# Informationally Complete Characters for Quark and Lepton Mixings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Minimal Informationally Complete Quantum Measurements

## 2. Informationally Complete Characters for Quark/Lepton Mixing Matrices

#### 2.1. Groups in the Series $\Delta (6{n}^{2})$ and More Groups

#### 2.2. Exceptional Subgroups of $SU(3)$

## 3. Generalized CP Symmetry, CP Violation

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Small groups considered in our Table 3. For each group and each character, the table provides the dimension of the representation and the rank of the Gram matrix obtained under the action of the corresponding Pauli group. Bold characters are for faithful representations. According to our demands, each selected group has both 2- and 3-dimensional characters (with at least one of them faithful) that are magic states for an informationally complete POVM (or MIC), with the rank of Gram matrix equal to ${d}^{2}$. The Pauli group performing this action is in general a d-dit but is a 2-qutrit (2QT) for the group $(120,5)=SL(2,5)=2I$, a 3-qutrit (2QT) for the group $(360,51)={\mathbb{Z}}_{3}\times SL(2,5)$ or may be a three-qubit/qutrit (3QB-QT) for the groups $(648,532)$ and $(648,533)$.

Group | d | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(24,12) | 5 | 1 | 1 | 2 | 3 | 3 | ||||||||||

5-dit | 5 | 21 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||||

(120,5) | 9 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | ||||||

9-dit | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 79 | ${d}^{2}$ | |||||||

2QT | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||

(150,5) | 13 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | ||

13-dit | 13 | 157 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||

(72,42) | 15 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

15-dit | 15 | 203 | 209 | 209 | 195 | 195 | 219 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

(216,95) | 19 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

19-dit | 19 | 343 | 357 | 359 | 355 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 6 | 6 | 6 | |||||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||||||

(294,7) | 20 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

20-dit | 20 | 349 | 388 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 6 | ||||||||||||

390 | 390 | 390 | 398 | 398 | ||||||||||||

(72,3) | 21 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |

21-dit | 21 | 405 | 405 | 421 | 421 | 421 | 421 | 421 | 421 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

2 | 2 | 2 | 3 | 3 | 3 | |||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||||

(162,12) | 22 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

22-dit | 22 | 446 | 463 | 463 | 463 | 463 | 473 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 6 | ||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 198 | ||||||||||

(162,14) | 22 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

22-dit | 22 | 444 | 461 | 463 | 461 | 463 | 473 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 6 | ||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 198 | ||||||||||

(648,532) | 24 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 |

24-dit | 24 | 527 | 527 | 562 | ${d}^{2}$ | ${d}^{2}$ | 560 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3QB-QT | 24 | 500 | 500 | 476 | 568 | 568 | 448 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 8 | 8 | 8 | 9 | 9 | ||||||||

24-dit | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 564 | ${d}^{2}$ | ${d}^{2}$ | 552 | 552 | |||||||

3QB-QT | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 448 | 560 | 560 | 510 | 510 | |||||||

(648,533) | 24 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 |

24-dit | 24 | 539 | 539 | 562 | ${d}^{2}$ | ${d}^{2}$ | 514 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 574 | 574 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3QB-QT | 24 | 532 | 532 | 481 | 572 | 572 | 452 | 572 | 568 | 568 | 570 | 570 | 572 | 575 | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 8 | 8 | 8 | 9 | 9 | ||||||||

24-dit | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 563 | ${d}^{2}$ | ${d}^{2}$ | 478 | 478 | |||||||

3QB-QT | ${d}^{2}$ | 573 | 573 | 575 | 488 | 560 | 560 | 520 | 520 |

**Table A2.**The following up of Table A1.

Group | d | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(120,37) | 25 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |

25-dit | 25 | 601 | 601 | 601 | 601 | 601 | 601 | 601 | 601 | 601 | 623 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||

(360,51) | 27 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

3QT | 27 | 613 | 613 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | |||||

727 | 725 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | 727 | |||||

(162,44) | 30 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

30-dit | 31 | 826 | 861 | 871 | 861 | 871 | 883 | 877 | 879 | 883 | 898 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 898 | |

