We consider the action of the group of local unitary transformations, U(m) x U(n), on the set of (mixed) states W of the bipartite m x n quantum system. We prove that the generic U(m) x U(n)--orbits in W have dimension m^2+n^2-2. This problem was mentioned (and left open) by Kus and Zyczkowski in their paper Geometry of entangled states. The proof can be extended to the case of arbitrary finite-dimensional multipartite quantum systems.
@article{arxiv.quant-ph/0608124,
title = {Dimensions of generic local orbits of multipartite quantum systems},
author = {Dragomir Z. Djokovic},
journal= {arXiv preprint arXiv:quant-ph/0608124},
year = {2007}
}