# The Geometry of Quantum Coherence for Two Qubit $X$ States

## TL;DR

This paper explores the geometric structure of quantum coherence in two-qubit Bell-diagonal states, revealing how coherence measures relate to entanglement and quantum discord, and analyzing coherence dynamics under local channels.

## Contribution

It provides a geometric visualization of coherence measures for Bell-diagonal states, establishes conditions for quantum discord and coherence equality, and studies coherence decay under local channels.

## Key findings

- Coherence measures increase from zero to one across state regions.
- Quantum discord equals relative entropy of coherence under specific conditions.
- Coherence decreases under local nondissipative channels.

## Abstract

We plot the geometry of several distance-based quantifiers of coherence for Bell-diagonal states. We find that along with both $l_{1}$ norm and relative entropy of coherence changes continuously from zero to one, their surfaces move from the separable regions to the entangled regions. Based on this fact, it is more illuminating to use an intuitive geometry to explain quantum states with nonzero coherence can be used for entanglement creation, rather than the other way around. We find the necessary and sufficient conditions that quantum discord of Bell-diagonal states equal to its relative entropy of coherence and depict the surfaces of the equality. We give surfaces of relative entropy of coherence for $X$ states. We show the surfaces of dynamics of relative entropy of coherence for Bell-diagonal states under local nondissipative channels and find that all coherence under local nondissipative channels decrease.