Exact SIC fiducial vectors

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):

AllSICs.zip

Non-sporadic SIC fiducial vectors

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.

Dim Type Galois Orbits Degree Citation
4 Fz 4a 16 Z99, RBSC04
5 Fz 5a 32 Z99
6 Fz 6a 48 G04
7 Fz 7a 48 A05
7b 24 A05
8 Fz 8a 128 SG10
8b 32 SG10
9 Fz 9a, b 144 SG10
10 Fz 10a 192 SG10
11 Fz 11a, b 320 SG10
11c 160 SG10
12 Fz 12a 192 G08
Fa 12b 64 G05
13 Fz 13a, b 384 SG10
14 Fz 14a, b 576 SG10
15 Fz 15a, c 384 ACFW17
15b 192 ACFW17
15d 96 SG10
16 Fz 16a, b 1024 ABBG+12
17 Fz 17a, b 768 ACFW17
17c 384 ACFW17
18 Fz 18a, b 864 ACFW17
19 Fz 19b, c 864 ACFW17
19a 432 ACFW17
19d 216 ACFW17
19e 72 A05
20 Fz 20a, b 1536 ACFW17
21 Fz 21a, b, c, d 1152 ACFW17
Fa 21e 384 ACFW17
24 Fz 24a, b 1536 ACFW17
24c 384 SG10
28 Fz 28a, b 2304 ACFW17
28c 576 ABBE+14
30 Fz 30a, b, c 3456 ACFW17
Fa 30d 1152 ACFW17
31 Fz 31a ACFW17
31c, d ACFW17
31b, e, f, g ACFW17
35 Fz 35b, c, d, g 4608 ACFW17
35a, f 2304 ACFW17
35e 1152 ACFW17
35h 1152 ACFW17
35i 576 ACFW17
35j 288 SG10
37 Fz 37a, b, c, d ACFW17
39 Fz 39a, c, d, e 4608 ACFW17
39b, f 2304 ACFW17
Fa 39g, h 1536 ACFW17
39i, j 768 ACFW17
43 Fz 43a, d, e, f ACFW17
43b, c ACFW17
48 Fz 48a, b, c, d 12288 ACFW17
Fa 48e 4096 ACFW17
Fz 48f 1536 ACFW17
Fa 48g 512 SG10
124 Fz 124a 2880 GS17
323 Fz 323c 10368 GS17

Sporadic SICs

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.

Dim Type Galois Orbits Degree Citation
2 Fz 2a -
3 Fz 3a ∞ Z99, RBSC04
3b Z99, RBSC04
Fa 3c Z99, RBSC04
8 * 8H H98

Bibliography

[C40] H. S. M. Coxeter, “The Polytope 221 Whose Twenty-Seven Vertices Correspond to the Lines to the General Cubic Surface," American Journal of Mathematics 62, 457 (1940).
[DGS75] P. Delsarte, J. M. Goethals, and J. J. Seidel, “Bounds for systems of lines and Jacobi polynomials,” Philips Res. Rep. 30, 91 (1975).
[H98] S. G. Hoggar, “64 lines from a quaternionic polytope,” Geom. Dedic. 69, 287 (1998).
[Z99] G. Zauner, “Quantendesigns - Grundzüge einer nichtkommutativen Designtheorie," Ph.D. thesis, University of Vienna, 1999. Available in English translation as “Quantum Designs: Foundations of a Noncommutative Design Theory,” Int. J. Quant. Info. 9, 445 (2011).
[RBSC04] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric Informationally Complete Quantum Measurements,” J. Math. Phys. 45, 2171 (2004).
[G04] M. Grassl, “On SIC-POVMs and MUBs in Dimension 6,” in Proceedings of the ERATO Conference on Quantum Information Science, p. 60 (Tokyo, 2004).
[A05] D. M. Appleby, “Symmetric Informationally Complete Positive-Operator Valued Measures and the Extended Clifford group,” J. Math. Phys. 46, 052107 (2005).
[G05] M. Grassl, “Tomography of quantum states in small dimensions,” Electronic Notes in Discrete Mathematics, 20:151–164, 2005.
[G08] M. Grassl, “Computing equiangular lines in complex space,” In Jacques Calmet, Willi Geiselmann, and Jörn Müller-Quade, editors, Mathematical Methods in Computer Science: Essays in Memory of Thomas Beth, pp. 89–104. Springer, 2008.
[SG10] A. J. Scott and M. Grassl, “Symmetric informationally complete positive-operator-valued measures: A new computer study,” J. Math. Phys. 51, 042203 (2010).
[ABBG+12] D. M. Appleby, I. Bengtsson, S. Brierley, M. Grassl, D. Gross, J.-Å. Larsson, “The monomial representations of the Clifford group,” Quantum Inf. Comput. 12, 404 (2012).
[AYZ13] D. M. Appleby, H. Yadsan-Appleby, and G. Zauner, “Galois automorphisms of a symmetric measurement,” Quantum Inf. Comput. 13, 672 (2013).
[ABBE+14] D. M. Appleby, I. Bengtsson, S. Brierley, Å. Ericsson, M. Grassl, J.-Å. Larsson, “Systems of Imprimitivity for the Clifford group,” Quantum Inf. Comput. 14, 339 (2014).
[S16] B. C. Stacey, “Sporadic SICs and the Normed Division Algebras,” arXiv:1605.01426, (2016).
[ACFW17] M. Appleby, T.-Y. Chien, S. Flammia, and S. Waldron, “Constructing exact symmetric informationally complete measurements from numerical solutions,” arXiv:1703.05981, 2017.
[GS17] M. Grassl and A. J. Scott, “"Fibonacci-Lucas SIC-POVMs", arXiv:1707.02944, 2017.