Sufficient conditions for additivity of the zero-error classical capacity of quantum channels
Jeonghoon Park, Jeong San Kim
TL;DR
This paper addresses the additivity of the zero-error classical capacity of quantum channels by linking it to the multiplicativity of the independence number of noncommutative graphs. It develops sufficient conditions for independence-number multiplicativity, yielding additive behavior for both the one-shot capacity $\mathcal{C}_0^{(1)}$ and the asymptotic capacity $\mathcal{C}_0$, and extends the analysis to a block form of noncommutative graphs. The authors provide explicit quantum-channel examples, including noiseless and qubit cases, to illustrate the criteria and broadens the framework with block-noncommutative-graphs to establish multiplicativity in more structured settings.
Abstract
The one-shot zero-error classical capacity of a quantum channel is the amount of classical information that can be transmitted with zero probability of error by a single use. Then the one-shot zero-error classical capacity equals to the logarithmic value of the independence number of the noncommutative graph induced by the channel. Thus the additivity of the one-shot zero-error classical capacity of a quantum channel is equivalent to the multiplicativity of the independence number of the noncommutative graph. The independence number is not multiplicative in general, and it is not clearly understood when the multiplicativity occurs. In this work, we present sufficient conditions for multiplicativity of the independence number, and we give explicit examples of quantum channels. Furthermore, we consider a block form of noncommutative graphs, and provide conditions when the independence number is multiplicative.