A conjecture about bases of finite n-dimensional 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 $U(n-1)$. As such, it gives a recursive definition of the unitary matrices in terms of complex phases only.