Counterexamples to additivity of minimum output p-rényi entropy for p close to 0

Abstract

Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Rényi entropies of channels are not generally additive for p > 1, we demonstrate here by a careful random selection argument that also at p = 0, and consequently for sufficiently small p, there exist counterexamples. An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Rényi entropy is non-additive for all p ≃ 0.11. We conjecture however that violations of additivity exist for all p < 1. © 2008 Springer-Verlag.