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We present a description of locally equivalent states in terms of symplecticgeometry. Using the moment map between local orbits in the space of states andcoadjoint orbits of the local unitary group we reduce the problem of localunitary equivalence to an easy part consisting of identifying the propercoadjoint orbit and a harder problem of the geometry of fibers of the momentmap. We give a detailed analysis of the properties of orbits of "equallyentangled states". In particular we show connections between certain symplecticproperties of orbits such as their isotropy and coisotropy with effectivecriteria of local unitary equivalence.

Geometry of the local equivalence of states