# Zauner’s conjecture is true in dimensions 18, 20, 21, 30, 31, 37, 39 and 43

Two years ago, I blogged about Tuan-Yow Chien’s PhD thesis, which proved Zauner’s conjecture in dimension 17. The idea was to exploit certain conjectures on the field structure of SIC-POVM fiducial vectors so as to round numerical solutions to exact solutions. This week, the arXiv announced Chien’s latest paper (coauthored with Appleby, Flammia and Waldron), which extends this work to find exact solutions in 8 new dimensions.

The following line from the introduction caught my eye:

For instance the print-out for exact fiducial 48a occupies almost a thousand A4 pages (font size 9 and narrow margins).

As my previous blog entry illustrated, the description length of SIC-POVM fiducial vectors appears to grow rapidly with $d$. However, it seems that the rate of growth is much better than I originally thought. Here’s a plot of the description lengths of the known fiducial vectors (the new ones due to ACFW17available here — appear in red):

Note that the vertical axis has logarithmic scale. Unlike my interpretation from two years ago, the description lengths appear to exhibit subexponential growth in $d$. Putting the horizontal axis in log scale says even more:

The dotted line depicts $d^4$. This suggests that the description length scales with the number of entries in the Gram matrix.

For context, let’s consider the more general problem of constructing equiangular tight frames (ETFs) of $n$ vectors in dimension $d$; see this paper for a survey. In the real case, it suffices to determine the sign pattern of an ETF’s Gram matrix, which can be naively described in $O(n^2)$ bits. However, there are several infinite families of real ETFs with much shorter description length. Indeed, the sign patterns are determined by certain strongly regular graphs, many of which enjoy a straightforward algebro-combinatorial construction.

In the case of SIC-POVMs, the Gram matrix is complex, so it doesn’t correspond to a strongly regular graph in the same way, but the conjectures used in ACFW17 suggest that the Gram matrix may be selected so as to satisfy certain group and number theoretic properties. But even after reducing to such specific structure, the description length appears to scale with the size of the Gram matrix (i.e., the naive scaling in the real case). As such, an infinite family of explicit SIC-POVMs will likely require the identification of additional structure. This is shocking, considering the conjectured structures that are currently used already seem miraculous.