Catalog of CHM
Appendix D: Multiunitary (complex) Hadamard Matrices
created: 2023-05-04
updated: 2024-06-07 by W. Bruzda
Collection of (complex) Hadamard matrices with additional symmetries based on Open Syst. Inf. Dyn. 31(2), 2250001 (2024). We distinguish here three subclasses in the set of complex Hadamard matrices H(N) of squared dimension N = d2 for d ≥ 2
- self R-dual: H = HR ∈ H(N),
- self Γ-dual: H = HΓ ∈ H(N),
- 2-unitary matrices: H and HR and HΓ are CHM,
where operations of reshuffling (R) and partial transpose (Γ) are defined by the following relations: ⟨ab|HR|cd⟩ := ⟨ac|H|bd⟩ and ⟨ab|HΓ|cd⟩ := ⟨ad|H|cb⟩, where abcd is four-index of H of dimension d × d.
Fourh case, that is
- strong 2-unitary: H = HR = HΓ ∈ H(N)
does not occur in Nature (theorem).
Filename scheme:
| realignment type | duality | self (strong) duality |
|---|---|---|
| R | RDN,d,#Λ | SRN,d,#Λ |
| Γ | GDN,d,#Λ | SGN,d,#Λ |
| 2-unitarity: both R and Γ | TUN,d,#Λ | ∅ |
Contents:
- self R-dual CHM
- self Γ-dual CHM
- 2-unitary CHM
- classification of Butson matrices (unfinished)
Self R-Dual CHM; H = HR↑
• SR9,16,3A
= exp(2j*pi*[
0 0 0 0 0 0 0 0 0 ;
0 0 0 1 1 1 2 2 2 ;
0 0 0 2 2 2 1 1 1 ;
0 1 2 0 1 2 0 1 2 ;
0 1 2 1 2 0 2 0 1 ;
0 1 2 2 0 1 1 2 0 ;
0 2 1 0 2 1 0 2 1 ;
0 2 1 1 0 2 2 1 0 ;
0 2 1 2 1 0 1 0 2 ;
]/3;
It is a self R-dual symmetric Butson-type matrix BH(9, 3) with S = (1, 1, 0).
• SR9,16,3B
= exp(2j*pi*[
0 0 0 0 0 0 0 0 0 ;
0 0 0 2 2 2 1 1 1 ;
0 0 0 1 1 1 2 2 2 ;
0 2 1 0 2 1 0 2 1 ;
0 2 1 2 1 0 1 0 2 ;
0 2 1 1 0 2 2 1 0 ;
0 1 2 0 1 2 0 1 2 ;
0 1 2 2 0 1 1 2 0 ;
0 1 2 1 2 0 2 0 1 ;
]/3;
Almost like in SR9,16,3A with swapped 1s and 2s; it is derived from the Karlsson's orbit K9(2)(ζ=3).
• SR9,4,57(p1, p2, p3, p4)
Matrix SR9,4,57 stems form the four-parameter Fourier matrix F9 transformed by appropriate diagonal unitary matrices. Its entropies read S = (1, 1, f) where f = f(p1, p2, p3, p4) ∈ [0, 1) is a certain function of free parameters (one can recover its analytic form if it has any significance...). Family SR9,4,57 is self R-dual for any parameters pj ∈ [0, 1) with generic values of defect = 4 and #Λ = 57.
Self Γ-Dual CHM; H = HΓ↑
• SG9,0,6
= exp(1j*pi*[
0 0 0 0 2 0 0 2 4 ;
0 5 3 2 1 3 2 3 3 ;
0 3 3 0 3 5 4 3 1 ;
0 2 0 2 2 2 4 0 2 ;
2 1 3 2 1 5 0 1 1 ;
0 3 5 2 5 5 2 1 5 ;
0 0 2 4 4 0 2 0 2 ;
0 3 3 4 1 1 0 5 5 ;
2 3 1 0 1 5 2 5 3 ;
]/3;
This is an isolated self Γ-dual Butson matrix BH(9, 6) with S = (1, 20/27, 1), where 20/27 = 0.(740).
