Extremal Polarization States

References

Gunnar Björk, Markus Grassl, Pablo de la Hoz, Gerd Leuchs and Luis L. Sánchez-Soto,
"Stars of the quantum Universe: extremal constellations on the Poincaré sphere",
Physica Scripta, vol. 90, no. 10, September 2015, 108008.
DOI: 10.1088/0031-8949/90/10/108008
G. Björk, A. B. Klimov, P. de la Hoz, M. Grassl, G. Leuchs, L. L. Sánchez-Soto,
"Extremal quantum states and their Marjorana constellations",
Physical Review A, vol. 92, no. 3, September 2015, 031801(R).
DOI: 10.1103/PhysRevA.92.031801
arXiv:1502.03446 [quant-ph]

General remarks

In the following, we give explicit examples of constellations for extremal, unpolarized, pure states for S = 1 to S = 10, corresponding to N = 2 to N = 20 photons. For all examples, the degree M of unpolarization, i.e., A^(S)_M = 0, is the highest we found, and it matches the largest degree t for which a spherical t design with N points is known to exist. For the parameters S listed in Table 1 of the article, we provide the details for the very same states. For the other cases, we list a state for which A^(S)_M = 0 and A^(S)_M+1 is non-zero, but minimal among the states found so far.

The state |Ψ^(S)⟩ = ∑^S_{m = -S} Ψ_m|S,m⟩ is given by the row vector with the 2S+1 coefficients [ Ψ_-S,Ψ_-S+1,…,Ψ_S ]. We also computed the cumulative squared lengths of the multipoles A^(S)_M. While we give exact algebraic expressions (with i = sqrt(-1)) for the state in most cases, we use floating point numbers for the constellation with N = 2S points. Finally, we provide information about the symmetry group of the constellation.

More examples can be found here.

Erratum

Unfortuantely, there are some typos in the printed versions of the articles for S=2,5/2,3,5,9. Please let us know in case you noticed further typos.

Extremal Constellations for S=1,…,10 (click on the image for more information)

S=1, M=1	S=3/2, M=1	S=2, M=2	S=5/2, M=1

S=3, M=3	S=7/2, M=2	S=4, M=3	S=9/2, M=2

S=5, M=3	S=11/2, M=3	S=6, M=5	S=13/2, M=3

S=7, M=4	S=15/2, M=3	S=8, M=5	S=17/2, M=4

S=9, M=5	S=19/2, M=4	S=10, M=5

S = 1, M = 1

state:
```
  |Ψ⁽¹⁾⟩ = [ 0, 1, 0 ]
	
```
(unique solution, M = 1 is maximal)
cumulative squared lengths:
```
  A⁽¹⁾_M = [ 0, 2/3 ]
	
```

constellation (radial line):

  [ 0.0, 0.0,  1.0 ]
  [ 0.0, 0.0, -1.0 ]

symmetry group: Z₂ of order 2

S = 3/2, M = 1

state:

  |Ψ^(3/2)⟩ = [ sqrt(2)/2, 0, 0, sqrt(2)/2 ]

(unique solution, M = 1 is maximal)

cumulative squared lengths:
```
  A^(3/2)_M = [ 0, 1/4, 3/4  ]
	
```

constellation (equatorial triangle):

  [  0.5, -0.866025403784439, 0.0 ]
  [ -1.0,  0.0,               0.0 ]
  [  0.5,  0.866025403784439, 0.0 ]

symmetry group: S₃ of order 6

S = 2, M = 2

state:

  |Ψ⁽²⁾⟩ = [ 0, sqrt(6)/3, 0, 0, sqrt(3)/3 ]

(unique solution, M = 2 is maximal)

cumulative squared lengths:
```
  A⁽²⁾_M = [ 0, 0, 1/2, 4/5 ]
	
```

constellation (tetrahedron):

  [  0.0,                0.0,                1.0               ]
  [  0.471404520791032,  0.816496580927726, -0.333333333333333 ]
  [ -0.942809041582063,  0.0,               -0.333333333333333 ]
  [  0.471404520791032, -0.816496580927726, -0.333333333333333 ]

symmetry group: S₄ of order 24

S = 5/2, M = 1

state:

  |Ψ^(5/2)⟩ = [ 0, sqrt(2)/2, 0, 0, sqrt(2)/2, 0 ]

(M = 1 is maximal)

cumulative squared lengths:

