Coherence measures induced by norm functions

TL;DR

This paper investigates which matrix norms can serve as proper measures of quantum coherence, identifying specific conditions under which certain norms induce valid coherence measures and providing a unified framework for their analysis.

Contribution

It characterizes the classes of matrix norms that induce quantum coherence measures, especially highlighting the role of the ll_{q,p}-norms and establishing new, computable coherence measures.

Findings

Unitary similarity invariant norms do not induce coherence measures.

The ll_{q,p}-norm induces a coherence measure only if q=1 and 1 q p q 2.

The results unify and extend previous understanding of norm-induced coherence measures.

Abstract

Which matrix norms induce proper measures for quantifying quantum coherence? We study this problem for two important classes of norms and show that (i) coherence measures cannot be induced by any unitary similarity invariant norm, and (ii) the $\ell_{q,p}$-norm induces a coherence measure if and only if $q=1$ and $1 \leq p \leq 2$, thus giving a new class of coherence measures with simple closed forms that are easy to compute. These results extend and unify previously known facts about norm-induced coherence measures, and lead to a broader framework for understanding what functionals can be coherence measures.