symmetry
S S
Article
Informationally Complete Characters for Quark and
Lepton Mixings
Michel Planat 1,*
, Raymond Aschheim 2
, Marcelo M. Amaral 2
and Klee Irwin 2
1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15 B Avenue des
Montboucons, F-25044 Besançon, France
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; raymond@QuantumGravityResearch.org (R.A.);
Marcelo@quantumgravityresearch.org (M.M.A.); Klee@quantumgravityresearch.org (K.I.)
* Correspondence: michel.planat@femto-st.fr
Received: 13 May 2020; Accepted: 9 June 2020; Published: 11 June 2020






Abstract: A popular account of the mixing patterns for the three generations of quarks and leptons
is through the characters κ of a finite group G. Here, we introduce a d-dimensional Hilbert space
with d = cc(G), the number of conjugacy classes of G. Groups under consideration should follow
two rules, (a) the character table contains both two- and three-dimensional representations with at
least one of them faithful and (b) there are minimal informationally complete measurements under
the action of a d-dimensional Pauli group over the characters of these representations. Groups with
small d that satisfy these rules coincide in a large part with viable ones derived so far for reproducing
simultaneously the CKM (quark) and PNMS (lepton) mixing matrices.
Keywords:
informationally complete characters; quark and lepton mixings; CP violation;
Pauli groups
PACS: 03.67.-a; 12.15.Ff; 12.15.Hh; 03.65.Fd; 98.80.Cq
1. Introduction
In the standard model of elementary particles and according to current experiments, there exist
three generations of matters but we do not know why. Matter particles are fermions of spin 1/2
and comprise the quarks (responsible for the strong interactions) and leptons (responsible for the
electroweak interactions as shown in Table 1 and Figure 1).
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, T3 and YW are for charge, isospin and weak hypercharge,
respectively. They satisfy the equation Q = T3 + 12 YW .
Matter
Type 1
Type 2
Type 3
Q
T3
YW
(1) quarks
u
c
t
2/3
1/2
1/3
d
s
b
−1/3 −1/2
1/3
(2) leptons
e
µ
τ
−1
−1/2 −1
νe
νµ
ντ
0
1/2
−1
Symmetry 2020, 12, 1000; doi:10.3390/sym12061000
www.mdpi.com/journal/symmetry
Symmetry 2020, 12, 1000
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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
1
3 and −1, respectively.
In order to explain the CP-violation (the non-invariance of interactions under the combined
action of charged-conjugation (C) and parity (P) transformations) in quarks, Kobayashi and Maskawa
introduced the so-called Cabibbo-Kobayashi-Maskawa unitary matrix (or CKM matrix) that describes
the probability of transition from one quark i to another j. These transitions are proportional to |Vij|2,
where the Vij’s are entries in the CKM matrix [1,2]
UCKM =
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
with |UCKM| ≈
0.974 0.225 0.004
0.225 0.973 0.041
0.009 0.040 0.999
 .
There is a standard parametrization of the CKM matrix with three Euler angles θ12 (the Cabbibo
angle), θ13, θ23, and the CP-violating phase δCP. Taking sij = sin(θij) and cij = cos(θij), the CKM
matrix reads
1
0
0
0
c23
s23
0 −s23
c23


c13
0
s13e−iδCP
0
1
0
−s13eiδCP
0
c13

 c12
s12 0
−s12
c12 0
0
0
1
 .
Similarly, the charged leptons e, µ and τ are related to three generations of flavors of neutrinos
νe, νµ and ντ in the charged-current weak interaction. Neutrino mass mi can be deduced with
probability |Uαi|2, where the Uαi’s are the amplitudes of mass eigenstates i in flavor α. The so-called
Pontecorvo–Maki–Nakagawa–Sakata unitary matrix (or PMNS matrix) is as follows [3]
UPMNS =
Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ1 Uτ2 Uτ3

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with |UPMNS| ≈
0.799→ 0.844 0.516→ 0.582 0.141→ 0.156
0.242→ 0.494 0.467→ 0.678 0.639→ 0.774
0.284→ 0.521 0.490→ 0.695 0.615→ 0.754
 ,
where the entries in the matrix mean the range of values allowed by present day experiments.
