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

TL;DR

This paper demonstrates that the additivity of minimum output p-Renyi entropy fails for small p, providing explicit counterexamples and numerical evidence suggesting non-additivity for all p<0.11, extending known results beyond p>1.

Contribution

It provides the first explicit construction of channels showing non-additivity of minimum output p-Renyi entropy at p close to 0, expanding understanding of quantum entropy additivity.

Findings

Counterexamples for p=0 and small p demonstrating non-additivity.

Explicit construction of channels with non-multiplicative minimum output rank.

Numerical evidence indicating non-additivity for all p<0.11.

Abstract

Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Renyi 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-Renyi entropy is non-additive for all p < 0.11. We conjecture however that violations of additivity exist for all p<1.