  A_M^(5/2) = [ 0, 1/84,  59/252, 5/9, 5/6 ]
         ≈ [ 0, 0.011905, 0.23413, 0.55556, 0.83334 ]

constellation (equatorial triangle and poles):

  [  0.0,  0.0,                1.0 ]
  [  0.5, -0.866025403784438,  0.0 ]
  [ -1.0,  0.0,                0.0 ]
  [  0.5,  0.866025403784438,  0.0 ]
  [  0.0,  0.0,               -1.0 ]

symmetry group: dihedral group D₆ of order 12

S = 3, M = 3

state:

  |Ψ⁽³⁾⟩ = [ 0, sqrt(2)/2,  0, 0, 0, -sqrt(2)/2, 0 ]

(unique solution, M = 3 is maximal)

cumulative squared lengths:

  A⁽³⁾_M = [ 0, 0, 0, 6/11, 6/11, 5/7 ]

constellation (octahedron):

  [  0.0,  0.0,  1.0 ]
  [  1.0,  0.0,  0.0 ]
  [  0.0,  1.0,  0.0 ]
  [ -1.0,  0.0,  0.0 ]
  [  0.0, -1.0,  0.0 ]
  [  0.0,  0.0, -1.0 ]

symmetry group: Z₂ × S₄ of order 48

S = 7/2, M = 2

state:

  |Ψ^(7/2)⟩ = [ sqrt(2)/3, 0, 0, sqrt(14)/6, 0, 0, sqrt(14)/6, 0 ]

(M = 2 is maximal)

cumulative squared lengths:

  A^(7/2)_M = [  0, 0, 49/264, 49/264, 49/143, 821/1144, 7/8 ]
         ≈ [ 0, 0, 0.18561, 0.18561, 0.34266, 0.71766, 0.875 ]

constellation (two aligned triangles and pole):

  [  0.406731490944482,  0.704479607354083,  0.581616692587408 ]
  [ -0.813462981888963,  0.000000000000000,  0.581616692587408 ]
  [  0.406731490944482, -0.704479607354083,  0.581616692587408 ]
  [  0.485089093681821,  0.840198956454453, -0.242392831502497 ]
  [ -0.970178187363642,  0.000000000000000, -0.242392831502497 ]
  [  0.485089093681821, -0.840198956454453, -0.242392831502497 ]
  [  0.0,                0.0,               -1.0               ]

symmetry group: S₃ of order 6

S = 4, M = 3

state:

  |Ψ⁽⁴⁾⟩ = [ sqrt(30)/12, 0, 0, 0, sqrt(21)/6, 0, 0, 0, sqrt(30)/12 ]

(M = 3 is maximal)

cumulative squared lengths:

  A⁽⁴⁾_M = [ 0, 0, 0,  98/429,   98/429,  74/117,  74/117,   8/9 ]
       ≈ [  0, 0, 0, 0.22844, 0.22844, 0.63248,  0.63248, 0.88889 ]

constellation (cube):

  [  0.577350269189626, -0.577350269189626,  0.577350269189626 ]
  [  0.577350269189626,  0.577350269189626,  0.577350269189626 ]
  [ -0.577350269189626, -0.577350269189626,  0.577350269189626 ]
  [ -0.577350269189626,  0.577350269189626,  0.577350269189626 ]
  [  0.577350269189626, -0.577350269189626, -0.577350269189626 ]
  [  0.577350269189626,  0.577350269189626, -0.577350269189626 ]
  [ -0.577350269189626, -0.577350269189626, -0.577350269189626 ]
  [ -0.577350269189626,  0.577350269189626, -0.577350269189626 ]

symmetry group: Z₂ × S₄ of order 48

S = 9/2, M = 2

state:

  |Ψ^(9/2)⟩ = [ sqrt(6)/6, 0, 0, -sqrt(3)/3, 0, 0, -sqrt(3)/3, 0, 0, sqrt(6)/6 ]

(M = 2 is maximal)

cumulative squared lengths:

  A^(9/2)_M ≈ [ 0, 0, 0.0064155, 0.028793, 0.35314, 0.45769, 0.45882, 0.73190, 0.9 ]

constellation (three triangles):

