Notes on Zauner’s conjecture

Last week, I visited Joey Iverson at the University of Maryland, and we spent a lot of time working through different approaches to Zauner’s conjecture. In general, my relationship with this problem is very similar to Steve Flammia‘s description, as paraphrased by smerkel on Physics Stack Exchange:

[Flammia described] 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.

Case in point, Joey and I identified a promising approach involving ideas from our association schemes paper. We were fairly optimistic, and Joey even bet me $5 that our approach would work. Needless to say, I now have this keepsake from Joey:

While our failure didn’t offer any great insights (as Flammia predicted), the experience forced me to review the literature on Zauner’s conjecture a lot more carefully. A few things caught my eye, and I’ll discuss them here. Throughout, SIC denotes “symmetric informationally complete line set” and WH denotes “the Weyl-Heisenberg group.”

1. WH over produces a SIC only if .

Of course, this works for since this reduces to WH over a cyclic group. The case corresponds to the famous Hoggar lines (introduced here). Last week, I learned that Godsil and Roy proved that this doesn’t work in general (see Lemma 3.1 here). What’s the obstruction? The system of equations doesn’t have a solution except for these two special cases. Sadly, this is not terribly enlightening.

Before seeing this result, I had assumed that SICs would arise from all WHs over finite abelian groups. After seeing Godsil and Roy’s result, I wrote an interpretation of the numerical optimization code briefly described in the computer study paper, and I failed to find SICs from WHs over other small non-cyclic abelian groups. Apparently, the cyclic groups and are special, but I have no idea why.

2. SICs can be generated by groups other than WH.

Back in 2003, Renes et al performed numerical optimization to determine whether SICs could be obtained from non-WH groups. To this end, they churned through certain members of the SmallGroups Library and found that the groups G(36,11), G(36,14), G(64,8) and G(81,9) all lead to SICs. Later, Grassl found exact SICs for the first and third of these groups by computing the appropriate Groebner bases (he also points out that G(36,14) is actually WH). For the record, the exact coordinates in these cases are about as ugly as the coordinates for the WH SICs. But as far as I can tell, these alternative constructions (which reside in dimensions 6 and 8) have been forgotten by the modern SIC literature. For example, they do not appear as “sporadic SICs” in the Exact SICs table.

3. Prime-dimensional group-generated SICs are necessarily generated by WH.

This was established by Huangjun Zhu in this paper back in 2010, and it suggests that WH is the “right” group to work with (if the substantial evidence in favor of Zauner’s conjecture weren’t enough). Unfortunately, the description length of exact fiducial vectors over WH scales poorly with the dimension. One is inclined to compress these descriptions into shorter, workable representations before attempting pattern recognition for theorem discovery. Based on my experience with constructing infinite families of ETFs, this is the most promising approach for a constructive proof of Zauner’s conjecture.

4. It looks like WH SICs are always determined by explicit equations, instead of .

Fuchs et al recently posed what they call the 3d conjecture, which asserts that the WH SICs are precisely the solutions to equations they give in (27)–(29) of their paper. The conjecture holds for , and it’s held up to numerical scrutiny for . This suggests a couple of new approaches: (1) Prove the 3d conjecture. (2) Prove that 3d implies Zauner. I wouldn’t be surprised if it’s easier to determine whether 3d admits solutions, so this could be an interesting conditional proof of Zauner.

5. A constructive proof of Zauner’s conjecture may require progress on Hilbert’s 12th problem.

A constructive proof requires a finite-length description of an infinite family of SICs, since the proof would contain such a description. For all known non-maxial ETFs (see this paper for a survey), the Gram matrix can always be phased in such a way that all of the entries are cyclotomic, and furthermore, expressing the Gram matrix entries in this way allows patterns to emerge that enables both a short description and a proof of ETF-ness for an infinite family. (For an illustrative example, consider the harmonic ETFs.)

As established back in 2012, all of the known exact WH SICs have the property that the orthogonal projection onto the line spanned by the fiducial vector has matrix entries that lie in an abelian extension of . Since they lie in an abelian extension of an abelian extension of , these entries are expressible by radicals, and this is the representation of choice in the Exact SICs table. However, Hilbert’s 12th problem suggests that a better representation might be available. By analogy, the Kronecker–Weber theorem gives that every abelian extension of is cyclotomic, and the Kronecker Jugendtraum gives that every abelian extension of an imaginary quadratic field can be obtained with values of certain elliptic functions. By contrast, we are looking at abelian extensions of a real quadratic field, which is not a solved case of Hilbert’s 12th. Still, one might leverage the Stark conjectures to find a suitable basis. Apparently, the computer algebra system PARI/GP makes this a plausible enterprise, but I haven’t found the time to write the necessary code. (I’m still recoiling from my latest Zauner burn with Joey.)