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

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.

Estimation of the geometric measure of entanglement with Wehrl moments through artificial neural networks

“…8 by red squares. This time, the error is systematically higher than that obtained for the depolarised states (38), but it remains at an acceptable level for k = 0.05 and q max ⩾ 4.…”

“…8 shows that Wehrl moments remain useful quantities for predicting entanglement of mixed states in multiqubit systems. It is interesting to note that even for highly mixed states of the form (38), ANNs are still able to predict with high accuracy the GME. This is shown in Fig.…”

“…,(37)is Uhlmann's fidelity between any two mixed states ρ and σ. The form(36) allows us to compute the GME of mixed states using a semidefinite program, as we explain in Appendix E (see also[38]). …”

The Tightness of Multipartite Coherence from Spectrum Estimation

“…The accuracy estimated bounds is indicated by P C = l max C /C. Note that C r and C l 1 of ρ ψ expt can be calculated directly according to the definition in Equations ( 1) and (2), while the calculations of C g (•) and C R (•) require converting them to the convex optimization problem [19,20,59] and the corresponding solution [60][61][62]. The calculation of C f , Cl 1 requires optimizing all pure state decomposition, and there is no general method for analytical and numerical solutions except a few special cases.…”