As for the CKM matrix, the three mixing angles are denoted θ12, θ13, θ23, and the CP-violating
phase is called δCP.
The current experimental values of angles for reproducing entries in the CKM and PMNS matrices
are in Table 2.
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)
θ12
θ13
θ23
δCP
quark mixings
13.04
0.201
2.38
71
lepton mixings
33.62
8.54
47.2 −90
Over the last twenty years, a paradigm has emerged wherein there may exist an underlying
discrete symmetry jointly explaining the mixing patterns of quarks and leptons [4,5]. This assumption
follows from the fact that the CKM matrix is found to be closed to the identity matrix and the entries in
the PMNS matrix are found to be of order 1 except for the almost vanishing Ue3. A puzzling difference
between quark and lepton mixing lies in the fact that there is much more neutrino mixing than mixing
between the quark flavors. Up and down quark matrices are only slightly misaligned, while there
exists a strong misalignment of charged leptons with respect to neutrino mass matrices. A valid model
should account for these features.
The standard model essentially consists of two continuous symmetries, the electroweak
symmetry SU(2)×U(1) (that unifies the electromagnetic and weak interactions) and the quantum
chromodynamics symmetry SU(3) (that corresponds to strong interactions). There are several
puzzles not explained within the standard model, including the flavor mixing patterns, the fermion
masses, and the CP violations in the quark and lepton sectors. There are astonishing numerical
coincidences such as the Koide formula for fermion masses [6,7], the quark-lepton complementarity
relations θquark
12 + θ
lepton
12
≈ π/4, θquark
23 ± θ
lepton
23
≈ π/4 [8] and efficient first order models such as
the tribimaximal model [9–12] and the “Golden ratio” model [13,14]. For instance, tribimaximal
mixing gives values of angles as θlepton
12 = sin
−1( 1√
3
) ≈ 35.3◦, θlepton
23 = 45
◦, θlepton
13 = 0 and δCP = 0,
compatible with earlier data. Such a model could be made more realistic by taking two CP-phases
instead of one [12]. In Reference [14], the conjecture is that reality is information-theoretic as its core
and the Golden Ratio is the fundamental dimensionless constant of Nature.
Currently, many discrete models of quark-lepton mixing patterns are based on the representations
of finite groups that are both subgroups of U(2) and U(3) [15–22]. In the same spirit, we add to this
body of knowledge by selecting valid subgroups of unitary groups with a criterion borrowed from the
theory of generalized quantum measurements.
One needs a quantum state (called a fiducial state) and one also requires that such a state
be informationally complete under the action of a d-dimensional Pauli group Pd. If such a state
is not an eigenstate of a d-dimensional Pauli group, it allows one to perform universal quantum
computation [23–25]. In the above papers, valid states belong to the eigenstates of mutually commuting
permutation matrices in a permutation group derived from the coset classes of a free group with
relations. From here, the fiducial state will have to be selected from the characters κ of a finite
group G with the number of conjugacy classes d = cc(G) defining the Hilbert space dimension.
Groups under consideration should obey two rules (a) the character table of G contains both 2-
and 3-dimensional representations with at least one of them faithful and (b) there are minimal
informationally complete measurements under the action of a d-dimensional Pauli group over the
Symmetry 2020, 12, 1000
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characters of these representations. The first criterion is inspired by the current understanding of
quark and lepton mixings (and the standard model) and the second one by the theory of magic states
in quantum computing [23]. Since matter particles are spin 1/2 fermions, it is entirely consistent to see
them under the prism of quantum measurements.
In the rest of this introduction we recall what we mean by a minimal informationally complete
quantum measurement (or MIC). In Section 2, we apply criteria (a) and (b) to groups with small cc ≤ 36,
where we can perform the calculations. Then we extrapolate to some other groups with cc > 36. Most
groups found from this procedure fit the current literature as being viable for reproducing lepton and
quark mixing patterns. In Section 3, we examine the distinction between generalized CP symmetry
and CP violation and apply it to our list of viable groups.