  [ -0.354635777076689, -0.614247184078495,  0.704935360489777 ]
  [  0.709271554153378,  0.000000000000000,  0.704935360489777 ]
  [ -0.354635777076689,  0.614247184078495,  0.704935360489777 ]
  [  0.5,                0.866025403784438,  0.0               ]
  [ -1.0,                0.0,                0.0               ]
  [  0.5,               -0.866025403784438,  0.0               ]
  [ -0.354635777076689, -0.614247184078495, -0.704935360489777 ]
  [  0.709271554153378,  0.000000000000000, -0.704935360489777 ]
  [ -0.354635777076689,  0.614247184078495, -0.704935360489777 ]

symmetry group: dihedral group D₆ of order 12

S = 5, M = 3

state:

  |Ψ⁽⁵⁾⟩ = [ sqrt(5)/5, 0, 0, 0, 0, sqrt(15)/5,  0, 0, 0, 0, sqrt(5)/5 ]

(M = 3 is maximal)

cumulative squared lengths:

  A⁽⁵⁾_M ≈ [ 0, 0, 0, 0.12587, 0.18126, 0.21014, 0.53593, 0.60719, 0.70601, 0.90909 ]

constellation (pentagonal prism):

  [  0.659009385323091,  0.478798344791898,  0.580050665959984 ]
  [ -0.251719186288245,  0.774711995630481,  0.580050665959984 ]
  [ -0.814580398069693,  0.0,                0.580050665959984 ]
  [ -0.251719186288245, -0.774711995630481,  0.580050665959984 ]
  [  0.659009385323091, -0.478798344791898,  0.580050665959984 ]
  [  0.659009385323091,  0.478798344791898, -0.580050665959984 ]
  [ -0.251719186288245, -0.774711995630481, -0.580050665959984 ]
  [ -0.814580398069693,  0.0,               -0.580050665959984 ]
  [ -0.251719186288245,  0.774711995630481, -0.580050665959984 ]
  [  0.659009385323091, -0.478798344791898, -0.580050665959984 ]

symmetry group: dihedral group D₁₀ of order 20

S = 11/2, M = 3

state:

  |Ψ^(11/2)⟩ = [ sqrt(17)/12, 0, 0, i sqrt(55)/12, 0, 0, 0, 0, i sqrt(55)/12, 0, 0, sqrt(17)/12 ]

(M = 3 is maximal)

cumulative squared lengths:

  A^(11/2)_M ≈ [ 0, 0, 0, 0.042193, 0.11612, 0.32001, 0.37254, 0.44398, 0.64454, 0.74368, 0.91666 ]

constellation:

  [ -0.540468153848830,  0.312669120556627,  0.781109592647269 ]
  [  0.001113638755064, -0.625130604696015,  0.780519369958951 ]
  [  0.542329414861365,  0.312456964568204,  0.779903488304132 ]
  [  0.808618504015847,  0.588101567578426,  0.016512455146701 ]
  [ -0.309745649254401, -0.950765489949400,  0.010129950110765 ]
  [ -1.0,                0.0,                0.                ]
  [ -0.309745649254401,  0.950765489949400, -0.010129950110765 ]
  [  0.808618504015847, -0.588101567578426, -0.016512455146701 ]
  [  0.542329414861365, -0.312456964568204, -0.779903488304132 ]
  [  0.001113638755064,  0.625130604696015, -0.780519369958951 ]
  [ -0.540468153848830, -0.312669120556627, -0.781109592647269 ]

symmetry group: Z₂ of order 2

S = 6, M = 5

state:

  |Ψ⁽⁶⁾⟩ = [  0, sqrt(7)/5, 0, 0, 0, 0, sqrt(11)/5, 0, 0, 0, 0, -sqrt(7)/5, 0 ]

(unique solution, M = 5 is maximal)

cumulative squared lengths:

  A⁽⁶⁾_M = [ 0, 0, 0, 0, 0, 121/323, 121/323, 121/323, 121/323, 328/437, 328/437, 12/13 ]
       ≈ [ 0, 0, 0, 0, 0, 0.37461, 0.37461, 0.37461, 0.37461, 0.75057, 0.75057, 0.92308 ]

constellation (dodecahedron):

