Revealing the Two-Fold Ambiguity: Tau Momentum Reconstruction and Its Impact on Entanglement Observables

The neutrinos produced in $\tau$ decays cannot be directly detected, making the reconstruction of $\tau$ kinematics challenging and affecting measurements of quantum correlations such as spin entanglement. For the process $e^+e^- \to \tau^+\tau^- \to \pi^+ \bar{\nu}_\tau\pi^-\nu_\tau$, the kinematic constraints allow the $\tau$ momenta to be reconstructed up to a well-known two-fold ambiguity, regardless of the presence of an intermediate resonance state. In this paper, we present a geometric interpretation of this ambiguity and propose a numerical reconstruction method based on singular value decomposition (SVD). Using only the information from visible final-state particles and decay kinematics, the method reconstructs the two possible solutions for the $\tau^+\tau^-$ pair. The reconstruction performance is validated with Monte Carlo simulations in typical collider environments. We further investigate the impact of the spurious solution on spin-entanglement measurements and show that reliable entanglement signals can still be extracted even when the true and spurious solutions cannot be experimentally distinguished. This work provides a practical approach for $\tau$-lepton kinematic reconstruction and spin-entanglement measurements in $e^+e^-$ collider experiments.