Minimal Informationally Complete Quantum Measurements
Let Hd be a d-dimensional complex Hilbert space and {E1, . . . , Em} be a collection of positive
semi-definite operators (POVM) that sum to the identity. Taking the unknown quantum state as a rank
1 projector ρ = |ψ〉 〈ψ| (with ρ2 = ρ and tr(ρ) = 1), the i-th outcome is obtained with a probability
given by the Born rule p(i) = tr(ρEi). A minimal and informationally complete POVM (or MIC)
requires d2 one-dimensional projectors Πi = |ψi〉 〈ψi|, with Πi = dEi, such that the rank of the Gram
matrix with elements tr(ΠiΠj), is precisely d2.
With a MIC, the complete recovery of a state ρ is possible at a minimal cost from the probabilities
p(i). In the best case, the MIC is symmetric and called a SIC with a further relation
∣∣〈ψi|ψj〉∣∣2 =
tr(ΠiΠj) =
dδij+1
d+1 so that the density matrix ρ can be made explicit [26,27].
In our earlier references [23,24], a large collection of MICs are derived. They correspond to
Hermitian angles
∣∣〈ψi|ψj〉∣∣i
6=j ∈ A = {a1, . . . , al} belonging to a discrete set of values of small
cardinality l. They arise from the action of a Pauli group Pd [28] on an appropriate magic state
pertaining to the coset structure of subgroups of index d of a free group with relations.
Here, an entirely new class of MICs in the Hilbert space Hd, relevant for the lepton and quark
mixing patterns, is obtained by taking fiducial/magic states as characters of a finite group G possessing
d conjugacy classes and using the action of a Pauli group Pd on them.
2. Informationally Complete Characters for Quark/Lepton Mixing Matrices
The standard classification of small groups is from their cardinality. Finite groups relevant to
quark and lepton mixings are listed accordingly [9,15,18]. We depart from this habit by classifying the
small groups G of interest versus the number d = cc(G) of their conjugacy classes. This motivation is
due to the application of criterion (b), where we need to check whether the action of a Pauli group in
the d-dimensional Hilbert spaceHd results in a minimal informationally complete POVM (or MIC).
A list of finite groups G according to the number of their conjugacy classes (complete only
up to d ≤ 12) is in Ref. [29]. It can also be easily recovered with a simple code in MAGMA [30].
For our application to quark and lepton mixings, we need much higher d. In practice, we use existing
tables of subgroups of U(3) (of cardinality up to 2000 in [9,15,18] and up to 1025 in [21] to select our
group candidates.
Table 3 gives the list of 16 + 2 small groups with cc ≤ 36 found to satisfy the two following rules:
(a) the character table of G contains both 2- and 3-dimensional representations with at least one of
them faithful and (b) the quantum measurement is informationally complete under a d-dimensional
Pauli group.
According to the quoted references in column 5 of Table 3, the 16 groups lead to good models for
the absolute values of entries in the CKM and PMNS matrices except for the ones that have the factor
SL(2, 5) in their signature. The two extra groups (294, 7) = ∆(6× 72) and (384, 568) = ∆(6× 82) arise
when one takes into account the generalized CP symmetry, as in Section 3.
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Table 3. List of the 16 + 2 groups with the number of conjugacy classes cc ≤ 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× l letters stabilizes the n-partite graph
Klk given in the fourth column. The group ∆(6× n
2) is isomorphic to Z2n o S3. 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
cc Graph
Ref.
SmallGroup(24,12)
S4, ∆(6× 22)
5
K4
[15]
SmallGroup(120,5)
2I, SL(2, 5)
9
K35
[20] †,‡
SmallGroup(150,5)
∆(6× 52)
13
K35
[2,15,16]
SmallGroup(72,42)
Z4 × S4
15
K43
[9]
SmallGroup(216,95)
∆(6× 62)
19
K36
[15]
SmallGroup(294,7)
∆(6× 72)
20
?
