is true that many of the entanglement measures used for bi-partite states carry straightforward generalizations to the multi-partite setting. This is particularly true for distance-based entanglement measures, such as the relative entropy of entanglement [@RelEnt] or the closely related geometric entanglement [@Geometric]: the entanglement of a state is then simply quantified in terms of the minimum distance of that state from the set of all multi-partite separable states, rather than the set of all bi-partite separable states. However, it is one thing to generalize the axiomatic definitions of entanglement measures to include the multi-partite case, but quite another to still be able to compute these in practice. Moreover, the theory of multi-partite entanglement is still plagued by a host of different candidates for suitable entanglement measures, even for the case of pure states (see Ref. [@EntMeasures] for more details).

In view of these difficulties, attention has shifted to include other, potentially related, means of characterizing QPTs [@Zanardi]. One such approach centres around the notion of *geometric phase* [@Berry; @Resta; @Carollo1; @Carollo2; @Hamma], and it is this approach which we wish to pursue in the remainder of this work.

Geometric Phase {#subsection:GeometricPhase}
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In his seminal paper [@Berry], Berry investigated the 