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.
Highlights
Quantifying coherence and entanglement is extremely important in quantum information processing
For the special kind of d-dimensional mixed states, we obtain an analytical solution of its geometric measure of coherence. We propose another algorithm for the geometric measure of entanglement, which can obtain the geometric measure of entanglement for arbitrary two-qubit and qubit-qutrit states, and some special kinds of higher dimensional mixed states
A lower bound of the geometric measure of entanglement can be acquired by using the algorithm
Summary
Quantifying coherence and entanglement is extremely important in quantum information processing. The analytical solutions of the coherence and entanglement measures are not available, so numerical algorithms must be applied. In Refs.[62,63], a semidefinite program (SDP) was proposed to calculate the fidelity between two states, which inspire us to apply this semidefinite program to numerically obtain the geometric measures of coherence and entanglement.
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