Evaluating the geometric measure of multiparticle entanglement

Abstract

We present an analytical approach to evaluate the geometric measure of multiparticle entanglement for mixed quantum states. Our method allows the computation of this measure for a family of multiparticle states with a certain symmetry and delivers lower bounds on the measure for general states. It works for an arbitrary number of particles, for arbitrary classes of multiparticle entanglement, and can also be used to determine other entanglement measures.

Nonlocality of three-qubit Greenberger-Horne-Zeilinger–symmetric states

“…Here we analyze whether it is sufficient for this class of tripartite mixed quantum states to obtain standard nonlocality. The criterion of biseparability of this class of states is [36,39]:…”

“…As an important class of mixed states from quantum theoretical perspective, GHZ-symmetric three-qubit states have been paid much attention [36][37][38][39]. In particular, in the eight dimensional state space of three qubit states, the set of GHZ-symmetric states defines a twodimensional affine section, specifically a triangle of the full eight dimensional set of states [40].…”

The Separability versus Entanglement Problem

“…Until now, in the case of multiparty mixed states, we have considered only the distance-based measures. However, it is also possible to use the convex-roof approach to define entanglement measures for multiparty mixed states, after choosing a certain measure for pure states [94,95]. Another classification is possible by considering interconversion of states through stochastic local operations and classical communication (SLOCC) [96], i.e, through LOCC but with a non-unit probability.…”

Analytically Computable Symmetric Quantum Correlations

“…[1][2][3] and references therein); however, only limited progress has been made toward the most essential goal, that is, the development of a good quantifier for a general quantum system. [12][13][14][15][16][17][18][19][20][21][22][23] In addition, as a quantum correlation that extends beyond entanglement, the quantum discord (QD), [24,25] which quantifies the discrepancy between the DOI: 10.1002/andp.201800178 quantum versions of two classically equivalent pieces of mutual information, can only be analytically calculated for a general state in 2 ⊗ n dimensions. Needless to say, multipartite quantum states include many inequivalent classes of entanglement, [10,11] and the quantification of a bipartite highdimensional state is usually related to some complex optimization; consequently, it is difficult to formulate an analytical or even an economic and effective numerical way to quantify the entanglement of a general state.…”