A conjecture about bases of finite ndimensional Hilbert spaces appears to have finally been proved. Given two bases, it's always possible to choose a state that has equal transition probabilities to all the 2n basis states:
http://math.stackexchange.com/q/28413/8536
I've been working on this for three years. The guy who proved it is a Stanford math postdoc specializing in symplectic topology and Hamiltonian dynamics:
http://math.stanford.edu/~lisi/
As soon as I get the (rather subtle I think) proof verified I'm going to rewrite a paper of mine and send it in to J. Math. Phys.:
http://brannenworks.com/Gravity/qioumm_view.pdf
The conjecture shows that the unitary group $U(n)$ can be described as the arbitrary complex phases acting on a subgroup which is isomorphic to $U(n1)$. As such, it gives a recursive definition of the unitary matrices in terms of complex phases only.
New decomposition for unitary matrices!

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