  [  0.0,                0.0,                1.0               ]
  [  0.723606797749979,  0.525731112119134,  0.447213595499958 ]
  [ -0.276393202250021,  0.850650808352040,  0.447213595499958 ]
  [ -0.894427190999916,  0.0,                0.447213595499958 ]
  [ -0.276393202250021, -0.850650808352040,  0.447213595499958 ]
  [  0.723606797749979, -0.525731112119134,  0.447213595499958 ]
  [ -0.723606797749979,  0.525731112119134, -0.447213595499958 ]
  [  0.276393202250021,  0.850650808352040, -0.447213595499958 ]
  [  0.894427190999919,  0.0,               -0.447213595499958 ]
  [  0.276393202250021, -0.850650808352040, -0.447213595499958 ]
  [ -0.723606797749979, -0.525731112119134, -0.447213595499958 ]
  [  0.0,                0.0,               -1.0               ]

symmetry group: Z₂ × A₅ of order 120

S = 13/2, M = 3

state:

  |Ψ^(13/2)⟩ = [ sqrt(15)/15, 0, 0, i sqrt(390)/30, 0, 0, 0, 0, 0, 0, i sqrt(390)/30, 0, 0, sqrt(15)/15 ]

cumulative squared lengths:

  A^(13/2)_M = [ 0, 0, 0, 247/5610, 247/5610, 656123/3391500, 994123/3391500, 169/510, 16393/41055, 863083/1560090, 19999967/27858750, 459353/603750, 13/14 ]
          ≈  [ 0, 0, 0, 0.044028, 0.044028, 0.19346, 0.29312, 0.33137, 0.39929, 0.55322, 0.71790, 0.76083, 0.92857 ]

constellation (bent heptagon and two twisted triangles above & below):

  [ -0.456786580721863,  0.263710287026873,  0.849589844683088 ]
  [  0.456807478105276,  0.263753513416359,  0.849565190028184 ]
  [  0.0,               -0.527463766250099,  0.849577527106852 ]
  [  0.900976971397839,  0.433818742643517,  0.006464947174969 ]
  [  0.222653721362577, -0.974883834307386,  0.005179765392943 ]
  [ -0.623400079035304,  0.781897753601443,  0.002871301411168 ]
  [ -1.0,                0.0,                0.0               ]
  [ -0.623400079035304, -0.781897753601443, -0.002871301411168 ]
  [  0.222653721362577,  0.974883834307386, -0.005179765392943 ]
  [  0.900976971397839, -0.433818742643517, -0.006464947174970 ]
  [  0.0,                0.527463766250099, -0.849577527106852 ]
  [  0.456807478105276, -0.263753513416359, -0.849565190028184 ]
  [ -0.456786580721863, -0.263710287026873, -0.849589844683088 ]

symmetry group: Z₂ of order 2

S = 7, M = 4

state:

  |Ψ⁽⁷⁾⟩ = [ 0.0, 0.484038792672727, 0.0, 0.0, -0.217868005816070 + 0.228902507261237 i, 0.0, 0.0, 0.277562504208005 + 0.504624909270376 i, 0.0, 0.0, -0.217868005816070 + 0.228902507261237 i, 0.0, 0.0, 0.484038792672727, 0.0 ]

cumulative squared lengths:

  A⁽⁷⁾_M ≈  [ 0, 0, 0, 0, 0.022088, 0.16322, 0.24969,  0.33424,  0.38239,  0.44336,  0.54533, 0.71629, 0.79733,  0.93333 ]

constellation (four triangles and the two poles):

  [  0.0,                0.0,                1.0               ]
  [  0.768016179259059,  0.239363858169746,  0.594016912047470 ]
  [ -0.591303271552385,  0.545439592670935,  0.594016912047470 ]
  [ -0.176712907706674, -0.784803450840680,  0.594016912047470 ]
  [  0.117497703154367,  0.945457109922784,  0.303817614778843 ]
  [  0.760041023804564, -0.574484550779396,  0.303817614778843 ]
  [ -0.877538726958931, -0.370972559143387,  0.303817614778843 ]
  [  0.117497703154367, -0.945457109922784, -0.303817614778843 ]
  [  0.760041023804564,  0.574484550779396, -0.303817614778843 ]
  [ -0.877538726958931,  0.370972559143387, -0.303817614778843 ]
  [ -0.591303271552385, -0.545439592670935, -0.594016912047470 ]
  [ -0.176712907706674,  0.784803450840680, -0.594016912047470 ]
  [  0.768016179259059, -0.239363858169746, -0.594016912047470 ]
  [  0.0,                0.0,               -1.0               ]

symmetry group: Z₂ of order 2

S = 15/2, M = 3

state:

  |Ψ^(15/2)⟩ = [ 3/44 sqrt(22), 0, 0, 0, sqrt(770)/44, 0, 0, 0, 0, 0, 0, sqrt(770)/44, 0, 0, 0, 3/44 sqrt(22)]

cumulative squared lengths:

