# Quantum coherence fraction

## TL;DR

This paper introduces the quantum coherence fraction as a measure of how close a quantum state is to maximally coherent states, explores its relationship with the robustness of coherence, and provides analytical results for low-dimensional systems.

## Contribution

The paper defines the quantum coherence fraction, relates it to the robustness of coherence, and proves its equivalence to the normalized robustness for qubit and qutrit states.

## Key findings

- $C_{ ext{F}}$ approximates $ar{C}_{ ext{R}}$ well in low dimensions
- $C_{ ext{F}}$ equals $ar{C}_{ ext{R}}$ for qubits and qutrits
- Numerical simulations support the theoretical relationships

## Abstract

As an analogy of fully entangled fraction in the framework of entanglement theory, we have introduced the notion of quantum coherence fraction $C_{\mathcal{F}}$, which quantifies the closeness between a given state and the set of maximally coherent states. By providing an alternative formulation of the robustness of coherence $C_{\mathcal{R}}$, we have elucidated the relationship between quantum coherence fraction and the normalized version of $C_{\mathcal{R}}$ (i.e., $\overline{C}_{\mathcal{R}}$), where the role of genuinely incoherent operations (GIO) is highlighted. Numerical simulation shows that though as expected $C_{\mathcal{F}}$ is upper bounded by $\overline{C}_{\mathcal{R}}$, $C_{\mathcal{F}}$ constitutes a good approximation to $\overline{C}_{\mathcal{R}}$ especially in low-dimensional Hilbert spaces. Even more intriguingly, we can analytically prove that $C_{\mathcal{F}}$ is exactly equivalent to $\overline{C}_{\mathcal{R}}$ for qubit and qutrit states. Moreover, some intuitive properties and implications of $C_{\mathcal{F}}$ are also indicated.