🤖 AI Summary
This work addresses the limitation of conventional confusability graphs in fully characterizing the indistinguishability of input states in quantum channels. To overcome this, the authors propose a novel “quantum confusability multigraph” model that embeds channel output information into a graph structure, where multiple edges between vertices recover classical confusability relations. Building upon Weaver’s theory of quantum relations, they establish a rigorous mathematical framework for this model. The study establishes, for the first time, a one-to-one correspondence between quantum confusability multigraphs and quantum channels, and provides necessary and sufficient conditions for the existence of such multigraphs. This result fully characterizes the structural properties of confusability multigraphs realizable by quantum channels, thereby extending the applicability of quantum graph theory to problems of information distinguishability.
📝 Abstract
We introduce a new approach to confusability in a quantum channel, namely quantum confusability multigraph, which incorporates the output information into the graphical structure. By``counting" the edges between two vertices of this confusability multigraph, one recovers the traditional confusability ``single-edged" graph of the channel. With this physical motivation, we therefore develop a theory of quantum multigraphs from Weaver's quantum relations point of view and explore its quantum graph theoretic properties. Finally, we provide a necessary and sufficient condition characterizing those quantum multigraphs that arise as quantum confusability multigraphs.