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The SIC-POVM problem is remarkably easy to state given that it has not yet been solved. It goes like this. With dim($\mathcal H$) $=d$, find states $|\psi_k\rangle\in\mathcal H$, $k=1,\ldots,d^2$ such that $|\langle \psi_k|\psi_j\rangle|=\frac{1}{d+1}$ for all $k\neq j$.

The state of the art on the solution I believe is here: http://arxiv.org/abs/0910.5784. Various constructive conjectures have been given but what existence proofs have been tried and why have they failed? What insight has been distilled from these attempts?

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There is an item devoted to the problem here –  Alex 'qubeat' Oct 30 '11 at 15:16
    
Thanks for the link Alex. But, again, it lists numerical results and connections to other conjectures. My question is why, for example, does induction on $d$ not work? It is possible to prove an inductive proof is impossible? –  Chris Ferrie Oct 30 '11 at 16:27
    
Constructions of SIC for consequent $d$ too different to hope on induction, e.g. see TABLE I in e-print you cited: for $d=3$ there are infinite number of SIC, but for other $d$ only finite number (and the numbers of SIC have rather unpredictable behavior). –  Alex 'qubeat' Oct 30 '11 at 21:08
    
Just mentioned related question on MO mathoverflow.net/questions/2897/… –  Alex 'qubeat' Nov 22 '11 at 10:12
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Het Chris, For more analytic arguments about SIC's you may want to check out http://arxiv.org/abs/1001.0004 .

I got interested in this problem at some point and talked to Steve. He warned me off, describing the SIC-POVM problem as a "heartbreaker" because every approach you take seems super promising but then inevitably fizzles out without really giving you a great insight as to why.

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Thanks Seth! That's definitely some useful information. Side note: the thicker papers always seem to find their way to the bottom of the reading pile. I'm going to blame gravity. –  Chris Ferrie Nov 7 '11 at 12:11
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