[31]
SmallGroup(72,3)
Q8 oZ9
21
K32
[9]
SmallGroup(162,12)
Z23 o (Z23 oZ2)
22
K39
[2,15,18]
SmallGroup(162,14)
Z23 o (Z23 oZ2), D
(1)
9,3
22
K39
[2,15,19]
SmallGroup(384,568) ∆(6× 82)
24
?
[31]
SmallGroup(648,532) Σ(216× 3), Z3 o (Z3 o SL(2, 3))
24
?
[15,22]
SmallGroup(648,533)
Q(648) , Z3 o (Z3 o SL(2, 3))
24
?
[15,17]
SmallGroup(120,37)
Z5 × S4
25
K45
†
SmallGroup(360,51)
Z3 × SL(2, 5)
27
K612
†
SmallGroup(162,44)
Z23 o (Z23 oZ2)
30
K39
[15]
SmallGroup(600,179) ∆(6× 102)
33
K310
[2,15,16]
SmallGroup(168,45)
Z7 × S4
35
K47
†
SmallGroup(480,221) Z8.A5, SL(2, 5).Z4
36
K68
‡
Details are in Table 4 for the first three groups and the group (294, 7). Full results are found in
Tables A1 and A2 of the Appendix A.
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 d2. 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
d2
d2
d2
(120,5)
9
1
2
2
3
3
4
4
5
6
9-dit
9
d2
d2
d2
d2
d2
d2
79
d2
2QT
9
d2
d2
d2
d2
d2
d2
d2
d2
(150,5)
13
1
1
2
3
3
3
3
3
3
3
3
6
6
13-dit
13
157
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
(294,7)
20
1
1
2
3
3
3
3
3
3
3
3
3
3
20-dit
20
349
388
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
6
6
6
6
6
390
390
390
398
398
Table 5 gives an extrapolation to groups with higher cc where criterion (a) is satisfied but where
(b) could not be checked. Most groups in the two tables have been found to be viable models, and
several of them belong to known sequences.
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In Tables 3 and 5, the first column is the standard small group identifier in which the first entry
is the order of the group (as in [15]). In the second column, one finds the signature in terms of a
direct product (with the symbol ×), a semidirect product (with the symbol o), a dot product (with
the symbol .) or a member of a sequence of groups such as the ∆(6× n2) sequence found to contain
many viable groups for quark and lepton mixings. The third column gives the number of irreducible
characters/conjugacy classes. Other information is about the geometry of the group. To obtain this
geometry, one first selects the smallest permutation representation on k× l letters of G. Then one looks
at the two-point stabilizer subgroup Gs of smallest cardinality in the selected group G. The incidence
matrix of such a subgroup turns out to be the l-partite graph Klk that one can identify from the graph
spectrum. Such a method is already used in our previous papers about magic state type quantum
computing [23–25], where other types of geometries have been found. Finally, column 5 refers to
papers where the group under study leads to a viable model both for quark and lepton mixing patterns.
The recent reference [21] is taken separately from the other references singled out with the index † in
the tables. It is based on the alternative concept of a two-Higgs-doublet model.
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)
∆(6× 112)
38
K311
[15,18]
SmallGroup(648,259)
(Z18 ×Z6)o S3, D
(1)
18,6
49
K318
[2,15,18,19]
SmallGroup(648,260)
Z23 o SmallGroup(72, 42)
49
K318
[2,15,18,19]
SmallGroup(648,266)
Z23 o SmallGroup(72, 42)
49
K36
[15]
SmallGroup(1176,243)
∆(6× 142)
59
K314
[15,18]
SmallGroup(972,64)
Z29 oZ12
62
K336
[15,18]
SmallGroup(972,245)
Z29 o (Z2 × S3)
62
K318
[18]
SmallGroup(1536,408544632)
∆(6× 162)
68
?
[2,15,16]
SmallGroup(1944,849)
∆(6× 182)
85
K318
[15,18]
2.1. Groups in the Series ∆(6n2) and More Groups
An important paper dealing with the series ∆(6n2) ∼= Z2n o S3 as a good model for lepton mixing
is [16]. A group in this series has to be spontaneously broken into two subgroups, one abelian subgroup
ZTm in the charged lepton sector and a Klein subgroup ZS2 ×ZU2 in the neutrino sector (with neutrinos
seen as Majorana particles). The superscripts S, T and U refer to the generators of their corresponding
Zm group in the diagonal charged lepton basis. In this particular model, there is trimaximal lepton
mixing with (so called reactor angle) θ13 fixed up to a discrete choice, an oscillation phase zero or π
and the (so-called atmospheric angle) θ23 = 45◦ ± θ13/
√
2.
