Quantum relations

TL;DR

This paper introduces the concept of quantum relations on von Neumann algebras, establishing their properties, correspondence with classical relations, and generalizations of classical structures like graphs and orders within a quantum framework.

Contribution

It defines quantum relations in a representation-independent way, links them to classical relations, and extends classical structures to the quantum setting with intrinsic characterizations.

Findings

Quantum relations correspond to subsets of X^2 in abelian cases.

Quantum relations can be characterized by projections in M f B(l^2).

Generalization of classical structures like graphs and orders to quantum relations.

Abstract

We define a "quantum relation" on a von Neumann algebra M \subset B(H) to be a weak* closed operator bimodule over its commutant M'. Although this definition is framed in terms of a particular representation of M, it is effectively representation independent. Quantum relations on l^\infty(X) exactly correspond to subsets of X^2, i.e., relations on X. There is also a good definition of a "measurable relation" on a measure space, to which quantum relations partially reduce in the general abelian case. By analogy with the classical setting, we can identify structures such as quantum equivalence relations, quantum partial orders, and quantum graphs, and we can generalize Arveson's fundamental work on weak* closed operator algebras containing a masa to these cases. We are also able to intrinsically characterize the quantum relations on M in terms of families of projections in M \otimes B(l^2).