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 vanishes, 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         2 1 3 1 4 2 5 1 6 3 7 2 8 3 9 2

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

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

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

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

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

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

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

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

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

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

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

N M 120 15     132 15     212 19

References:

• 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