  The large number of $\tau$ leptons available at the Belle II experiment makes
it possible to study their properties and the extent of their compositeness.
Our strategy relies on three observables defined in terms of elements of the
polarization density matrix of the produced $\tau$ pairs, which we obtain via
quantum tomography. The observables are able to explore values of the magnetic
dipole moment down to $6.3 \times 10^{-4}$, which is two order of magnitude
better than the current experimental limit, constrain the electromagnetic
radius below $4.3 \times 10^{-3}$ fm and exclude an electric dipole moment
larger than $1.7 \times 10^{-17}$ e cm. The quoted limits are at the $95\%$
confidence level and are obtained for a benchmark integrated luminosity of 1
ab$^{-1}$.