• SG9,0,18 := DL · N9(0) · DR
This is an isolated matrix, where DL = diag[1, 1, 1, 1, -y4, -y3, 1, y, 1] and DR = diag[1, 1, 1, 1, -1, -y, -y3, ξ, ξ y] with y = -1/4 + sqrt(-15)/4 and ξ = 7/128 + 33*sqrt(-15)/128. Number ξ is very nice. As much as S = (1, 22259/31104, 1). :)
• SG9,16,3A(p1) := diag[1, 1, e2 i π p1] ⊗ I3 · F3 ⊗ F3
This is a one-parametric family of self Γ-dual matrices with S = (1, 0, 1), stemming (for p1 = 0) from a symmetric Butson matrix BH(9, 3). I3 is the identity matrix of size 3.
• SG9,16,3B := D · K9(2)(3) · D
Log-form of this symmetric matrix with S = (1, 3/4, 1) roughly looks like
≃ [
0 0 0 0 1 2 1 0 2 ;
0 0 0 1 2 0 0 2 1 ;
0 0 0 2 0 1 2 1 0 ;
0 1 2 0 2 1 1 1 1 ;
1 2 0 2 1 0 1 1 1 ;
2 0 1 1 0 2 1 1 1 ;
1 0 2 1 1 1 2 0 1 ;
0 2 1 1 1 1 0 1 2 ;
2 1 0 1 1 1 1 2 0 ;
]
where diagonal matrix D = diag[1, 1, 1, 1, ω2, ω, ω, ω2, 1] with ω = exp(i π/3). Note that there is conjugate on the right-hand side in matrix D. It seems that this matrix might be hidden in the previous family as a special case for some p1, but there is a little discrepancy in the form they are presented: this one is obviously undephased whereas the the previous one is only "partially" dephased. Moreover, their entropies are different.
• SG16,49,18(p1, p2) := F4(p1) ⊗ F4(p2)
This is a two-parametric family of self Γ-dual matrices with S = (1, 0, 1), stemming (for p1 = p2 = 0) from a symmetric Butson matrix BH(16, 4). No additional diagonal matrices is required.
2-Unitary CHM; all three: H, HR and HΓ are CHM↑
• TU9,2,89 := K9(2)(ζ) · P9
Matrix TU9,2,89 is a 2-unitary matrix with S = (1, 1, 1) for any valid value of ζ ⊂ D ⊂ C, where P9 is a permutation matrix representation of AME(4, 3) state.
• TU9,4,57A(p1, p2, p3, p4) := DL · F9(4)(p1, p2, p3, p4) · DR
Matrix TU9,4,57A is a 2-unitary matrix stemming from a matrix ≃ BH(9, 9) with S = (1, 1, 1), where
DL = diag(exp(2j*π*[0, 1, 1, 1, 1, 0, 0, 2, 0]/3)); DR = diag(exp(2j*π*[0, 1, 1, 1, 0, 1, 2, 2, 1]/3));
• TU9,4,57B(p1, p2, p3, p4)
This family requires attention...
• TU9,10,21(p1, p2)
This family requires attention...
• TU9,16,3A := DL · F3 ⊗ F3 · DR
Matrix TU9,16,3A is a 2-unitary Butson matrix BH(9, 3)
≃ [
0 1 1 1 0 1 2 2 1
1 0 1 2 2 1 0 1 1
1 1 0 2 0 0 0 2 0
1 2 2 0 2 0 2 2 1
1 0 1 0 0 2 2 3 3
0 0 2 2 0 0 1 3 1
0 1 1 0 2 0 3 3 2
2 1 2 2 2 1 2 3 3
0 0 2 0 1 1 3 2 3
]
with S = (1, 1, 1), where two diagonal unitary matrices are defined above for SG9,4,57.
• TU9,16,3B := D · BH9,16,3 · D†
Matrix TU9,16,3B is a 2-unitary Butson matrix BH(9, 3) with S = (1, 1, 1), where D = diag(exp(2j*π*[0, 0, 0, 0, 4, 2, 0, 2, 4]/3)).
• TU16,10,48(p1, p2)
This family requires attention...
• TU16,12,90(p1, p2, p3)
This family requires attention...
