Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product

TL;DR

This paper introduces a new geometric framework for quantifying entanglement in multiparticle pure states using Lagrange's identity and wedge products, extending the concept of concurrence beyond two qubits.

Contribution

It develops a generalized concurrence measure for multiparticle pure states in arbitrary dimensions, connecting it to existing measures and revealing the geometric nature of entanglement.

Findings

Provides a faithful entanglement measure for multiparticle states

Unifies and extends previous concurrence concepts

Highlights the geometric structure of quantum entanglement

Abstract

Concurrence, introduced by Hill and Wootters [Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is faithful: strictly positive for entangled states and vanishing for all separable states. Such a measure captures the entire content of entanglement, providing necessary and sufficient conditions for separability. We present an extension of concurrence to multiparticle pure states in arbitrary dimensions by a new framework using the Lagrange's identity and wedge product representation of separability conditions, which coincides with the "I-concurrence" of Rungta et al. [Phys. Rev. A 64, 042315 (2001)] who proposed by extending Wootters's spin-flip operator to a so-called universal inverter superoperator. Our framework exposes an inherent geometry of entanglement, and may be useful for the further extensions to mixed and continuous variable states.