New decomposition for unitary matrices!

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New decomposition for unitary matrices!

Postby CarlBrannen » Wed Mar 30, 2011 12:22 pm

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:

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:

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.:

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.

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