Extending additivity from symmetric to asymmetric channels

We prove a lemma which allows one to extend results about the additivity of the minimal output entropy from highly symmetric channels to a much larger class. A similar result holds for the maximal output p-norm. Examples are given showing its use in a variety of situations. In particular, we prove the additivity and the multiplicativity for the shifted depolarising channel. A natural and important class of measures of the noisiness of a quantum channel are based on optimal output purity, i.e., how close can the output be to a pure state as measured by the minimal ouput entropy (MOE) or the maximal output p-norm. It is then natural to ask if a tensor product of two channels can ever be less noisy in the sense that some entangled input can have its output closer to a pure state than the product of the optimal inputs of the two channels. This leads to the conjectures of the additivity of the MOE and the multiplicativity of the maximal output p-norm. Although the additivity of the MOE has been shown [1] to be equivalent globally to several other fundamental conjectures in quantum information theory; the additivity of the Holevo capacity, the additivity of the entanglement of formation and the strong superadditivity of the entanglement of formation, we consider only the output purity.