This webpage hosts a collection of exact fiducial vectors for symmetric informationally complete quantum measurements (SICs, or SICPOVMs). The solutions are the complete solutions analyzed in the paper [ACFW17] cited below.
The format is a text file, readable by Magma, in terms of the number fields defined in the tables of the aforementioned paper. One must first define the relevant number fields in order to be able to read in the files. The provided Magma code will build the relevant field towers in terms of the field generators as described in [ACFW17]. The code for generating the fields for a specific dimension can be downloaded by clicking on the link in the Dim column, while the fiducials themselves can be found by clicking on the corresponding link under the Galois Orbits column.
All SIC fiducial vectors (except d=124 and 323) and field definitions in a single zip file (16.7 MB):
The table below organizes SIC fiducials primarily by their dimension Dim. Each new row of the table is a different Galois orbit, following the labeling established by Scott and Grassl [SG10]. Absent from this list are the so-called sporadic SICs; these are listed in a separate table below. The Type column describes the stabilizer type. If only one type is listed for several rows, the subsequent rows are understood to be of that same type. For example, both dimension 7 solutions are of type Fz. The Degree column lists the absolute degree of the defining number field.
Important note: As of this writing (14 March 2017), the table below does not contain links to the solutions in d = 4, ..., 14 nor of d = 31, 37, 43. A complete analysis of the number fields is still lacking for these dimensions. However, exact fiducial vectors have been calculated for all of these dimensions and solutions for d ≤ 14 are available from Gerhard Zauner's website following the conventions in [AYZ13]. Once the analysis is complete, those solutions will be hosted here as well. Update (21 Sep 2017): Added new exact solutions in d=124 and d=323 by Grassl and Scott.
|21||Fz||21a, b, c, d||1152||ACFW17|
|30||Fz||30a, b, c||3456||ACFW17|
|31b, e, f, g||ACFW17|
|35||Fz||35b, c, d, g||4608||ACFW17|
|37||Fz||37a, b, c, d||ACFW17|
|39||Fz||39a, c, d, e||4608||ACFW17|
|43||Fz||43a, d, e, f||ACFW17|
|48||Fz||48a, b, c, d||12288||ACFW17|
Following Stacey [S16], we consider the following sporadic SICs here for completeness. Solutions in d = 2 and d = 3 were presumably known to many people; for example, Coxeter [C40] discusses the case d = 3 as early as 1940, and the d = 2 case must have been well known much earlier. To our knowledge, these geometric objects were first discussed explicitly in the framework of equiangular lines by Delsarte, Goethals, and Seidel [DGS75, Example 6.4]. However, no explicit solution was given in that reference, and we cite Zauner [Z99] and Renes et al. [RBSC04] for discussing the complete solutions. The exceptional SIC in d = 8 is the famous Hoggar lines [H98]. Links to these exact fiducial vectors and (with the exception of the trancendental 3a) their number fields will be added soon.