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.”