2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ||

898 | 898 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

(600,179) | 33 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

33-dit | 33 | 1041 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

6 | 6 | 6 | ||||||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||||||||||||||

(168,45) | 35 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |

35-dit | 35 | 1175 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | 1191 | ${d}^{2}$ | |

2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||

3 | 3 | 3 | 3 | 3 | ||||||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||||||||||||

(480,221) | 36 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 |

36-dit | 36 | 36 | 1085 | 1185 | 1184 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 1278 | 1278 | 1278 | |

3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | ||

1278 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 1275 | 1278 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 1277 | 1273 | ||

5 | 5 | 6 | 6 | 6 | 6 | |||||||||||

1294 | 1294 | 1295 | 1295 | 1295 | 1295 |

## References

- Ramond, P. The five instructions. In The Dark Secrets of the Terascale (TASI 2011), Proceedings of the 2011 Theoretical Advanced Study Institute in Elementary Particle PhysicsBoulder, Boulder, CO, USA, 6 June–1 July 2011; World Scientific: Singapore, 2011. [Google Scholar]
- Holthausen, M.; Lim, K.S. Quark and leptonic mixing patterns from the breakdown of a common discrete flavor symmetry. Phys. Rev. D
**2013**, 88, 033018. [Google Scholar] [CrossRef][Green Version] - Esteban, I.; Gonzalez Garcia, C.; Hernandez Cabezudo, A.; Maltoni, M.; Martinez Soler, I.; Schwetz, T. Leptonic Mixing Matrix (January 2018). Available online: http://www.nu-fit.org/?q=node/166 (accessed on 1 March 2020).
- Frampton, P.H.; Kephart, T.W. Flavor symmetry for quarks and leptons. J. High Energy Phys.
**2007**, 9, 110. [Google Scholar] [CrossRef] - Xing, Z.Z. Flavor structures of charged fermions and massive neutrinos. Phys. Rep.
**2020**, 854, 1–147. [Google Scholar] [CrossRef][Green Version] - Brannen, C. Spin path integrals and generations. Found. Phys.
**2010**, 40, 1681–1699. [Google Scholar] [CrossRef][Green Version] - Sheppeard, M.; Te Atatu Peninsula, W. Lepton mass phases and CKM matrix. arXiv
**2017**, arXiv:1711.0336. Available online: https://pdfs.semanticscholar.org/7dd0/7f4995e0e239905e72ce396ee4ce0cf91be2.pdf (accessed on 3 April 2020). - Minakata, H.; Smirnov, A.Y. Neutrino mixing and quark-lepton Complementarity. Phys. Rev. D
**2004**, 70, 073009. [Google Scholar] [CrossRef][Green Version] - Parattu, K.M.; Wingerter, A. Tri-bimaximal mixing from small groups. Phys. Rev. D
**2011**, 84, 013011. [Google Scholar] [CrossRef][Green Version] - Harrison, P.F.; Perkins, D.H.; Scott, W.G. Tribimaximal mixing and the neutrino oscillation data. Phys. Lett. B
**2002**, 530, 167. [Google Scholar] [CrossRef][Green Version] - King, S.F. Tri-bimaximal-Cabibbo mixing. Phys. Lett. B
**2012**, 718, 136–142. [Google Scholar] [CrossRef][Green Version] - Chen, P.; Chulia, S.C.; Ding, G.J.; Srivastava, R.; Valle, J.W.F. Realistic tribimaximal neutrino mixing. Phys. Rev. D
**2018**, 98, 055019. [Google Scholar] [CrossRef][Green Version] - Kajiyama, Y.; Raidal, M.; Strumia, A. The Golden ratio prediction for the solar neutrino mixing. Phys. Rev. D
**2007**, 76, 117301. [Google Scholar] [CrossRef][Green Version] - Irwin, K.; Amaral, M.M.; Aschheim, R.; Fang, F. Quantum walk on a spin network and the Golden ratio as the fundamental constants of nature. In Proceedings of the Fourth International Conference on the Nature and Ontology of Spacetime, Varna, Bulgaria, 30 May–2 June 2016; pp. 117–160. [Google Scholar]
- Jurciukonis, D.; Lavoura, L. Group-theoretical search for rows or columns of the lepton mixing matrix. J. Phys. G Nucl. Part. Phys.