It is shown in [2] (Table I) that two groups in this series with n = 10 and n = 16 provide
leading order leptonic mixing patterns within 3-sigma of current best fit with acceptable entries in
the CKM matrix. The small group (648, 259) = D(1)
18,6 also satisfies this requirement. Additionally, if
one accepts that neutrinos are Dirac particles, the residual symmetry group of neutrino masses is no
longer restricted to the Klein group but may be any abelian group. In such a case, four small groups
which are ∆(6× 52) and the small groups (162, 10), (162, 12) and (162, 14) = D(1)
9,3 predict acceptable
entries for the quark and lepton mixing matrices [2] (Table II). It is noticeable that our small selection
of groups (from requirements (a) and (b)) include all of them except for the group (162, 10) whose
two-dimensional representations are not MICs.
Still assuming that neutrinos are Dirac particles and with loose enough constraints on Vus,
paper [18] includes ∆-groups with n = 9 (it does not lie in our Table 3) and n = 14 in their selection, as
well as groups (648, 259), (648, 260) and (648, 266), the latter groups are in our Table 5. Additional
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material [18] provides very useful information about the ability of a group to be a good candidate for
modeling the mixing patterns. According to this reference, the groups ∆(6× n2) with n = 10, 11, 14
and 18, and small groups (972, 64) and (972, 245), that are in our tables, also match Dirac neutrinos
with a 3-sigma fit and quark mixing patterns for triplet assignment.
Three extra groups (120, 5) (the binary icosahedral group SL(2, 5) = 2I), (360, 51) = Z3× SL(2, 5)
and (480, 221) = SL(2, 5).Z4 in our tables, whose signature has a factor equal to the binary icosahedral
group 2I, can be assigned with a doublet and a singlet for quarks but cannot be generated by the
residual symmetries in the lepton sector.
2.2. Exceptional Subgroups of SU(3)
The viability of so-called exceptional groups of SU(3) for lepton mixings have been studied in [22]
by assuming neutrinos to be either Dirac or Majorana particles. These subgroups are listed according
to the number of their conjugacy classes in Table 6. They are Σ(60) ∼= A5 (a subgroup of SO(3)),
Σ(168) ∼= PSL(2, 7), Σ(36× 3), Σ(72× 3), Σ(360× 3) and Σ(216× 3). Only group Σ(360× 3) has
Klein subgroups and thus supports a model with neutrinos as Majorana particles. Group Σ(216× 3) is
already in our Table 3 and potentially provides a valid model for quark/lepton mixings by assuming
neutrinos are Dirac particles.
According to our Table 6, all these exceptional groups have informationally complete characters in
regard to most of their faithful three-dimensional representations. Another useful piece of information
is about groups Σ(60) and Σ(360× 3) that are informationally complete in regard to their 5-dimensional
representations. Models based on the A5 family symmetry are in [31,32].
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), Σ(60)
5
1
3
3
4
5
5-dit
5
d2
d2
d2
d2
(168,42), Σ(168)
6
1
3
3
6
7
8
6-dit
6
d2
d2
33
33
33
(108,15), Σ(36× 3)
14
1
1
1
1
3
3
3
3
3
3
3
3
14-dit
14
166
181
181
195
195
d2
d2
d2
d2
d2
d2
4
4
154
154
(216,88), Σ(72× 3)
16
1
1
1
1
2
3
3
3
3
3
3
3
16-dit
16
175
175
157
233
d2
d2
d2
d2
d2
d2
d2
2Quartits
16
121
149
125
200
d2
d2
d2
d2
d2
d2
d2
3
3
3
8
16-dit
d2
222
222
144
2Quartits
d2
118
118
144
(1080,260), Σ(360× 3)
17
1
3
3
3
3
5
5
6
6
8
8
9
17-dit
17
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
9
9
10
15
15
d2
d2
d2
d2
d2
(648,532),Σ(216× 3)
24
1
1
1
2
2
2
3
3
3
3
3
3
24-dit
24
527
527
562
d2
d2
560
d2
d2
d2
d2
d2
3
6
6
6
6
6
6
8
8
8
9
9
d2
d2
d2
d2
d2
d2
d2
564
d2
d2
552
552
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3. Generalized CP Symmetry, CP Violation
Currently, many models focus on the introduction of a generalized CP symmetry in the lepton
mixing matrix [12,31,33]. The Dirac CP phase δCP = δ13 for leptons is believed to be around −π/2 [33].
