Connectivity for quantum graphs

TL;DR

This paper introduces a definition of connectedness for quantum graphs, extending classical concepts, and proves a quantum analogue of a classical tree-packing theorem, with implications for quantum information theory.

Contribution

It generalizes classical graph connectedness to quantum graphs and establishes a quantum version of a tree-packing theorem, including related notions like $k$-connectedness.

Findings

Defined quantum graph connectedness

Proved a quantum tree-packing theorem

Extended orthogonal representation concepts

Abstract

In quantum information theory there is a construction for quantum channels, appropriately called a quantum graph, that generalizes the confusability graph construction for classical channels in classical information theory. In this paper, we provide a definition of connectedness for quantum graphs that generalizes the classical definition. This is used to prove a quantum version of a particular case of the classical tree-packing theorem from graph theory. Generalizations for the related notions of $k$-connectedness and of orthogonal representation are also proposed for quantum graphs, and it is shown that orthogonal representations have the same implications for connectedness as they do in the classical case.