It is shown that the description of polarization based on quantization of classical Stokes parameters is incomplete in quantum domain. For example, the polarization of the electric dipole radiation of the atomic or molecular transitions is described by nine generators of the SU(3) subalgebra in the Weyl-Heisenberg algebra of photons, in other words, by nine independent Stokes operators instead of four dependent classical Stokes parameters. Although some of the Stokes operators have one and the same averages, they describe absolutely different physical quantities with strongly different quantum fluctuations. The quantum polarization properties of the electric dipole and quadrupole radiation are examined as a function of distance for the near, intermediate, and far zones. Application of results in the near-field optics and quantum entanglement research is discussed.