The maximal p-norm multiplicativity conjecture is false

TL;DR

This paper disproves the maximal p-norm multiplicativity conjecture for quantum channels when 1 < p < 2, showing that the minimal output Renyi entropy is not additive, which challenges a key assumption in quantum information theory.

Contribution

It provides explicit counterexamples demonstrating the failure of maximal p-norm multiplicativity for all 1 < p < 2 in quantum channels.

Findings

Existence of quantum channels with non-multiplicative maximal p-norms for 1 < p < 2

Large violations of multiplicativity are demonstrated

Counterexamples show additivity cannot be proved via maximal p-norm multiplicativity

Abstract

For all 1 < p < 2, we demonstrate the existence of quantum channels with non-multiplicative maximal p-norms. Equivalently, the minimum output Renyi entropy of order p of a quantum channel is not additive for all 1 < p < 2. The violations found are large. As p approaches 1, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p=1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of maximal p-norm multiplicativity.