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

## TL;DR

This paper establishes a unique axiomatic characterization of the $l_1$-norm of coherence, linking it to negativity in entanglement theory, and provides an operational interpretation involving entanglement generation through incoherent operations.

## Contribution

It introduces a novel axiomatic framework for the $l_1$-norm of coherence and connects it to negativity, offering an operational interpretation and explicit calculations for certain bipartite states.

## Key findings

- The $l_1$-norm of coherence is uniquely characterized by simple axioms.
- It is analogous to negativity in entanglement theory.
- All derived entangled states are distillable with explicit entanglement measures.

## Abstract

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. We also provide an operational interpretation of the $l_1$-norm of coherence as the maximum entanglement, measured by the negativity, produced by incoherent operations acting on the system and an incoherent ancilla. To achieve this goal, we clarify the relation between the $l_1$-norm of coherence and negativity for all bipartite states, which leads to an interesting generalization of maximally correlated states. Surprisingly, all entangled states thus obtained are distillable. Moreover, their entanglement cost and distillable entanglement can be computed explicitly for a qubit-qudit system.