  We show that the minimum Renyi entropy output of a quantum channel is locally
additive for Renyi parameter alpha>1. While our work extends the results of
[10] (in which local additivity was proven for alpha=1), it is based on several
new techniques that incorporate the multiplicative nature of p-norms, in
contrast to the additivity property of the von-Neumann entropy. Our results
demonstrate that the counterexamples to the Renyi additivity conjectures
exhibit global effects of quantum channels. Interestingly, the approach
presented here can not be extended to Renyi entropies with parameter alpha<1.
