Concurrence fill and mode distribution of entanglement in neutrino oscillation

TL;DR

This work treats a single neutrino undergoing three-flavor oscillations as a multimode, single-particle entangled state across the electron, muon, and tau flavor modes. By mapping the system to a three-qubit single-excitation framework and using a density-matrix formalism, the authors derive entanglement diagnostics—tangle, partial tangles, and concurrence fill—that can be expressed entirely in terms of experimentally measurable oscillation probabilities. A key result is that the tangle vanishes for all flavors, ruling out GHZ-type entanglement and indicating a W-type or biseparable structure, while the concurrence patterns reveal how entanglement is distributed predominantly between specific flavor pairs, constrained by monogamy. Using DUNE/GLoBES simulations, they show that the concurrence triangle area (concurrence fill) varies with energy, peaking at energies where all three flavors participate, and that a W-class inequality is satisfied across the energy range, providing a robust quantum-information perspective on flavor evolution with direct experimental relevance.

Abstract

In the framework of three flavor neutrino oscillation, we demonstrate that the measures of entanglement can be expressed in terms of experimentally accessible appearance and disappearance probabilities. We explicitly show here that the genuine tripartite entanglement measure, i.e., the tangle vanishes identically for all flavors signifying that three flavor neutrino system form a W-type entangled state. Further, we investigate alternative measures of tripartite entanglement like the partial tangle and the concurrence fill which capture the total sharing of entanglement beyond pairwise correlations. In terms of bipartite and bi-partitioned entanglement measures, we derive the symmetric invariant and the concurrence fill, which quantify the distributed entanglement completely expressible in terms of flavor transition probabilities. These entanglement measures display distinct energy dependent patterns across the oscillation window which can be experimentally accessible in the long baseline experiments like DUNE providing an alternative quantum information perspective on flavor evolution. We use GLobal Long Baseline Experiment Simulator (\textsf{GLoBES}) simulations within the DUNE set up to investigate these tripartite entanglement measures in terms of neutrino energy and the length of the baseline. It is observed that, at the point of maximal mixing, these measures show near maximal entanglement between the muon and the tau flavor modes establishing entanglement monogamy. Within the DUNE set up, the wide band of energy and expected higher sensitivity to CP-violation at second oscillation maximum provide a unique advantage to explore the quantum correlation effects across a broader energy window.

Figures (6)

• Figure 1: Geometric illustration of the tripartite entanglement, biseparability, and fully separable (product) states.
• Figure 2: Probability plots for $P_{\mu\mu}$, $P_{\mu e}$, and $P_{\mu\tau}$ (left panel) and the corresponding neutrino flux (shown up to a scale) alongside $P_{\mu e}$ (right panel) are presented for the DUNE experiment at three different values of the CP-violating phase $\delta$.
• Figure 3: Bipartite entanglement structure of the three flavor neutrino system as a function of energy, shown for both vacuum (magenta) and matter (turquoise) propagation. The left panel displays the three bipartite concurrences $C^{2}_{\mu|e\tau}$ (solid green), $C^{2}_{e|\mu\tau}$ (dashed green), and $C^{2}_{\tau|\mu e}$ (dotted green), while the right panel shows the corresponding squared concurrences for the alternative bipartitions considered. Together, these curves map how quantum correlations are dynamically redistributed among the flavor modes across the full energy range relevant to long-baseline oscillations. Vertical markers highlight representative energies corresponding to the first (2.6 GeV), intermediate (1.7 GeV and 1.3 GeV), and second (0.8 GeV) oscillation maxima. The behavior of the concurrences demonstrates that entanglement is not uniformly shared: different flavor pairs dominate at different energies, and matter effects further reshape this distribution by enhancing or suppressing selected bipartite correlations. Taken together, the two panels provide a complementary view of the flavor-mode entanglement landscape and illustrate how bipartite quantum correlations evolve under conditions relevant for experiments such as DUNE.
• Figure 4: Energy dependence of the partial tangles $\tau^{2}_{i|jk}$ in the three-flavor neutrino system for vacuum (dashed) and matter (solid) propagation. The curves correspond to the three bipartitions $\tau^{2}_{\mu e}$ (tuquoise), $\tau^{2}_{e\tau}$ (purple), and $\tau^{2}_{\mu\tau}$ (yellow), and quantify how the residual entanglement associated with each flavor mode is shared among the remaining two modes. Vertical markers indicate representative energies (2.6, 1.7, 1.3, and 0.8 GeV) corresponding to the first, intermediate, and second oscillation maxima. The variation of the partial tangles demonstrates that the distribution of residual entanglement evolves non-uniformly with energy: in vacuum, the structure reflects the coherent interplay among flavor components, whereas matter effects further reshape this pattern by enhancing or suppressing individual bipartitions. Overall, the figure provides a compact view of how tripartite entanglement is dynamically reorganized as the neutrino propagates, offering insight into the quantum structure relevant for long-baseline experiments such as DUNE.
• Figure 5: Concurrence $Q$ (red), concurrence fill $C_F$ (blue), and concurrence triangles illustrating the energy dependence of bipartite and tripartite entanglement in the three flavor neutrino system. The left panel shows $Q$ and $C_F$ in vacuum (dashed curve) and matter (solid curve), with vertical markers indicating the first, intermediate, and second oscillation maxima. Matter effects modify the distribution of quantum correlations, leading to characteristic enhancements or suppressions relative to vacuum. The right panel displays concurrence triangles constructed from $(C_{e\mu}, C_{e\tau}, C_{\mu\tau})$, where the enclosed area provides a geometric indicator of W type tripartite entanglement. Together, the panels summarize the dynamical redistribution of entanglement across energies relevant to long-baseline experiments such as DUNE.