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What proof techniques have failed for solving the SIC-POVM problem and what new insights have been gleaned from them?

+ 14 like - 0 dislike
581 views

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?

This post has been migrated from (A51.SE)
asked Oct 30, 2011 in Theoretical Physics by Chris Ferrie (645 points) [ no revision ]
Just mentioned related question on MO http://mathoverflow.net/questions/2897/a-group-action-of-the-heisenberg-group-with-special-symmetries

This post has been migrated from (A51.SE)
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).

This post has been migrated from (A51.SE)
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?

This post has been migrated from (A51.SE)
There is an item devoted to the problem [here](http://qig.itp.uni-hannover.de/qiproblems/SIC_POVMs_and_Zauner's_Conjecture)

This post has been migrated from (A51.SE)

1 Answer

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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.

This post has been migrated from (A51.SE)
answered Nov 6, 2011 by smerkel (70 points) [ no revision ]
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.

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