Axiomatic and operational connections between $l_1$-norm of coherence and negativity

Huangjun Zhu, Masahito Hayashi, Lin Chen

Published 2017-04-10Version 1

Quantum coherence plays a central role in various research areas. The $l_1$-norm of coherence is one of the most important coherence measures that are easily computable, but it is not easy to find a simple interpretation. We show that the $l_1$-norm of coherence is uniquely characterized by a few simple axioms, which demonstrates in a precise sense that it is the analog of negativity in entanglement theory and sum negativity in the resource theory of magic state quantum computation. Furthermore, we provide an operational interpretation of $l_1$-norm of coherence as the maximum entanglement, measured by negativity, produced by (strictly) incoherent operations acting on our system and an incoherent ancilla. To achieve this goal, we clarify the relation between $l_1$-norm of coherence and negativity for all bipartite states, which leads to an interesting generalization of maximally correlated states.