# Numerical and analytical results for geometric measure of coherence and geometric measure of entanglement

## TL;DR

This paper develops numerical algorithms and analytical solutions to quantify geometric measures of coherence and entanglement in quantum states, enabling better analysis of quantum information resources.

## Contribution

It introduces semidefinite programming algorithms for calculating geometric coherence and entanglement, along with analytical solutions for specific mixed states.

## Key findings

- Algorithms successfully compute geometric measures for various states.
- Numerical results match analytical solutions where available.
- Lower bounds are provided for complex states.

## Abstract

Quantifying coherence and entanglement is extremely important in quantum information processing. Here, we present numerical and analytical results for the geometric measure of coherence, and also present numerical results for the geometric measure of entanglement. On the one hand, we first provide a semidefinite algorithm to numerically calculate geometric measure of coherence for arbitrary finite-dimensional mixed states. Based on this semidefinite algorithm, we test randomly generated single-qubit states, single-qutrit states, and a special kind of $d$-dimensional mixed states. Moreover, we also obtain an analytical solution of geometric measure of coherence for a special kind of mixed states. On the other hand, another algorithm is proposed to calculate the geometric measure of entanglement for arbitrary two-qubit and qubit-qutrit states, and some special kinds of higher dimensional mixed states. For other states, the algorithm can get a lower bound of the geometric measure of entanglement. Randomly generated two-qubit states, the isotropic states and the Werner states are tested. Furthermore, we compare our numerical results with some analytical results, which coincide with each other.