  Characterizing and quantifying quantum correlations in states of
many-particle systems is at the core of a full understanding of phase
transitions in matter. In this work, we continue our investigation of the
notion of generalized entanglement [Barnum et al. Phys. Rev. A 68, 032308
(2003)] by focusing on a simple Lie-algebraic measure of purity of a quantum
state relative to an observable set. For the algebra of local observables on
multi-qubit systems, the resulting local purity measure is equivalent to a
recently introduced global entanglement measure [Meyer and Wallach, J. Math.
Phys. 43, 4273 (2002)]. In the condensed-matter setting, the notion of
Lie-algebraic purity is exploited to identify and characterize the quantum
phase transitions present in two exactly solvable models: the
Lipkin-Meshkov-Glick model, and the spin-1/2 anisotropic XY model in a
transverse magnetic field. For the latter, we argue that a natural fermionic
observable-set arising after the Jordan-Wigner transformation, better
characterizes the transition than alternative measures based on qubits. This
illustrates the usefulness of going beyond the standard subsystem-based
framework while providing a global disorder parameter for this model. Our
results show how generalized entanglement leads to useful tools for
distinguishing between the ordered and disordered phases in the case of broken
symmetry quantum phase transitions. Additional implications and possible
extensions of concepts to other systems of interest in condensed matter physics
are also discussed.