**2017**, 44, 045003. [Google Scholar] [CrossRef][Green Version] - King, S.F.; Neder, T.; Stuart, A.J. Lepton mixing predictions from Δ(6n
^{2}) family symmetry. Phys. Lett. B**2013**, 726, 312. [Google Scholar] [CrossRef][Green Version] - King, S.F.; Ludl, P.O. Direct and semi-direct approaches to lepton mixing with a massless neutrino. J. High Energy Phys.
**2016**, 6, 147. [Google Scholar] [CrossRef][Green Version] - Yao, C.Y.; Ding, G.J. Lepton and quark mixing patterns from finite flavor symmetries. Phys. Rev. D
**2015**, 92, 096010. [Google Scholar] [CrossRef][Green Version] - Li, C.C.; Yao, C.Y.; Ding, G.J. Lepton Mixing Predictions from InfiniteGroup Series ${D}_{9n,3n}^{(1)}$ with Generalized CP. J. High Energy Phys.
**2016**, 1605, 007. [Google Scholar] [CrossRef][Green Version] - Hashimoto, K.; Okada, H. Lepton flavor model and decaying dark matter in the binary icosahedral group symmetry. arXiv
**2011**, arXiv:1110.3640. [Google Scholar] - Chaber, P.; Dziewit, B.; Holeczek, J.; Richter, M.; Zajac, S.; Zralek, M. Lepton masses and mixing in two-Higgs-doublet model. Phys. Rev. D
**2018**, 98, 055007. [Google Scholar] [CrossRef][Green Version] - Hagedorn, C.; Meronic, A.; Vitale, L. Mixing patterns from the groups Σ(nϕ). J. Phys. A Math. Theor.
**2014**, 47, 055201. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Gedik, Z. Magic informationally complete POVMs with permutations. R. Soc. Open Sci.
**2017**, 4, 170387. [Google Scholar] [CrossRef][Green Version] - Planat, M. The Poincaré half-plane for informationally complete POVMs. Entropy
**2018**, 20, 16. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Universal quantum computing and three-manifolds. Symmetry
**2018**, 10, 773. [Google Scholar] [CrossRef][Green Version] - Fuchs, C.A. On the quantumness of a Hibert space. Quant. Inf. Comp.
**2004**, 4, 467–478. [Google Scholar] - DeBrota, J.B.; Fuchs, C.A.; Stacey, B.C. Analysis and synthesis of minimal informationally complete quantum measurements. arXiv
**2018**, arXiv:1812.08762. [Google Scholar] - Planat, M. Pauli graphs when the Hilbert space dimension contains a square: Why the Dedekind psi function? J. Phys. A Math. Theor.
**2011**, 44, 045301. [Google Scholar] [CrossRef][Green Version] - López, A.V.; López, J.V. Classification of finite groups according to the number of conjugacy classes. Israel J. Math.
**1985**, 51, 305. [Google Scholar] [CrossRef] - Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions Edition 2.23; University of Sidney: Sidney, Australia, 2017; p. 5914. [Google Scholar]
- Li, C.C.; Ding, G.J. Lepton mixing in A
_{5}family symmetry and generalized CP. J. High Energy Phys.**2015**, 1505, 100. [Google Scholar] [CrossRef][Green Version] - Varzielas, I.M.; Lavoura, L. Golden ratio lepton mixing and non zero reactor angle with A
_{5}. J. Phys. G Nucl. Part. Phys.**2014**, 41, 055005. [Google Scholar] [CrossRef][Green Version] - Rong, S.J. Lepton mixing patterns from PSL
_{2}(7) with a generalized CP symmetry. Adv. High Energy Phys.**2020**, 2020, 6120803. [Google Scholar] [CrossRef] - Yao, C.Y.; Ding, G.J. CP Symmetry and Lepton Mixing from a Scan of Finite Discrete Groups. Phys. Rev. D
**2016**, 94, 073006. [Google Scholar] [CrossRef][Green Version] - Chen, M.C.; Fallbacher, M.; Mahanthappa, K.; Ratz, M.; Trautner, A. CP violation from finite groups. Nucl. Phys. B
**2014**, 883, 267. [Google Scholar] [CrossRef][Green Version] - Karozas, A.; King, S.F.; Leontaris, G.L.; Meadowcroft, A.K. Phenomenological implications of a minimal F-theory GUT with discrete symmetry. J. High Energy Phys.
**2015**, 10, 41. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**An angular picture of the three generations of quarks and leptons. The blue and black pancakes have isospin $1/2$ and $-1/2$, respectively. The inner and outer rings have weak hypercharges $\frac{1}{3}$ and $-1$, respectively.

