  J-PET is a detector optimized for registration of photons from the
electron-positron annihilation via plastic scintillators where photons interact
predominantly via Compton scattering. Registration of both primary and
scattered photons enables to determinate the linear polarization of the primary
photon on the event by event basis with a certain probability. Here we present
quantitative results on the feasibility of such polarization measurements of
photons from the decay of positronium with the J-PET and explore the physical
limitations for the resolution of the polarization determination of 511 keV
photons via Compton scattering. For scattering angles of about 82 deg (where
the best contrast for polarization measurement is theoretically predicted) we
find that the single event resolution for the determination of the polarization
is about 40 deg (predominantly due to properties of the Compton effect).
However, for samples larger than ten thousand events the J-PET is capable of
determining relative average polarization of these photons with the precision
of about few degrees. The obtained results open new perspectives for studies of
various physics phenomena such as quantum entanglement and tests of discrete
symmetries in decays of positronium and extend the energy range of polarization
measurements by five orders of magnitude beyond the optical wavelength regime.