  A^(15/2)_M ≈ [ 0, 0, 0, 0.0039210, 0.0039210, 0.13265, 0.20292, 0.29853, 0.34431, 0.39883, 0.46949, 0.57603, 0.75148, 0.79966, 0.93750 ]

constellation (heptagon and two aligned square above & below):

  [  0.433217774817543,  0.433301918821350,  0.790298555438306 ]
  [  0.433217774817543, -0.433301918821350,  0.790298555438306 ]
  [ -0.433368690130957, -0.433284237986233,  0.790225504224502 ]
  [ -0.433368690130957,  0.433284237986233,  0.790225504224502 ]
  [  0.219585359474376,  0.975593291235907,  0.0               ]
  [ -0.622122481482076,  0.782919930794065,  0.0               ]
  [  0.902631297489889,  0.430414614983877,  0.0               ]
  [ -1.0,                0.0,                0.0               ]
  [  0.902631297489889, -0.430414614983877,  0.0               ]
  [ -0.622122481482076, -0.782919930794065,  0.0               ]
  [  0.219585359474376, -0.975593291235907,  0.0               ]
  [  0.433217774817543,  0.433301918821350, -0.790298555438306 ]
  [  0.433217774817543, -0.433301918821350, -0.790298555438306 ]
  [ -0.433368690130957, -0.433284237986233, -0.790225504224502 ]
  [ -0.433368690130957,  0.433284237986233, -0.790225504224502 ]

symmetry group: Z₂ × Z₂ of order 4

S = 8, M = 5

state:

  |Ψ⁽⁸⁾⟩ = [ sqrt(69)/32, 0, -7/1472 i sqrt(3358)+17/1472 sqrt(690), 0, 3/4784 i sqrt(152789)-1/1196 sqrt(31395), 0, -1/210496 i sqrt(50420370)-279/210496 sqrt(46046), 0, 7/26312 i sqrt(480194)-137/105248 sqrt(98670), 0, 1/210496 i sqrt(50420370)+279/210496 sqrt(46046), 0, 3/4784 i sqrt(152789)-1/1196 sqrt(31395), 0, 7/1472 i sqrt(3358)-17/1472 sqrt(690), 0, sqrt(69)/32 ]

cumulative squared lengths:

  A⁽⁸⁾_M ≈ [ 0, 0, 0, 0, 0, 0.066374, 0.21902, 0.24823, 0.31960, 0.36874, 0.37228,  0.51437, 0.66655, 0.70602, 0.82630, 0.94118 ]

constellation:

  [ -0.146859793851903,  0.403430399755151,  0.903147891268742 ]
  [  0.146859793851903, -0.403430399755151,  0.903147891268742 ]
  [  0.816496580927726,  0.0,                0.577350269189626 ]
  [ -0.816496580927726,  0.0,                0.577350269189626 ]
  [  0.510336695378834,  0.766907301282083,  0.389113927520006 ]
  [ -0.510336695378834, -0.766907301282083,  0.389113927520006 ]
  [ -0.363476901526931,  0.913767095133985,  0.181422815287328 ]
  [  0.363476901526931, -0.913767095133985,  0.181422815287328 ]
  [ -0.913767095133985,  0.363476901526931, -0.181422815287328 ]
  [  0.913767095133985, -0.363476901526931, -0.181422815287328 ]
  [  0.766907301282083,  0.510336695378834, -0.389113927520006 ]
  [ -0.766907301282083, -0.510336695378834, -0.389113927520006 ]
  [  0.0,                0.816496580927726, -0.577350269189626 ]
  [  0.0,               -0.816496580927726, -0.577350269189626 ]
  [ -0.403430399755151,  0.146859793851903, -0.903147891268742 ]
  [  0.403430399755151, -0.146859793851903, -0.903147891268742 ]

symmetry group: A₄ of order 12

S = 17/2, M = 4

state:

  |Ψ^(17/2)⟩ = [ 0.0, 0.435874159108882, 0.0, 0.0, -0.158009774587788 + 0.300431770262860 i, 0.0, 0.0, 0.438536379196716 - 0.0497315201215425 i, 0.0, 0.0, 0.438536379196716 - 0.0497315201215425 i, 0.0, 0.0, -0.158009774587788 + 0.300431770262860 i, 0.0, 0.0, 0.435874159108882, 0.0 ]

cumulative squared lengths:

