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Random tensor networks with nontrivial links

Newton Cheng, Cécilia Lancien, Geoff Penington, Michael Walter, Freek Witteveen

TL;DR

This work extends random tensor network models by allowing nontrivial link spectra and develops a rigorous, two-regime framework for their entanglement structure. In the bounded-spectral-variation regime, Rényi-entropies analyzed via replicas and free probability yield a clean limiting spectrum: a free product of edge-cut spectra with a Marchenko-Pastur component, with explicit forms for single- versus two-cut geometries; entanglement negativity is likewise characterized. In the unbounded-spectral-variation regime, the authors deploy one-shot entropy tools and decoupling/recovery theorems to bound the spectrum of boundary subsystems and to describe the competing-min-cut physics through smooth min-entropies, enabling a min-structure description of entanglement via a min- vs- min convolution. Together, these results sharpen the connection between RTN models and holographic gravity, elucidating how bulk entropy, minimal-surface competition, and one-shot information tasks shape entanglement spectra and negativity in toy quantum-gravity models.

Abstract

Random tensor networks are a powerful toy model for understanding the entanglement structure of holographic quantum gravity. However, unlike holographic quantum gravity, their entanglement spectra are flat. It has therefore been argued that a better model consists of random tensor networks with link states that are not maximally entangled, i.e., have nontrivial spectra. In this work, we initiate a systematic study of the entanglement properties of these networks. We employ tools from free probability, random matrix theory, and one-shot quantum information theory to study random tensor networks with bounded and unbounded variation in link spectra, and in cases where a subsystem has one or multiple minimal cuts. If the link states have bounded spectral variation, the limiting entanglement spectrum of a subsystem with two minimal cuts can be expressed as a free product of the entanglement spectra of each cut, along with a Marchenko-Pastur distribution. For a class of states with unbounded spectral variation, analogous to semiclassical states in quantum gravity, we relate the limiting entanglement spectrum of a subsystem with two minimal cuts to the distribution of the minimal entanglement across the two cuts. In doing so, we draw connections to previous work on split transfer protocols, entanglement negativity in random tensor networks, and Euclidean path integrals in quantum gravity.

Random tensor networks with nontrivial links

TL;DR

This work extends random tensor network models by allowing nontrivial link spectra and develops a rigorous, two-regime framework for their entanglement structure. In the bounded-spectral-variation regime, Rényi-entropies analyzed via replicas and free probability yield a clean limiting spectrum: a free product of edge-cut spectra with a Marchenko-Pastur component, with explicit forms for single- versus two-cut geometries; entanglement negativity is likewise characterized. In the unbounded-spectral-variation regime, the authors deploy one-shot entropy tools and decoupling/recovery theorems to bound the spectrum of boundary subsystems and to describe the competing-min-cut physics through smooth min-entropies, enabling a min-structure description of entanglement via a min- vs- min convolution. Together, these results sharpen the connection between RTN models and holographic gravity, elucidating how bulk entropy, minimal-surface competition, and one-shot information tasks shape entanglement spectra and negativity in toy quantum-gravity models.

Abstract

Random tensor networks are a powerful toy model for understanding the entanglement structure of holographic quantum gravity. However, unlike holographic quantum gravity, their entanglement spectra are flat. It has therefore been argued that a better model consists of random tensor networks with link states that are not maximally entangled, i.e., have nontrivial spectra. In this work, we initiate a systematic study of the entanglement properties of these networks. We employ tools from free probability, random matrix theory, and one-shot quantum information theory to study random tensor networks with bounded and unbounded variation in link spectra, and in cases where a subsystem has one or multiple minimal cuts. If the link states have bounded spectral variation, the limiting entanglement spectrum of a subsystem with two minimal cuts can be expressed as a free product of the entanglement spectra of each cut, along with a Marchenko-Pastur distribution. For a class of states with unbounded spectral variation, analogous to semiclassical states in quantum gravity, we relate the limiting entanglement spectrum of a subsystem with two minimal cuts to the distribution of the minimal entanglement across the two cuts. In doing so, we draw connections to previous work on split transfer protocols, entanglement negativity in random tensor networks, and Euclidean path integrals in quantum gravity.
Paper Structure (32 sections, 28 theorems, 481 equations, 8 figures)

This paper contains 32 sections, 28 theorems, 481 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Random tensor network states
  4. The replica trick for random tensor networks
  5. Maximally entangled link states and minimal cuts
  6. The replica trick for general background states
  7. Normalization of random tensor network states
  8. Link states with bounded spectral variation
  9. Random matrices, free probability and non-crossing partitions
  10. Random matrix theory and Wishart matrices
  11. Free probability
  12. Non-crossing partitions
  13. Entanglement spectrum of random tensor network states as a free product
  14. Nontrivial link states and entanglement negativity
  15. Link states with unbounded spectral variation
  16. ...and 17 more sections

Key Result

Theorem 1

Consider a family of link states in the bounded spectral variation limit. If there is a unique minimal cut $\gamma_A$ for a boundary subsystem $A$, then $\mu_A^{(D)}$ converges weakly, in probability, to $\mu_{\gamma_A}$, while if there are exactly two non-intersecting minimal cuts $\gamma_{A,1}$ an

Figures (8)

  • Figure 1: The basic structure of a random tensor network.
  • Figure 2: Tensor networks with one and two minimal cuts.
  • Figure 3: The structure of a (purified) random tensor network with a general background state.
  • Figure 4: Tensor networks with one and two minimal cuts. The relevant ground state configuration domains are denoted by $\Gamma_A$.
  • Figure 5: Illustration of the proof of \ref{['thm:measure convergence surface transition']}.
  • ...and 3 more figures

Theorems & Definitions (54)

  • Theorem : Informal
  • Theorem : Informal
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 44 more