Response of Hellinger-distance based coherence to weak decoherence in two-flavor neutrino oscillations

Abstract

We evaluate the Hellinger-distance coherence of the two-flavor neutrino state subject to Lindblad damping of the mass-eigenstate interference term, and obtain a closed expression in terms of the flavor-basis density-matrix elements. In the two-flavor vacuum treatment with negligible wave-packet separation, the undamped state is pure, and the smallest eigenvalue of the damped state grows linearly with $ε=1-κ$, where $κ=e^{-ΓL}$. An expansion about the pure state then yields a nonanalytic $\sqrtε$ correction to the Hellinger coherence, with coefficient $K=2\sqrt{2}\,u\sin2θ/\sqrt{1-2u}$, whereas the flavor-transition probability, the $l_1$ coherence, the concurrence, the entanglement of formation and the local quantum Fisher information all respond linearly in $ε$. Using representative Daya Bay, KamLAND and MINOS working points together with a current $90\%$ C.L. bound on energy-independent damping, we find fractional changes in the Hellinger coherence of $0.18\%$, $9.40\%$ and $21.51\%$, against $0.0009\%$, $0.43\%$ and $1.81\%$ for the concurrence. The comparison concerns the response of the quantifiers to the damping parameter and carries no implication about experimental resolution. The same mechanism operates, with a different coefficient, for a correlated dephasing channel acting on the flavor coherence, where the non-Markovian regime produces damped revivals.

The paper asks how Hellinger-distance coherence responds to weak decoherence in two-flavor neutrino oscillations, compared with oscillation probabilities and other quantum-information quantifiers. It models mass-basis Lindblad dephasing through $\kappa=e^{-\Gamma L}$ and evaluates coherence in the flavor basis. Reducing the density matrix to its occupied $2\times2$ block gives the exact expression $C_{\rm sk}=2|z|^2/(1+2\sqrt{\det\varrho})$ and $C_H=2(1-\sqrt{1-C_{\rm sk}})$. For a pure undamped state, dephasing lifts the zero eigenvalue linearly in $\epsilon=1-\kappa$, yielding $C_H=C_H^{(0)}-K\sqrt{\epsilon}+O(\epsilon)$, where $K=2\sqrt{2}u\sin2\theta/\sqrt{1-2u}$ and $u=P_{\alpha\alpha}(1-P_{\alpha\alpha})$. At the working points considered, transition probabilities, concurrence, entanglement of formation and local quantum Fisher information instead have linear leading responses. The formulas are checked against direct matrix calculations. A flavor-basis correlated-dephasing example reproduces the square-root scaling and exhibits damped revivals. Using the energy-independent bound $\Gamma\leq5.1\times10^{-24}$ GeV, Hellinger coherence decreases by $0.18\%$, $9.40\%$ and $21.51\%$ at representative Daya Bay, KamLAND and MINOS points, versus concurrence changes of $0.0009\%$, $0.43\%$ and $1.81\%$. These are model benchmarks, not experimental sensitivity estimates. The analysis assumes two-flavor vacuum propagation and negligible wave-packet separation; pre-existing mixedness regularizes the square-root response to additional damping. Three-flavor propagation, matter effects and experimental reconstruction are not treated.