  A^(17/2)_M ≈ [ 0, 0, 0, 0, 0.024845, 0.090384, 0.11317, 0.20352, 0.26268, 0.32590, 0.40828, 0.48103, 0.52997, 0.60888, 0.75945, 0.83311, 0.94444 ]

constellation (five triangles plus the two poles):

  [  0.0,                0.0,                1.0               ]
  [ -0.602115369310431,  0.430511934776098,  0.672396130309734 ]
  [  0.673891956803705,  0.306191238443833,  0.672396130309734 ]
  [ -0.071776587493275, -0.736703173219931,  0.672396130309734 ]
  [ -0.162582048449742,  0.902577354866469,  0.398649214227414 ]
  [  0.862945942419795, -0.310488493276445,  0.398649214227414 ]
  [ -0.700363893970053, -0.592088861590024,  0.398649214227414 ]
  [  0.5,                0.866025403784438,  0.0               ]
  [ -1.0,                0.0,                0.0               ]
  [  0.5,               -0.866025403784438,  0.0               ]
  [ -0.700363893970053,  0.592088861590024, -0.398649214227414 ]
  [  0.862945942419795,  0.310488493276445, -0.398649214227414 ]
  [ -0.162582048449742, -0.902577354866469, -0.398649214227414 ]
  [ -0.071776587493275,  0.736703173219931, -0.672396130309734 ]
  [  0.673891956803705, -0.306191238443833, -0.672396130309734 ]
  [ -0.602115369310431, -0.430511934776098, -0.672396130309734 ]
  [  0.0,                0.0,               -1.0               ]

symmetry group: S₃ of order 6

S = 9, M = 5

state:

  |Ψ⁽⁹⁾⟩ = [ sqrt(455)/65, 0, 0, 0, 0, i sqrt(6630)/130, 0, 0, 0, 0, 0, 0, 0, i sqrt(6630)/130, 0, 0, 0, 0, sqrt(455)/65 ]

cumulative squared lengths:

  A⁽⁹⁾_M ≈ [ 0, 0, 0, 0, 0, 0.081787, 0.081787, 0.21370, 0.21370, 0.32413, 0.32413, 0.45360, 0.45360, 0.58772, 0.58772, 0.82262, 0.82262, 0.94736 ]

constellation (slightly bent equatorial octagon and two flat pentagons above & below):

  [ -0.600064784923290, -0.194953054131990,  0.775832172946943 ]
  [  0.0,               -0.630894385012959,  0.775868722761216 ]
  [  0.600064784923290, -0.194953054131990,  0.775832172946943 ]
  [  0.370865130063146,  0.510400251630871,  0.775854779219919 ]
  [ -0.370865130063146,  0.510400251630871,  0.775854779219919 ]
  [ -0.923876304082174, -0.382688993090229,  0.001307410818663 ]
  [ -0.382677490170340, -0.923881835161757, -0.000541456436505 ]
  [  0.382677490170340, -0.923881835161757, -0.000541456436505 ]
  [  0.923876304082174, -0.382688993090229,  0.001307410818663 ]
  [  0.923876304082174,  0.382688993090229, -0.001307410818663 ]
  [  0.382677490170340,  0.923881835161757,  0.000541456436505 ]
  [ -0.382677490170340,  0.923881835161757,  0.000541456436505 ]
  [ -0.923876304082174,  0.382688993090229, -0.001307410818663 ]
  [ -0.600064784923290,  0.194953054131990, -0.775832172946943 ]
  [  0.0,                0.630894385012959, -0.775868722761216 ]
  [  0.600064784923290,  0.194953054131990, -0.775832172946943 ]
  [  0.370865130063146, -0.510400251630871, -0.775854779219919 ]
  [ -0.370865130063146, -0.510400251630871, -0.775854779219919 ]

symmetry group: S₃ of order 6

S = 19/2, M = 4

state:

  |Ψ^(19/2)⟩ = [ 0.0, 0.394790292433844, 0.0, 0.0, 0.0, 0.533445903570947 + 0.0699184736916915 i, 0.0, 0.0, 0.0, -0.0634904625134868 + 0.0797470135479915 i, 0.0, 0.0, 0.0, 0.650420537037385, 0.0, 0.0, 0.0, 0.0, 0.0, 0.348209706929603 ]

cumulative squared lengths:

