Random tensor networks with nontrivial links

TL;DR

This paper explores the entanglement properties of random tensor networks with nontrivial link spectra, using advanced mathematical tools to understand their spectra and connections to quantum gravity models.

Contribution

It systematically studies entanglement in tensor networks with non-maximally entangled links, introducing new analytical results for their spectra using free probability and random matrix theory.

Findings

Limiting entanglement spectrum expressed as a free product for bounded spectral variation.

Relation between unbounded spectral variation and minimal entanglement distribution.

Connections established to quantum gravity concepts and previous entanglement studies.

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.