A Max-Flow approach to Random Tensor Networks

TL;DR

This paper analyzes the entanglement entropy of random tensor networks using free probability theory, providing explicit formulas for eigenvalue distributions and insights into quantum entanglement in complex network models.

Contribution

It introduces a new model for RTNs, derives the eigenvalue distribution in the large dimension limit, and connects entanglement properties to a maximum flow problem on the network graph.

Findings

Eigenvalue distribution derived for large local dimensions.

Explicit formulas for series-parallel graphs.

Insights into finite-size corrections to entanglement entropy.

Abstract

We study the entanglement entropy of a random tensor network (RTN) using tools from free probability theory. Random tensor networks are simple toy models that help the understanding of the entanglement behavior of a boundary region in the ADS/CFT context. One can think of random tensor networks are specific probabilistic models for tensors having some particular geometry dictated by a graph (or network) structure. We first introduce our model of RTN, obtained by contracting maximally entangled states (corresponding to the edges of the graph) on the tensor product of Gaussian tensors (corresponding to the vertices of the graph). We study the entanglement spectrum of the resulting random spectrum along a given bipartition of the local Hilbert spaces. We provide the limiting eigenvalue distribution of the reduced density operator of the RTN state, in the limit of large local dimension. The limit value is described via a maximum flow optimization problem in a new graph corresponding to the geometry of the RTN and the given bipartition. In the case of series-parallel graphs, we provide an explicit formula for the limiting eigenvalue distribution using classical and free multiplicative convolutions. We discuss the physical implications of our results, allowing us to go beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN.