Quantum tomography with $\tau$ leptons at the FCC-ee

The Future Circular Collider (FCC) -- in its first incarnation as a lepton collider -- will produce, according to the proposed design, more than 100 billion pairs of $\tau$ leptons after working for four years at the energy of the $Z$-boson resonance. The $\tau$ lepton is special because its relatively long lifetime makes it possible to reconstruct the momenta of neutrinos emitted in the single pion decay mode. The resulting large number of events is an ideal source for a full quantum tomography of the process that will test quantum entanglement and the violation of Bell inequality with unprecedented precision. In addition, the study of polarizations and spin correlations can provide a competitive determination of the Weinberg angle $ \theta_W$ and constrain possible anomalous couplings in the neutral electroweak current. We utilize analytic results and Monte Carlo simulations to explore to what extent these goals might be accomplished.