**Table 1.**(1) The three generations of up-type quarks (up, charm and top) and of down-type quarks (down, strange and bottom) and, (2) the three generations of leptons (electron, muon and tau) and their partner neutrinos. The symbols Q, ${T}_{3}$ and ${Y}_{W}$ are for charge, isospin and weak hypercharge, respectively. They satisfy the equation $Q={T}_{3}+\frac{1}{2}{Y}_{W}$.

Matter | Type 1 | Type 2 | Type 3 | Q | ${\mathit{T}}_{3}$ | ${\mathit{Y}}_{\mathit{W}}$ |
---|---|---|---|---|---|---|

(1) quarks | u | c | t | 2/3 | 1/2 | 1/3 |

d | s | b | −1/3 | −1/2 | 1/3 | |

(2) leptons | e | $\mu $ | $\tau $ | −1 | −1/2 | −1 |

${\nu}_{e}$ | ${\nu}_{\mu}$ | ${\nu}_{\tau}$ | 0 | 1/2 | −1 |

**Table 2.**Experimental values of the angles in degrees for mixing patterns of quarks (in the CKM matrix) and leptons (in the PMNS matrix).

Angles (in Degrees) | ${\mathit{\theta}}_{12}$ | ${\mathit{\theta}}_{13}$ | ${\mathit{\theta}}_{23}$ | ${\mathit{\delta}}_{\mathit{CP}}$ |
---|---|---|---|---|

quark mixings | 13.04 | 0.201 | 2.38 | 71 |

lepton mixings | 33.62 | 8.54 | 47.2 | −90 |

**Table 3.**List of the $16+2$ groups with the number of conjugacy classes $cc\le 36$ that satisfy rules (a) and (b). As mentioned in Section 3, groups $(294,7)$ and $(384,568)$ need two CP phases to become viable models. The smallest permutation representation on $k\times l$ letters stabilizes the n-partite graph ${K}_{k}^{l}$ given in the fourth column. The group $\Delta (6\times {n}^{2})$ is isomorphic to ${\mathbb{Z}}_{n}^{2}\u22ca{S}_{3}$. A reference is given in the last column if a viable model for quark and/or lepton mixings can be obtained. The extra cases with reference † and ‡ can be found in [18,21], respectively.

Group | Name or Signature | $\mathit{cc}$ | Graph | Ref. |
---|---|---|---|---|

SmallGroup(24,12) | ${S}_{4}$, $\Delta (6\times {2}^{2})$ | 5 | ${K}_{4}$ | [15] |

SmallGroup(120,5) | 2I, SL(2, 5) | 9 | ${K}_{5}^{3}$ | [20] †,‡ |

SmallGroup(150,5) | $\Delta (6\times {5}^{2})$ | 13 | ${K}_{5}^{3}$ | [2,15,16] |

SmallGroup(72,42) | ${\mathbb{Z}}_{4}\times {S}_{4}$ | 15 | ${K}_{3}^{4}$ | [9] |

