# A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural irrep of $H_{d}$ acting on $\mathbb{C}^{d}$ 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:

Can you find a unit vector $v$ such that $|(v,U_g v)| = c$ for all g not in the center of $H_{d}\ ?$ One can solve for the constant: $c=\frac{1}{\sqrt{d+1}}$.

Numerics suggests that these vectors exist in all the dimensions $< 67$, hence they may exist in every dimension, but the form of the vectors contains no (obvious) hint as to how to prove this.

This problem seems extremely truculent and any help is greatly appreciated!

• Wiki'd as open problem, see David Speyer's answer and attached comments, below. – Scott Morrison Nov 23 '09 at 19:33
• Z_d is mildly ambiguous; do you mean Z/dZ? – Qiaochu Yuan Dec 1 '09 at 17:44
• For interesting algebraic properties of the solutions found so far see e.g. GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS VIA COMPLEX EQUIANGULAR LINES, arxiv.org/abs/1604.06098 . – jjcale Oct 27 '16 at 16:09

Just to check, this is the same as the conjecture of Zauner, yes?

• Yes; I'm surprised that you've heard of it. From my experience, people don't seem to know about this problem. – Steve Flammia Oct 27 '09 at 21:46
• I've heard about it from Henry Cohn, who thinks a lot about sphere packing. It was also mentioned in the comments to Scott Aaronson's post "The ten most annoying questions in quantum computing." scottaaronson.com/blog/?p=112 I have no ideas how to attack it, though. – David E Speyer Oct 27 '09 at 22:15
• I would be interested in knowing what progress has been made on this problem, and what are some reasonable approaches. How would you feel about refocusing the question that way and making this a community wiki, as proposed here tea.mathoverflow.net/discussion/8/… ? – David E Speyer Oct 29 '09 at 13:35
• Sure. Should I just edit the original statement, or start a new thread? – Steve Flammia Oct 29 '09 at 22:58

I just wanted to point out this paper 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.

I also want to use the word SIC-POVM, as that is what anyone searching this site for references will probably look for.

• David, have you considered posting about this problem on your blog? I would love to see this problem get a wider mathematical audience. I think it is very beautiful and seems to touch on lots of different areas of mathematics. I would be happy to assist you in anyway if you wanted to write such a post. (I have closely followed work on the problem, though regrettably I haven't contributed much.) – Steve Flammia Nov 3 '09 at 17:54
• I hesitate to do this because I don't think I'm the right person; I am aware of the problem, but I certainly don't have a good sense of the literature or of what has been tried. For example, I don't even know if there are good surveys already out there. – David E Speyer Nov 3 '09 at 18:19
• I've just spoken to Marcus Appleby, who knows as much or more about the problem than anyone else in the world. There is currently no review article, but he is considering writing one. – Steve Flammia Nov 3 '09 at 23:34