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The degree to which a pure quantum state is entangled can be characterized bythe distance or angle to the nearest unentangled state. This geometric measureof entanglement is explored for bi-partite and multi-partite pure and mixedstates. It is determined analytically for arbitrary two-qubit mixed states,generalized Werner, and isotropic states, and is also applied to certainmulti-partite mixed states, including two distinct multi-partite boundentangled states. Moreover, the ground-state entanglement of the XY model in atransverse field is calculated and shown to exhibit singular behavior near thequantum critical line. Along the way, connections are pointed out between thegeometric measure of entanglement, the Hartree approximation, entanglementwitnesses, correlation functions, and the relative entropy of entanglement.

Quantum entanglement: geometric quantification and applications to multi-partite states and quantum phase transitions