Geometric Phase Transition: direct measurement of a mathematical   abstraction
  By viewing entanglement as a state function, a new kind of phase transition takes place: the geometric phase transition. This phenomenon occurs due to singularities in the shape of the entangled states set. It is shown how this result can be carried to provide a better understanding of the geometry of entanglement. Surprisingly, this study can be done experimentally, what allows to determine the shape of different entangled states sets, a purely mathematical definition, in real experiments. 
Daniel Cavalcanti, Fernando G.S.L. Brandao, Marcelo O. Terra Cunha