A set of viable models with discrete symmetries including generalized CP symmetry has been derived
in [34], where full details about the so-called semidirect approach and its variant are provided. Most
finite groups used for quark/lepton mixings without taking into account the CP symmetry survive as
carrying generalized CP symmetries in the model described in [34]. It is found that two extra groups
(294, 7) = ∆(6× 72) and (384, 568) = ∆(6× 82), that have triplet assignments for the quarks, can be
added. This confirms the relevance of ∆ models in this context. Group (294, 7) was added to our short
Table 4, where we see that all of its 2- and 3-dimensional characters are informationally complete.
We follow Reference [35] in distinguishing generalized CP symmetry from a “physical” CP
violation. A “physical” CP violation is a prerequisite for baryogenesis that is the matter-antimatter
asymmetry of elementary matter particles. The generalized CP symmetry was introduced as a way
of reproducing the absolute values of the entries in the lepton and quark mixing matrices and, at the
same time, explaining or predicting the phase angles. A physical CP violation, on the other hand,
exchanges particles and antiparticles and its finite group picture had to be clarified.
It is known that the exchange between distinct conjugacy classes of a finite group G is controlled
by the outer automorphisms u of the group. Such (non trivial) outer automorphisms have to be
class-inverting to correspond to a physical CP violation [35]. This is equivalent to a relation obeyed by
the automorphism u : G → G that maps every irreducible representation ρri to its conjugate
ρri (u(g)) = Uri ρri (g)
∗U†r , ∀g ∈ G and ∀i,
with Uri a unitary symmetric matrix.
A criterion that ensures that this relation is satisfied is given in terms of the so-called twisted
Frobenius-Schur indicator over the character κri
FS(n)
u (ri) =
(dim ri)(n−1)
|G|n
∑
gi∈G
κri (g1u(g1) · · · gnu(gn)) = ±1, ∀i,
where n = ord(u)/2 if ord(u) is even and n = ord(u) otherwise.
Following this criterion, there are three types of groups
1. the groups of type I: there is at least one representation ri for which FS
(n)
u (ri) = 0, these groups
correspond to a physical CP violation,
2. groups of type II: for (at least) one automorphism u ∈ G the FSu’s for all representations are
non zero. The automorphism u can be used to define a proper CP transformation in any basis. There
are two sub-cases:
Case II A, all FSu’s are +1 for one of those u’s,
Case II B, some FSu’s are −1 for all candidates u’s.
A simple program written in the Gap software allows one to distinguish these cases [35]
(Appendix B).
Applying this code to our groups in Tables 3, 5 and 6, we find that all groups are of type II A or
type I. Type I groups corresponding to a physical CP violation are
(216, 95) = ∆(6× 62), (162, 44), (216, 88) = Σ(72× 3),
where we could check that our criteria (a) and (b) apply, the exceptional group (1080, 260) = Σ(360, 3)
in Table 6 and groups (972, 64), (972, 245), (1944, 849) = ∆(6× 183) of Table 5.
Symmetry 2020, 12, 1000
9 of 12
4. Conclusions
Selecting 2- and 3-dimensional representations of informationally complete characters has been
found to be efficient in the context of models of CKM and PMNS mixing matrices. Generalized
quantum measurements (in the form of MICs) are customary in the field of quantum information and
provide a Bayesian interpretation of quantum theory leading to an innovative approach of universal
quantum computing. The aim of this paper has been to see the mixing patterns of matter particles
through the prism of MICs. Our method has been shown to have satisfactorily predictive power
for predicting the appropriate symmetries used so far in modeling CKM/PMNS matrices and for
investigating the symmetries of CP phases.
