Extremal Polarization States

References

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) = ∑Sm = -S Ψm|S,m⟩ is given by the row vector with the 2S+1 coefficients [ Ψ-S-S+1,…,ΨS ]. 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. 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=1S=3/2, M=1S=2, M=2S=5/2, M=1
S=3, M=3S=7/2, M=2S=4, M=3S=9/2, M=2
S=5, M=3S=11/2, M=3S=6, M=5S=13/2, M=3
S=7, M=4S=15/2, M=3S=8, M=5S=17/2, M=4
 
S=9, M=4S=19/2, M=4S=10, M=5

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 = 4

S = 19/2, M = 4

S = 10, M = 5


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