  We introduce a methodology and investigate the feasibility of measuring
quantum properties of tau lepton pairs in the $H \to \tau^+ \tau^-$ decay at
future lepton colliders. In particular, observation of entanglement,
steerability and violation of Bell inequalities are examined for the ILC and
FCC-ee. We find that detecting quantum correlation crucially relies on precise
reconstruction of the tau lepton rest frame and a simple kinematics
reconstruction does not suffice due to the finite energy resolution of the
colliding beams and detectors. To correct for energy mismeasurements, a
log-likelihood method is developed that incorporates the information of impact
parameters of tau lepton decays. We demonstrate that an accurate measurement of
quantum properties is possible with this method. As a by-product, we show that
a novel model-independent test of CP violation can be performed and the
CP-phase of $H \tau \tau$ interaction can be constrained with an accuracy
comparable to dedicated analyses, i.e., up to $7.9^{\circ}$ and $5.4^{\circ}$
at ILC and FCC-ee, respectively.
