New decomposition for unitary matrices!

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CarlBrannen
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Joined: Mon May 24, 2010 11:34 pm

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:
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.



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