Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1

TL;DR

This paper constructs counterexamples showing that the maximal p-norm multiplicativity conjecture fails for all p > 1, indicating non-additivity of minimum output Renyi entropy in quantum channels and challenging existing conjectures in quantum information theory.

Contribution

The paper provides the first explicit counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, demonstrating non-additivity of quantum channel entropies.

Findings

Counterexamples exist for all p > 1 showing non-multiplicativity.

Minimum output Renyi entropy is not additive for these channels.

Channels previously studied for encryption also serve as counterexamples for p > 2.

Abstract

For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p >1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, 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 any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.