Noncommutative extensions of parameters in the asymptotic spectrum of graphs

TL;DR

This paper investigates the extension of classical graph parameters related to zero-error capacity into the quantum domain, revealing a dichotomy in the existence and nature of such extensions based on their properties.

Contribution

It establishes a characterization of all possible extensions of classical parameters to noncommutative graphs, identifying conditions under which these extensions exist and their properties.

Findings

Extensions are either unbounded or nonexistent.

Admissible exponents form an interval or are empty.

Certain parameters like Lovász number have maximal extensions.

Abstract

The zero-error capacity of a classical channel is a parameter of its confusability graph, and is equal to the minimum of the values of graph parameters that are additive under the disjoint union, multiplicative under the strong product, monotone under homomorphisms between the complements, and normalized. We show that any such function either has uncountably many extensions to noncommutative graphs with similar properties, or no such extensions at all. More precisely, we find that every extension has an exponent that characterizes its values on the confusability graphs of identity quantum channels, and the set of admissible exponents is either an unbounded subinterval of $[1,\infty)$ or empty. In particular, the set of admissible exponents for the Lov\'asz number, the projective rank, and the fractional Haemers bound over the complex numbers are maximal, while the fractional clique cover number does not have any extensions.