Quantum channels from association schemes

TL;DR

This paper explores quantum channels derived from association schemes, analyzing their properties through non-commutative graphs and demonstrating how entanglement can significantly enhance independence numbers in quantum information scenarios.

Contribution

It introduces a novel approach to constructing quantum channels from association schemes and provides bounds and formulas for independence numbers in this context.

Findings

Unitary entanglement-assisted independence number grows quadratically faster than independence number.

Provides bounds and closed formulas for independence numbers of non-commutative graphs.

Uses pseudocyclic association schemes as an illustrative example.

Abstract

We propose in this note the study of quantum channels from association schemes. This is done by interpreting the $(0,1)$-matrices of a scheme as the Kraus operators of a channel. Working in the framework of one-shot zero-error information theory, we give bounds and closed formulas for various independence numbers of the relative non-commutative (confusability) graphs, or, equivalently, graphical operator systems. We use pseudocyclic association schemes as an example. In this case, we show that the unitary entanglement-assisted independence number grows at least quadratically faster, with respect to matrix size, than the independence number. The latter parameter was introduced by Beigi and Shor as a generalization of the one-shot Shannon capacity, in analogy with the corresponding graph-theoretic notion.