A group action of the Heisenberg group with special symmetries - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:55:22Z http://mathoverflow.net/feeds/question/2897 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2897/a-group-action-of-the-heisenberg-group-with-special-symmetries A group action of the Heisenberg group with special symmetries Steve Flammia 2009-10-27T21:26:49Z 2009-12-02T05:55:50Z <p>Suppose we look at the Heisenberg group H<sub>d</sub> as a matrix group of upper triangular matrices over the ring &#8484;/d&#8484;. You can even choose d to be prime if you want. A natural irrep of H<sub>d</sub> acting on &#8450;<sup>d</sup> maps the group elements into the "shift" and "phase" operators, plus roots of unity. More specifically, the two natural generators map the orthonormal basis vectors from j \to j+1 mod d, and the Fourier transform of that operation, plus overall phases by roots of unity. The question is this:</p> <p>Can you find a unit vector v such that |(v,U<sub>g</sub> v)| = c for all g not in the center of H<sub>d</sub>? One can solve for the constant: c=1/sqrt(d+1).</p> <p>Numerics suggests that these vectors exist in all the dimensions &lt; 67, hence they may exist in every dimension, but the form of the vectors contains no (obvious) hint as to how to prove this.</p> <p>This problem seems extremely truculent and any help is greatly appreciated!</p> http://mathoverflow.net/questions/2897/a-group-action-of-the-heisenberg-group-with-special-symmetries/2901#2901 Answer by David Speyer for A group action of the Heisenberg group with special symmetries David Speyer 2009-10-27T21:42:29Z 2009-10-27T21:42:29Z <p>Just to check, this is the same as the <a href="http://en.wikipedia.org/wiki/SIC-POVM" rel="nofollow">conjecture of Zauner</a>, yes?</p> http://mathoverflow.net/questions/2897/a-group-action-of-the-heisenberg-group-with-special-symmetries/3949#3949 Answer by David Speyer for A group action of the Heisenberg group with special symmetries David Speyer 2009-11-03T16:14:19Z 2009-11-03T16:14:19Z <p>I just wanted to point out <a href="http://arxiv.org/abs/0910.5784" rel="nofollow">this paper</a> to anyone who is interested. The authors report on a massive computational test of Zauner's conjecture. Don't be intimidated by the length; there are 18 pages of math, the rest is all tables of data.</p> <p>I also want to use the word SIC-POVM, as that is what anyone searching this site for references will probably look for. </p>