Jennifer Zhu, University of Waterloo
Limits of Quantum Graphs
Quantum graphs were originally introduced as confusability graphs of quantum channels by Duan, Severini, and Winter. Weaver generalized a quantum graph to any weak-* closed operator system $\mathcal V \subseteq B(\mathcal H)$ that is bimodule over the commutant of some von Neumann algebra $\mathcal M \subseteq B(\mathcal H)$. To date, there seem to be two notions of quantum graph morphism. Weaver introduced and Daws extended a notion of CP morphism of quantum graphs. Musto, Reutter, and Verdon have also defined classical morphisms of quantum graphs in finite dimensions which agrees with CP morphisms in finite dimensions. Notably, however, these morphisms are not UCP maps between operator systems of the respective quantum graphs.
Using a characterization of quantum relations as left ideals in the extended Haagerup tensor product, we will obtain a notion of quantum graph morphism (and hence limit) using the categories of von Neumann algebras and operator spaces. Time permitting, we will show that this limit recovers profinite classical graphs.
QNC 1507