• TU16,17,20(p1)
Matrix TU16,17,20 is a one-parametric affine family of 2-unitary matrices with S = (1, 1, 1).
• TU16,17,146A(p1, p2, p3, p4)
This family requires attention...
• TU16,17,146B(p1, p2, p3, p4)
This family requires attention...
• TU16,19,26(p1, p2)
This family requires attention...
• TU16,23,26(p1, p2)
This family requires attention...
• TU16,31,20(p1)
This family requires attention...
• TU16,81,4(p1, p2) := DL(p1) · BH16,81,4 · P16 · DR(p2)
Matrix TU16,81,4 is a 2-unitary matrix with S = (1, 1, 1), where P16 is a permutation matrix representation of AME(4, 4) state and two diagonal unitary matrices read:
DL(p1) = diag([1, 1, 1, 1, 1, 1, exp(2j*π*p1), exp(2j*π*p1), ω, ω, -1, -1, -1, 1, -1, 1]); DR(p2) = diag([1, 1, 1, 1, 1, i, 1, i, exp(2j*π*p2), exp(2j*π*p2), exp(2j*π*p2), exp(2j*π*p2), 1, i, 1, i]);with ω = exp(2j*π/3).
• TU16,105,2 := BH16,105,2 · P16
Matrix TU16,105,2 is a 2-unitary Butson matrix BH(16, 2) with S = (1, 1, 1), where P16 is a permutation matrix representation of AME(4, 4) state.
Butson Classification... (unfinished)↑
show tables
contents:
BH(9, 3)
BH(9, 6)
BH(9, 9)
BH(9, 10)
BH(9, 12)
BH(9, 15)
********************************************************************************
There are 3 Butson matrices in the class BH(9, 3)
# d(B) #L(B) class D
--------------------------------------------------------------------------------
01 16 3 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
02 4 3 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
03 4 3 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
********************************************************************************
There are 17 Butson matrices in the class BH(9, 6)
# d(B) #L(B) class D
--------------------------------------------------------------------------------
01 10 6 .. ..
02 4 6 .. ..
03 4 6 .. ..
04 4 6 .. ..
05 10 6 .. ..
06 4 6 .. ..
07 0 6 CONJECTURE
08 0 6 CONJECTURE
09 16 3 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
10 4 6 1 :
11 4 3 1 :
12 8 6 1 :
13 4 3 1 :
14 4 6 1 :
15 4 6 1 :
16 12 6 1 :
17 4 6 1 :
********************************************************************************
There are 23 Butson matrices in the class BH(9, 9)
# d(B) #L(B) class D
--------------------------------------------------------------------------------
01 16 3 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
02 4 9 1 :
03 4 9 1
04 4 3 1
05 4 9 1
06 4 9 1
07 4 3 1
08 4 9 1
09 4 9 1
10 8 9 1
11 8 9 1
12 4 9 1
13 12 9 1
14 4 9 1
15 4 9 1
16 6 9 1
17 4 9 1
18 4 9 1
19 6 9 1
20 4 9 1
21 10 9 1
22 4 9 1
23 10 9 1
********************************************************************************
There is 1 Butson matrix in the class BH(9, 10)
# d(B) #L(B) class D
--------------------------------------------------------------------------------
01 2 10 .. ..
********************************************************************************
There are 65 Butson matrices in the class BH(9, 12)
# d(B) #L(B) class
--------------------------------------------------------------------------------
01 10 6 ..
02 4 6 ..
03 4 12 ..
04 4 12 ..
05 4 6 ..
06 4 6 ..
07 4 12 ..
08 4 12 ..
09 10 6 ..
10 4 6 ..
11 4 12 ..
12 4 12 ..
