Quantum Fisher Information Revealing Parameter Sensitivity in Long-Baseline Neutrino Experiments

Determination of the leptonic CP-violating phase $\delta_{\mathrm{CP}}$, the atmospheric mixing angle $\theta_{23}$, and the mass-squared difference $\Delta m_{31}^{2}$ constitutes a primary objective of current and next-generation long-baseline neutrino experiments. We employ QFI (QFI) to establish fundamental precision bounds on single-parameter estimation in three-flavor $\nu_\mu \to \nu_e$ oscillations, treating the neutrino as an evolving pure quantum state. Computing QFI as a function of the baseline-to-energy ratio $L/E$ for benchmark parameter sets from NuFit-6.0, we find distinct sensitivity hierarchies and $L/E$-dependent structures. Specifically, $\delta_{\mathrm{CP}}$ and $\theta_{23}$ exhibit bimodal QFI profiles with peaks at $L/E \sim 500$ and $1500~\mathrm{km/GeV}$, corresponding to the first and second oscillation maxima, reaching $F_Q(\delta_{\mathrm{CP}}) \sim 0.15$ and $F_Q(\theta_{23}) \sim 15$, respectively. In contrast, $\Delta m_{31}^{2}$ displays a unimodal structure peaking at $L/E \sim 1000$--$1200~\mathrm{km/GeV}$ with $F_Q(\Delta m_{31}^{2}) \sim 3 \times 10^{6}$, reflecting its role in setting the oscillation length scale.