# Coherence Fraction

## TL;DR

This paper introduces the coherence fraction as a new measure of quantum coherence, exploring its properties, relationships with existing measures, and behavior under quantum channels, with both theoretical analysis and numerical validation.

## Contribution

It generalizes entanglement fraction to define coherence fraction, establishing its properties, connections with $l_1$-norm coherence, and its dynamics under quantum channels.

## Key findings

- Coherence fraction quantifies proximity to maximally coherent states.
- Optimal coherence fraction relates inversely to decohering power of channels.
- Local coherence fractions are bounded by the global coherence fraction.

## Abstract

The concept of entanglement fraction is generalized to define coherence fraction of a quantum state. Precisely, it quantifies the proximity of a quantum state to maximally coherent state and it can be used as a measure of coherence. Coherence fraction has a connection with $l_1$-norm coherence and provides the criteria of coherence distillability. Optimal coherence fraction corresponding to a channel, defined from this new idea of coherence fraction, obeys a complementary relation with its decohering power. The connection between coherence fraction and $l_1$-norm coherence turns to hold for bipartite pure states and $X$ states too. The bipartite generalization shows that the local coherence fractions of a quantum state are not free and they are bounded by linear function of its global coherence fraction. Dynamics of optimal coherence fraction is also studied for single sided and both sided application of channels. Numerical results are provided in exploring properties of optimal coherence fraction.