It is admitted that the standard model has to be completed with discrete symmetries or/and
to be replaced by more general symmetries such as SU(5) or E8 ⊃ SU(5), as in F-theory [36],
to account for existing measurements on quarks, leptons and bosons, and the hypothetical dark
matter. Imposing the right constraints on quantum measurements of such particles happens to be a
useful operating approach.
Author Contributions: Conceptualization, M.P. and K.I.; methodology, M.P. and R.A.; software, M.P.; validation,
R.A. and M.M.A.; formal analysis, M.P. and M.M.A.; investigation, M.P. and M.M.A.; writing–original draft
preparation, M.P.; writing–review and editing, M.P.; visualization, R.A.; supervision, M.P. and K.I.; project
administration, K.I.; funding acquisition, K.I. All authors have read and agreed to the published version of
the manuscript.
Funding: Funding was obtained from Quantum Gravity Research in Los Angeles,CA
Conflicts of Interest: The authors declare no conflict 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 d2. 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) = Z3 × 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
d2
d2
d2
(120,5)
9
1
2
2
3
3
4
4
5
6
9-dit
9
d2
d2
d2
d2
d2
d2
79
d2
2QT
9
d2
d2
d2
d2
d2
d2
d2
d2
(150,5)
13
1
1
2
3
3
3
3
3
3
3
3
6
6
13-dit
13
157
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
(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
d2
d2
d2
d2
d2
d2
d2
d2
(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
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
3
6
6
6
d2
d2
d2
d2
(294,7)
20
1
1
2
3
3
3
3
3
3
3
3
3
3
3
3
20-dit
.
20
349
388
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
6
6
6
6
6
390
390
390
398
398
Symmetry 2020, 12, 1000
10 of 12
Table A1. Cont.
Group
d
(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
d2
d2
d2
d2
d2
d2
2
2
2
3
3
3
d2
d2
d2
d2
d2
d2
(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
d2
d2
d2
d2
d2
d2
d2
d2
3
3
3
3
3
3
6
d2
d2
d2
d2
d2
d2
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
d2
d2
d2
d2
d2
d2
d2
d2
3
3
3
3
3
3
6
d2
d2
d2
d2
d2
d2
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
d2
d2
560
d2
d2
d2
d2
d2
d2
d2
d2
3QB-QT
24
500
500
476
568
568
448
d2
d2
d2
d2
d2
d2
d2
d2
6
6
6
6
8
8
8
9
9
24-dit
d2
d2
d2
d2
564
d2
d2
552
552
3QB-QT
d2
d2
d2
d2
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
d2
d2
514
d2
d2
d2
574
574
d2
d2
d2
3QB-QT
24
532
532
481
572
572
452
572
568
568
570
570
572
575
d2
6
6
6
6
8
8
8
9
9
24-dit
d2
d2
d2
d2
563
d2
d2
478
478
3QB-QT
d2
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
d2
d2
d2
d2
3
3
3
3
3
3
3
3
3
3
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
(360,51)
27
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
3QT
27
613
613
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
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
d2
d2
d2
898
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
898
898
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
(600,179)
33
1
1
2
3
3
3
3
3
3
3
3
3
3
3
3
33-dit
33
1041
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
3
3
3
3
3
3
6
6
6
6
6
6
6
6
6
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
6
6
6
d2
d2
d2
(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
d2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
3
3
3
3
3
d2
d2
d2
d2
d2
(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
d2
d2
d2
d2
d2
d2
d2
1278
1278
1278
3
3
3
3
3
4
4
4
4
4
4
4
4
5
5
1278
d2
d2
d2
d2
1275
1278
d2
d2
d2
d2
d2
d2
1277
1273
5
5
6
6
6
6
1294
1294
1295
1295
1295
1295
Symmetry 2020, 12, 1000
11 of 12
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