Sufficient conditions for additivity of the zero-error classical capacity of quantum channels

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.