Multi-Photon States with Minimal Polarization

The table below summarizes the maximal degree M of extremal unpolarized, pure states with N photons (or, equivalently, spin S=N/2). For those states, all multipoles up to order M vanishe, corresponding to isotropic moments of the Stoke's variables up to Mth order. Furthermore, the cumulative squared lengths A^(S)_M vanishes.

So far, in all case for which we have found solutions, the maximal degree M of unpolarization matches the largest degree t for which a spherical t design with N points is known to exist.

Click on the number of photons to get more detailed information (more solutions will be added).

N M t             2 1 1 3 1 1 4 2 2 5 1 1 6 3 3 7 2 2 8 3 3 9 2 2

N M t 10 3 3 11 3 3 12 5 5 13 3 3 14 4 4 15 3 3 16 5 5 17 4 4 18 5 5 19 4 4

N M t 20 5 5 21 4 4 22 5 5 23 5 5 24 7 7 25 5 5 26 6 6 27 5 5 28 6 6 29   6

N M t 30 7 7 31 6 6 32 7 7 33 6 6 34 7 7 35 6 6 36 8 8 37   7 38 7 7 39 7 7

N M t 40 8 8 41 7 7 42 8 8 43   7 44 8 8 45   8 46 8 8 47   8 48 9 9 49   8

N M t 50 9 9 51   8 52   9 53   8 54 9 9 55   9 56 9 9 57   9 58   9 59   9

N M t 60 10 10 61   9 62   10 63   9 64   10 65   10 66   10 67   10 68   10 69   10

N M t 70 11 11 71   10 72 11 11 73   10 74 11 11 75   11 76   11 77   11 78 11 11 79   11

N M t 80 11 11 81   11 82   11 83   11 84 12 12 85   11 86   12 87   12 88   12 89   12

N M t 90   12 91   12 92   12 93   12 94   13 95   12 96   13 97   12 98   13 99   12

N M t 100   13 101     102     103     104     105     106     107     108   14 109

N M t 110     111     112     113     114     115     116     117     118     119

N M t 120 15 15 132 15 15 144   16 156   17 180   18 204   19 212 19   216   20 240   21

References:

• F. Bouchard, P. de la Hoz, G. Björk, R. W. Boyd, M. Grassl, Z. Hradil, E. Karimi, A. B. Klimov, G. Leuchs, J. Rehacek, and L. L. Sanchez-Soto,
  "Quantum metrology at the limit with extremal Majorana constellations",
  Optica, vol. 4, no. 11, pp. 1429-1432 (2017).
  DOI: 10.1364/OPTICA.4.001429
  arXiv:1612.06804 [quant-ph]
• G. Björk, A. B. Klimov, P. de la Hoz, M. Grassl, G. Leuchs, L. L. Sánchez-Soto.
  "Extremal quantum states and their Majorana constellations",
  Physical Review A, vol. 92, no. 3, September 2015, 031801(R).
  DOI: 10.1103/PhysRevA.92.031801
  arXiv:1502.03446 [quant-ph]
• 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