13 0 6 CONJECTURE
14 0 6 CONJECTURE
15 16 3 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
16 4 9 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
17 4 6 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
18 4 9 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
19 4 3 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
20 4 9 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
21 8 6 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
22 4 6 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
23 4 9 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
24 4 9 1 :
25 4 3 1
26 4 9 1
27 8 12 1
28 4 12 1
29 4 12 1
30 8 12 1
31 4 12 1
32 4 12 1
33 4 12 1
34 12 9 1
35 4 12 1
36 4 12 1
37 6 12 1
38 4 9 1
39 4 12 1
40 6 12 1
41 8 12 1
42 4 12 1
43 4 12 1
44 4 9 1
45 6 12 1
46 4 12 1
47 4 12 1
48 6 12 1
49 4 12 1
50 4 12 1
51 4 6 1
52 4 12 1
53 12 6 1
54 4 12 1
55 4 6 1
56 4 12 1
57 10 12 1
58 4 12 1
59 4 12 1
60 4 12 1
61 4 12 1 :
62 10 12 1 DL = exp(1j*pi*[0 0 0 0 4 2 0 2 4]/3), DR = DL'
63 2 12 ..
64 2 12 ..
65 2 12 ..
********************************************************************************
There are 93 Butson matrices in the class BH(9, 15)
# d(B) #L(B) class D q
--------------------------------------------------------------------------------
01 4 3 .. .. 3
02 4 9 1 .. 15 DL = exp(1j*pi*[0 0 0 0 2 4 0 4 2]/3), DR = DL (no conjugate!)
03 4 9 1 .. 15
04 4 15 1 .. 15
05 4 15 1 .. 15
06 4 9 .. .. 15
07 4 15 1 .. 15
08 4 9 .. .. 15
09 8 15 1 .. 15
10 4 15 1 .. 15
11 4 15 1 .. 15
12 16 3 .. .. 3
13 4 9 1 .. 15
14 4 9 1 .. 15
15 8 15 1 .. 15
16 4 15 1 .. 15
17 4 3 .. .. 3
18 4 9 1 .. 15
19 4 9 1 .. 15
20 4 15 1 .. 15
21 8 15 1 .. 15
22 4 9 .. .. 15
23 8 15 1 .. 15
24 4 9 .. .. 15
25 4 15 1 .. 15
26 4 9 .. .. 15
27 4 15 1 .. 15
28 4 9 .. .. 15
29 4 15 1 .. 15
30 8 15 1 .. 15
31 4 15 1 .. 15
32 4 15 1 .. 15
33 6 15 1 .. 15
34 4 15 .. .. 15
35 4 15 1 .. 15
36 4 15 1 .. 15
37 12 9 1 .. 15
38 4 15 1 .. 15
39 4 15 .. .. 15
40 6 15 1 .. 15
41 4 15 1 .. 15
42 4 9 1 .. 15
43 4 15 .. .. 15
44 4 15 1 .. 15
45 6 15 1 .. 15
46 4 9 1 .. 15
47 4 15 1 .. 15
48 6 15 1 .. 15
49 4 15 .. .. 15
50 12 9 1 .. 15
51 4 15 1 .. 15
52 6 15 1 .. 15
53 4 15 .. .. 15
54 4 15 1 .. 15
55 10 15 1 .. 15
56 4 15 .. .. 15
57 4 9 1 .. 15
58 6 15 1 .. 15
59 4 15 1 .. 15
60 4 15 .. .. 15
61 4 9 1 .. 15
62 4 15 1 .. 15
63 4 15 1 .. 15
64 10 15 .. .. 15
65 4 15 1 .. 15
66 4 15 1 .. 15
67 4 15 .. .. 15
68 4 15 1 .. 15
69 4 15 1 .. 15
70 4 15 .. .. 15
71 4 15 1 .. 15
72 4 15 1 .. 15
73 10 15 .. .. 15
74 4 15 1 .. 15
75 4 15 .. .. 15
76 4 15 1 .. 15
77 6 15 1 .. 15
78 4 15 .. .. 15
79 6 15 1 .. 15
80 4 15 1 .. 15
81 4 15 .. .. 15
82 10 15 1 .. 15
83 4 15 .. .. 15
84 4 15 1 .. 15
85 4 15 .. .. 15
86 8 15 1 .. 15
87 4 15 .. .. 15
88 4 15 .. .. 15
89 4 15 1 .. 15
90 4 15 .. .. 15
91 4 15 1 .. 15
92 4 15 .. .. 15
93 4 15 .. .. 15
********************************************************************************