SmallGroup(216,95) | $\Delta (6\times {6}^{2})$ | 19 | ${K}_{6}^{3}$ | [15] |

SmallGroup(294,7) | $\Delta (6\times {7}^{2})$ | 20 | ? | [31] |

SmallGroup(72,3) | ${Q}_{8}\u22ca{\mathbb{Z}}_{9}$ | 21 | ${K}_{2}^{3}$ | [9] |

SmallGroup(162,12) | ${\mathbb{Z}}_{3}^{2}\u22ca({\mathbb{Z}}_{3}^{2}\u22ca{\mathbb{Z}}_{2})$ | 22 | ${K}_{9}^{3}$ | [2,15,18] |

SmallGroup(162,14) | ${\mathbb{Z}}_{3}^{2}\u22ca({\mathbb{Z}}_{3}^{2}\u22ca{\mathbb{Z}}_{2})$, ${D}_{9,3}^{(1)}$ | 22 | ${K}_{9}^{3}$ | [2,15,19] |

SmallGroup(384,568) | $\Delta (6\times {8}^{2})$ | 24 | ? | [31] |

SmallGroup(648,532) | $\Sigma (216\times 3)$, ${\mathbb{Z}}_{3}\u22ca({\mathbb{Z}}_{3}\u22caSL(2,3))$ | 24 | ? | [15,22] |

SmallGroup(648,533) | Q(648), ${\mathbb{Z}}_{3}\u22ca({\mathbb{Z}}_{3}\u22caSL(2,3))$ | 24 | ? | [15,17] |

SmallGroup(120,37) | ${\mathbb{Z}}_{5}\times {S}_{4}$ | 25 | ${K}_{5}^{4}$ | † |

SmallGroup(360,51) | ${\mathbb{Z}}_{3}\times SL(2,5)$ | 27 | ${K}_{12}^{6}$ | † |

SmallGroup(162,44) | ${\mathbb{Z}}_{3}^{2}\u22ca({\mathbb{Z}}_{3}^{2}\u22ca{\mathbb{Z}}_{2})$ | 30 | ${K}_{9}^{3}$ | [15] |

SmallGroup(600,179) | $\Delta (6\times {10}^{2})$ | 33 | ${K}_{10}^{3}$ | [2,15,16] |

SmallGroup(168,45) | ${\mathbb{Z}}_{7}\times {S}_{4}$ | 35 | ${K}_{7}^{4}$ | † |

SmallGroup(480,221) | ${\mathbb{Z}}_{8}.{A}_{5}$, $SL(2,5).{\mathbb{Z}}_{4}$ | 36 | ${K}_{8}^{6}$ | ‡ |

**Table 4.**For each of the first three small groups considered in our Table 3 and the group $(294,7)$ added in Section 3, for each character, the table provides the dimension of the representation and the rank of the Gram matrix obtained under the action of the corresponding Pauli group. Bold characters are for faithful representations. According to our requirement, each selected group has both 2- and 3-dimensional characters (with at least one of them faithful) that are fiducial states for an informationally complete POVM (or MIC) with the rank of Gram matrix equal to ${d}^{2}$. The Pauli group performing this action is a d-dit or a 2-qutrit (2QT) for the group $(120,5)=SL(2,5)=2I$.

Group | d | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(24,12) | 5 | 1 | 1 | 2 | 3 | 3 | ||||||||

5-dit | 5 | 21 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||

(120,5) | 9 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | ||||

9-dit | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 79 | ${d}^{2}$ | |||||

2QT | 9 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||

(150,5) | 13 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 |

13-dit | 13 | 157 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

(294,7) | 20 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

20-dit | 20 | 349 | 388 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

6 | 6 | 6 | 6 | 6 | ||||||||||

390 | 390 | 390 | 398 | 398 |

**Table 5.**List of considered groups with number of conjugacy classes $cc>36$ that satisfy rule (a) (presumably (b) as well) and have been considered before as valid groups for quark/lepton mixing. A reference is given in the last column if a viable model for quark or/and lepton mixings can be obtained. The question mark means that the minimal permutation representation could not be obtained.

