  Bipartite maximally entangled states have the property that the largest
Schmidt coefficient reaches its lower bound. However, for multipartite states
the standard Schmidt decomposition generally does not exist. We use a
generalized Schmidt decomposition and the geometric measure of entanglement to
characterize three-qubit pure states and derive a single-parameter family of
maximally entangled three-qubit states. The paradigmatic
Greenberger-Horne-Zeilinger (GHZ) and W states emerge as extreme members in
this family of maximally entangled states. This family of states possess
different trends of entanglement behavior: in going from GHZ to W states the
geometric measure and the relative entropy of entanglement and the bipartite
entanglement all increase monotonically whereas the three-tangle and
bi-partition negativity both decrease monotonically.
