Quantum relations in the general setting: composition and adjacency operators

TL;DR

This paper explores the concept of quantum relations as bimodules over von Neumann algebras, extending classical relations, and develops tools to analyze their properties, including adjacency operators and connections to completely positive maps.

Contribution

It introduces a framework for quantum relations as bimodules, extends the concept to relations between two sets, and links these to adjacency operators and CP maps, generalizing prior work.

Findings

Quantum relations generalize classical relations using bimodules.

A functor from CP maps to quantum relations is established.

Adjacency operators are explicitly computed for *-homomorphisms.

Abstract

Quantum relations in the sense of Weaver are $M'$-bimodules, for a von Neumann algebra $M$, these generalising actual relations on a set $X$ when $M=\ell^\infty(X)$. Similarly, relations between two sets can be generalised as bimodules over the commutants of two algebras. We make an explicit study of this idea, developing some tools to check that constructions are well-defined. Motivation comes from Kornell's concept of a Quantum Set (for algebras which are sums of matrix algebras), and we find that $*$-homomorphisms correspond to certain quantum relations, extending unpublished work of Kornell. We find a functor from completely positive maps to quantum relations, related to the idea of taking a noisy communication channel and reducing it to its underlying ``relation''. As with Quantum Graphs, at least in finite-dimensions, quantum relations correspond to ``adjacency operators'', certain CP maps depending on a choices of faithful functional on the algebras. We develop some tools to deal with the non-Schur-idempotent case, and show links with our functor from CP maps, and work of Verdon. We explicitly compute the adjacency operator of a $*$-homomorphism.