Quantum Fisher Information Revealing Parameter Sensitivity in Long-Baseline Neutrino Experiments
Determination of the leptonic CP-violating phase $δ_{\rm CP}$, the atmospheric mixing angle $θ_{23}$, and the mass-squared difference $Δm_{31}^2$ constitutes a primary objective of current and next-generation long-baseline neutrino experiments. We employ Quantum Fisher Information (QFI) to quantify the maximum information that the neutrino quantum state contains about these oscillation parameters, treating the neutrino as an evolving pure quantum state. Computing the 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, $δ_{\rm CP}$ and $θ_{23}$ exhibit bimodal QFI profiles with peaks around $L/E \sim 500$ and $1500~\mathrm{km/GeV}$, reaching $F_Q(δ_{\rm CP}) \sim 0.15$ and $F_Q(θ_{23}) \sim 15$, respectively. In contrast, $F_Q(Δm_{31}^2)$ increases rapidly with $L/E$ over the range considered, reaching values of order $10^7$. This hierarchy indicates that, within the considered single-parameter framework, the neutrino state carries substantially greater intrinsic quantum sensitivity to $Δm_{31}^2$ than to $θ_{23}$ or $δ_{\rm CP}$. We further compare the QFI with the classical Fisher information (CFI) obtained from flavor-transition probabilities, showing how much of the available information can be extracted through this specific measurement. For a fixed DUNE baseline, matter effects substantially enhance the QFI for $δ_{\rm CP}$ in the few-GeV region, while producing only minor modifications for $θ_{23}$ and $Δm_{31}^2$. These results show that the information available for estimating the three oscillation parameters differs significantly, and that the flavor-probability measurement does not always extract all the information available in the quantum state.