Group | Name or Signature | cc | Graph | Ref. |
---|---|---|---|---|

SmallGroup(726,5) | $\Delta (6\times {11}^{2})$ | 38 | ${K}_{11}^{3}$ | [15,18] |

SmallGroup(648,259) | $({\mathbb{Z}}_{18}\times {\mathbb{Z}}_{6})\u22ca{S}_{3}$, ${D}_{18,6}^{(1)}$ | 49 | ${K}_{18}^{3}$ | [2,15,18,19] |

SmallGroup(648,260) | ${\mathbb{Z}}_{3}^{2}\u22caSmallGroup(72,42)$ | 49 | ${K}_{18}^{3}$ | [2,15,18,19] |

SmallGroup(648,266) | ${\mathbb{Z}}_{3}^{2}\u22caSmallGroup(72,42)$ | 49 | ${K}_{6}^{3}$ | [15] |

SmallGroup(1176,243) | $\Delta (6\times {14}^{2})$ | 59 | ${K}_{14}^{3}$ | [15,18] |

SmallGroup(972,64) | ${\mathbb{Z}}_{9}^{2}\u22ca{\mathbb{Z}}_{12}$ | 62 | ${K}_{36}^{3}$ | [15,18] |

SmallGroup(972,245) | ${\mathbb{Z}}_{9}^{2}\u22ca({\mathbb{Z}}_{2}\times {S}_{3})$ | 62 | ${K}_{18}^{3}$ | [18] |

SmallGroup(1536,408544632) | $\Delta (6\times {16}^{2})$ | 68 | ? | [2,15,16] |

SmallGroup(1944,849) | $\Delta (6\times {18}^{2})$ | 85 | ${K}_{18}^{3}$ | [15,18] |

**Table 6.**Exceptional subgroups of $SU(3)$. For each group and each character, the table provides the dimension of the representation and the rank of the Gram matrix obtained under the action of the corresponding Pauli group. Bold characters are for faithful representations.

Group | d | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(60,5), $\Sigma (60)$ | 5 | 1 | 3 | 3 | 4 | 5 | |||||||

5-dit | 5 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ||||||||

(168,42), $\Sigma (168)$ | 6 | 1 | 3 | 3 | 6 | 7 | 8 | ||||||

6-dit | 6 | ${d}^{2}$ | ${d}^{2}$ | 33 | 33 | 33 | |||||||

(108,15), $\Sigma (36\times 3)$ | 14 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

14-dit | 14 | 166 | 181 | 181 | 195 | 195 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

4 | 4 | ||||||||||||

154 | 154 | ||||||||||||

(216,88), $\Sigma (72\times 3)$ | 16 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

16-dit | 16 | 175 | 175 | 157 | 233 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

2Quartits | 16 | 121 | 149 | 125 | 200 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 3 | 3 | 8 | ||||||||||

16-dit | ${d}^{2}$ | 222 | 222 | 144 | |||||||||

2Quartits | ${d}^{2}$ | 118 | 118 | 144 | |||||||||

(1080,260), $\Sigma (360\times 3)$ | 17 | 1 | 3 | 3 | 3 | 3 | 5 | 5 | 6 | 6 | 8 | 8 | 9 |

17-dit | 17 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

9 | 9 | 10 | 15 | 15 | |||||||||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |||||||||

(648,532),$\Sigma (216\times 3)$ | 24 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |

24-dit | 24 | 527 | 527 | 562 | ${d}^{2}$ | ${d}^{2}$ | 560 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

3 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 9 | 9 | ||

${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 564 | ${d}^{2}$ | ${d}^{2}$ | 552 | 552 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K.
Informationally Complete Characters for Quark and Lepton Mixings. *Symmetry* **2020**, *12*, 1000.
https://doi.org/10.3390/sym12061000

**AMA Style**

Planat M, Aschheim R, Amaral MM, Irwin K.
Informationally Complete Characters for Quark and Lepton Mixings. *Symmetry*. 2020; 12(6):1000.
https://doi.org/10.3390/sym12061000

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin.
2020. "Informationally Complete Characters for Quark and Lepton Mixings" *Symmetry* 12, no. 6: 1000.
https://doi.org/10.3390/sym12061000