Adam Sawicki, Michał Oszmaniec, Marek Kuś
Using geometric invariant theory and momentum map geometry we show how to effectively find all stochastic local operations and classical communication (SLOCC) classes of pure states for both distinguishable and indistinguishable particles. We prove that they are parametrized by critical sets of the total variance of the state which can be regarded as an entanglement measure. Remarkably, the Morse index at the critical point have a nice interpretation of number of non-local directions in which such defined entanglement increases. We also introduce the SLOCC-invariant 'measure' of entanglement as square root of total variance of state at the critical point and show how it is connected to the distance of the SLOCC Kirwan polytope from the origin.
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http://arxiv.org/abs/1208.0556
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