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. We have worked provided Magma code that 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 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 F_{z}. 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.

Dim | Type | Galois Orbits | Degree | Citation |
---|---|---|---|---|

4 | F_{z} |
4a | 16 | Z99, RBSC04 |

5 | F_{z} |
5a | 32 | Z99 |

6 | F_{z} |
6a | 48 | G04 |

7 | F_{z} |
7a | 48 | A05 |

7b | 24 | A05 | ||

8 | F_{z} |
8a | 128 | SG10 |

8b | 32 | SG10 | ||

9 | F_{z} |
9a, b | 144 | SG10 |

10 | F_{z} |
10a | 192 | SG10 |

11 | F_{z} |
11a, b | 320 | SG10 |

11c | 160 | SG10 | ||

12 | F_{z} |
12a | 192 | G08 |

F_{a} |
12b | 64 | G05 | |

13 | F_{z} |
13a, b | 384 | SG10 |

14 | F_{z} |
14a, b | 576 | SG10 |

15 | F_{z} |
15a, c | 384 | ACFW17 |

15b | 192 | ACFW17 | ||

15d | 96 | SG10 | ||

16 | F_{z} |
16a, b | 1024 | ABBG+12 |

17 | F_{z} |
17a, b | 768 | ACFW17 |

17c | 384 | ACFW17 | ||

18 | F_{z} |
18a, b | 864 | ACFW17 |

19 | F_{z} |
19b, c | 864 | ACFW17 |

19a | 432 | ACFW17 | ||

19d | 216 | ACFW17 | ||

19e | 72 | A05 | ||

20 | F_{z} |
20a, b | 1536 | ACFW17 |

21 | F_{z} |
21a, b, c, d | 1152 | ACFW17 |

F_{a} |
21e | 384 | ACFW17 | |

24 | F_{z} |
24a, b | 1536 | ACFW17 |

24c | 384 | SG10 | ||

28 | F_{z} |
28a, b | 2304 | ACFW17 |

28c | 576 | ABBE+14 | ||

30 | F_{z} |
30a, b, c | 3456 | ACFW17 |

F_{a} |
30d | 1152 | ACFW17 | |

31 | F_{z} |
31a | ACFW17 | |

31c, d | ACFW17 | |||

31b, e, f, g | ACFW17 | |||

35 | F_{z} |
35b, c, d, g | 4608 | ACFW17 |

35a, f | 2304 | ACFW17 | ||

35e | 1152 | ACFW17 | ||

35h | 1152 | ACFW17 | ||

35i | 576 | ACFW17 | ||

35j | 288 | SG10 | ||

37 | F_{z} |
37a, b, c, d | ACFW17 | |

39 | F_{z} |
39a, c, d, e | 4608 | ACFW17 |

39b, f | 2304 | ACFW17 | ||

F_{a} |
39g, h | 1536 | ACFW17 | |

39i, j | 768 | ACFW17 | ||

43 | F_{z} |
43a, d, e, f | ACFW17 | |

43b, c | ACFW17 | |||

48 | F_{z} |
48a, b, c, d | 12288 | ACFW17 |

F_{a} |
48e | 4096 | ACFW17 | |

F_{z} |
48f | 1536 | ACFW17 | |

F_{a} |
48g | 512 | SG10 |

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 | F_{z} |
2a | - | |

3 | F_{z} |
3a | ∞ | Z99, RBSC04 |

3b | Z99, RBSC04 | |||

F_{a} |
3c | Z99, RBSC04 | ||

8 | * | 8H | H98 |

- [C40] H. S. M. Coxeter, “The Polytope 2

- [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.