literature, such as 1-uniform multipartite states, quantum states related to dual unitary operators and absolutely maximally entangled states (AME) corresponding to 2-unitary matrices. Original examples of presented graph-restricted tensors are motivated by tensor network models for the holographic principle. In concrete cases we find exact analytic solutions, demonstrating thereby that there exists a vast landscape of non-stabilizer tensors useful for the lattice models of holography.
Paper Structure (16 sections, 7 theorems, 44 equations, 10 figures, 1 table)

This paper contains 16 sections, 7 theorems, 44 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Graph-constrained tensors
  3. Planar pentagonal tensor
  4. Planar hexagonal tensors
  5. Necessary properties of the tensors
  6. Solutions to the isometry condition
  7. Type I. solutions
  8. Application in the AdS/CFT correspondence
  9. Results
  10. Concluding remarks
  11. Hypergraph-constrained tensors
  12. Solutions
  13. Type II
  14. Type III
  15. Isolated points
  16. ...and 1 more sections

Key Result

Proposition 1

Let $T^{(1)}$ and $T^{(2)}$ be two tensors constrained by graphs $G^{(1)}$, $G^{(2)}$. If $T^{(1)}$ and $T^{(2)}$ are contracted on some indices corresponding to one clique in each graph, $C^{(1)}, C^{(2)}$, then the resulting tensor is constrained by graph $G$ which is a disjoint union of $G^{(1)}$

Figures (10)

  • Figure 1: Pentagonal graph encoding constraints of planar 2-uniform tensor or order $5$planar_k_uniform_states.
  • Figure 2: A graph with 7 vertices encoding the constraints for the tensors \ref{['eq:tensor']}. The central node corresponds to the index $s_0$ from \ref{['eq:tensor']} while other nodes to the remaining indices.
  • Figure 3: Rényi-2 entropies \ref{['eq:renyi']} of the reduced density matrices $\rho_{013}$ and $\rho_{014}$ corresponding to $7$-index tensors $T_{s_1,s_2,s_3,s_4,s_5,s_6}^{s_0}$ that satisfy the isometry condition \ref{['eq:isom']} as well as rotational, spin-flip and spatial reflection invariance. Every point denotes a numerical solution to the imposed constraints. The colors blue, red and orange corresponds to tensors belonging to first, second and third type receptively, except of two isolated points P1 and P2. Inset: magnification of the same plot around the isolated point P1.
  • Figure 4: Contraction of bulk indices between planar hexagonal tensor (violet) and its conjugate (green), which appears while calculating correlation functions for bulk subsystems unoccupied by bulk operator \ref{['eq:gen_correlations']}
  • Figure 5: Contraction of the bulk and four pairs of boundary indices between planar hexagonal tensor (violet) and its conjugate (green), which happens for nodes on the path between distinct boundary indices while calculating two or three point correlation functions \ref{['eq:red_node']}.
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Lemma 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • ...and 3 more