  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.