  A^(19/2)_M ≈  [ 0, 0, 0, 0, 0.024892, 0.059595, 0.073605, 0.17021, 0.19811, 0.25225, 0.30899, 0.37421, 0.45820, 0.53688, 0.57819, 0.63019, 0.78372, 0.85237, 0.95000 ]

constellation (bent equatorial octagon, hexagon below, square above, and pole):

  [  0.0,                0.0,                1.0               ]
  [ -0.485306671181903,  0.514193954201776,  0.707164770310770 ]
  [  0.514193950997086,  0.485306679117874,  0.707164767194738 ]
  [  0.485306671181903, -0.514193954201776,  0.707164770310770 ]
  [ -0.514193950997086, -0.485306679117874,  0.707164767194738 ]
  [ -0.363347596778625,  0.924375137638232,  0.116228777983327 ]
  [  0.924499195547427,  0.363089900543646,  0.116047238465058 ]
  [  0.363347596778625, -0.924375137638232,  0.116228777983327 ]
  [ -0.924499195547427, -0.363089900543646,  0.116047238465058 ]
  [ -0.338774094909787, -0.940223561578101,  0.034810441988656 ]
  [ -0.940469942586444,  0.338121781084990,  0.034495626496208 ]
  [  0.338774094909787,  0.940223561578101,  0.034810441988656 ]
  [  0.940469942586444, -0.338121781084990,  0.034495626496208 ]
  [  0.000563459578012, -0.670741144911577, -0.741691309802137 ]
  [  0.580440369683931, -0.335915107176755, -0.741788391666796 ]
  [  0.580115578771631,  0.335075664091756, -0.742421857571505 ]
  [ -0.000563459578012,  0.670741144911577, -0.741691309802137 ]
  [ -0.580440369683931,  0.335915107176755, -0.741788391666796 ]
  [ -0.580115578771631, -0.335075664091756, -0.742421857571505 ]

symmetry group: Z₂ of order 2

S = 10, M = 5

state:

  |Ψ⁽¹⁰⁾⟩ = [ sqrt(561)/75, 0, 0, 0, 0, sqrt(209)/25, 0, 0, 0, 0,  sqrt(741)/75, 0, 0, 0, 0, -sqrt(209)/25, 0, 0, 0, 0, sqrt(561)/75 ]

cumulative squared lengths:

  A⁽¹⁰⁾_M ≈  [ 0, 0, 0, 0, 0, 0.093330, 0.093330, 0.093330, 0.093330, 0.27876, 0.27876, 0.43038, 0.43038, 0.43038, 0.43038, 0.65514, 0.65514, 0.83394, 0.83394, 0.95238 ]

constellation (dodecahedron):

  [  0.491123473188423,  0.356822089773090,  0.794654472291766 ]
  [ -0.187592474085080,  0.577350269189626,  0.794654472291766 ]
  [ -0.607061998206686,  0.0,                0.794654472291766 ]
  [ -0.187592474085080, -0.577350269189626,  0.794654472291766 ]
  [  0.491123473188423, -0.356822089773090,  0.794654472291766 ]
  [  0.794654472291766,  0.577350269189626,  0.187592474085080 ]
  [ -0.303530999103343,  0.934172358962716,  0.187592474085080 ]
  [ -0.982246946376846,  0.0,                0.187592474085080 ]
  [ -0.303530999103343, -0.934172358962716,  0.187592474085080 ]
  [  0.794654472291766, -0.577350269189626,  0.187592474085080 ]
  [ -0.794654472291766,  0.577350269189626, -0.187592474085080 ]
  [  0.303530999103343,  0.934172358962716, -0.187592474085080 ]
  [  0.982246946376846,  0.0,               -0.187592474085080 ]
  [  0.303530999103343, -0.934172358962716, -0.187592474085080 ]
  [ -0.794654472291766, -0.577350269189626, -0.187592474085080 ]
  [ -0.491123473188423,  0.356822089773090, -0.794654472291766 ]
  [  0.187592474085080,  0.577350269189626, -0.794654472291766 ]
  [  0.607061998206686,  0.0,               -0.794654472291766 ]
  [  0.187592474085080, -0.577350269189626, -0.794654472291766 ]
  [ -0.491123473188423, -0.356822089773090, -0.794654472291766 ]

symmetry group: Z₂ × A₅ of order 120

Markus Grassl (Markus.Grassl[at]mpl.mpg.de)