  Non-leptonic two-body weak decays of baryons are an important tool to probe
the combined charge-conjugation--parity symmetry (CP) violation. We explain why
the decays of strange baryons provide complementary information to the decays
of kaons. A model-independent parameterization of the non-leptonic decays of
the $\Lambda$- and $\Xi$-baryons is reviewed, and the amplitudes are updated
according to the latest experimental input. We demonstrate the potential of
performing precision tests in strange baryon decays at the next generation
electron-positron $J/\psi$ factories with luminosity of $10^{35}$
cm$^{-2}$s$^{-1}$. The copious production of spin-entangled hyperon-antihyperon
pairs via the $J/\psi$ resonance allows for a direct comparison of the baryon
and antibaryon decay properties. Using analytic approximations and numerical
calculations, we study the quantitative impact of spin correlations and
polarization in such CP tests. We show that by using a longitudinally-polarized
electron beam, the statistical precision of the CP tests can be significantly
improved compared to the experiments without polarized beams. Furthermore, we
map out further directions for possible improvements, like analysis of
incompletely reconstructed events or a combination of the isospin related
processes. Altogether, these methods are promising for the observation of a
statistically significant CP-violation signal with a strength corresponding to
the standard model predictions. Our conclusions should encourage more detailed
feasibility studies, including optimisation of the measurement methods and
studies of systematic effects. Finally, our results call for an update of the
theory predictions with increased precision.
