Canonical partial ordering from min-cuts and quantum entanglement in random tensor networks

TL;DR

This paper extends the max-flow min-cut theorem to a relation among partial orders in networks, applying it to random tensor networks to analyze entanglement entropy corrections via order morphisms and connections to free probability.

Contribution

It introduces a new partial order based on min-cut structures and relates entanglement entropy corrections to order morphisms and free probability measures.

Findings

Finite correction to entanglement Rényi entropy from min-cut degeneracy

Number of order morphisms linked to moments of a graph-dependent measure

Connections to free Bessel law in free probability theory

Abstract

The \emph{max-flow min-cut theorem} has been recently used in the theory of random tensor networks in quantum information theory, where it is helpful for computing the behavior of important physical quantities, such as the entanglement entropy. In this paper, we extend the max-flow min-cut theorem to a relation among different \emph{partial orders} on the set of vertices of a network and introduce a new partial order for the vertices based on the \emph{min-cut structure} of the network. We apply the extended max-flow min-cut theorem to random tensor networks and find that the \emph{finite correction} to the entanglement R\'enyi entropy arising from the degeneracy of the min-cuts is given by the number of \emph{order morphisms} from the min-cut partial order to the partial order induced by non-crossing partitions on the symmetric group. Moreover, we show that the number of order morphisms corresponds to moments of a graph-dependent measure which generalizes the free Bessel law in some special cases in free probability theory.