Logarithmic coherence: Operational interpretation of $\ell_1$-norm coherence

TL;DR

This paper establishes a fundamental operational interpretation of the $ ext{l}_1$-norm coherence measure, linking it to the distillable coherence and robustness of coherence, and clarifies their interrelations in quantum resource theory.

Contribution

It demonstrates that the $ ext{l}_1$-norm coherence measure bounds the distillable coherence up to a constant factor, connecting it to the robustness of coherence and providing tight bounds for specific states.

Findings

The $ ext{l}_1$-norm coherence bounds the distillable coherence.

Relationships between coherence measures are tight for pure and qubit states.

Constructs states with minimal distillable coherence for a given robustness.

Abstract

We show that the distillable coherence---which is equal to the relative entropy of coherence---is, up to a constant factor, always bounded by the $\ell_1$-norm measure of coherence (defined as the sum of absolute values of off diagonals). Thus the latter plays a similar role as logarithmic negativity plays in entanglement theory and this is the best operational interpretation from a resource-theoretic viewpoint. Consequently the two measures are intimately connected to another operational measure, the robustness of coherence. We find also relationships between these measures, which are tight for general states, and the tightest possible for pure and qubit states. For a given robustness, we construct a state having minimum distillable coherence.