Quantum graphs in infinite-dimensions: Hilbert--Schmidts and Hilbert modules

TL;DR

This paper introduces two new frameworks for quantum graphs in infinite-dimensional von Neumann algebras, connecting Hilbert--Schmidt operators and quantum adjacency operators, with detailed examples and links to finite-dimensional cases.

Contribution

It develops novel approaches to quantum graphs using Hilbert--Schmidt operators and quantum adjacency operators within von Neumann algebras, establishing bijections and symmetries.

Findings

Bijection between Hilbert--Schmidt quantum relations and projections in tensor products.

Construction of quantum adjacency operators with Kraus representations.

Analysis of symmetries when certain integrability conditions are met.

Abstract

We develop two approaches to Quantum (or Non-commutative) Graphs based on arbitrary von Neumann algebras $M\subseteq\mathcal B(H)$: one looking at operator bimodules of Hilbert--Schmidt (instead of bounded) operators, and the second looking at Quantum Adjacency Operators. Hilbert--Schmidt Quantum Graphs relate to Weaver's picture of Quantum Graphs in a complex way: by defining certain hull operations, we find a bijection between certain subsets of both objects. Given a nfs weight $\varphi$ on $M$ the operator-valued weight $\varphi^{-1}$ can be defined, as considered by Wasilewski for direct sums of matrix algebras. We show how to build a natural self-dual Hilbert $C^*$-module from this, which mediates a bijection between HS Quantum Relations and projections $e\in M\bar\otimes M^{\text{op}}$. When $e$ is integrable for the slice-map $\operatorname{id}\otimes\varphi^{\text{op}}$ there is a related normal CP map $A\colon M\to M$: this is a Quantum Adjacency Operator, which has a Kraus operator representation built from the HS Quantum Relation. When $e$ and its tensor swap map are both integrable, we find certain symmetries of $A$. We illustrate our theory by a careful consideration of certain examples, including detailed links with the finite-dimensional setting.