Residual Entanglement vs. Residual Coherence: A Wave Packet Approach in Neutrino Oscillation with Matter

Abhishek Kumar Jha; Bhavna Yadav; Lekhashri Konwar

REVIEW

Residual Entanglement vs. Residual Coherence: A Wave Packet Approach in Neutrino Oscillation with Matter

Lekhashri Konwar, Abhishek Kumar Jha, Bhavna Yadav

High Energy Physics - Phenomenology · Submitted 12 Nov 2026 · 81 references

Pith's verdict: A technically sound extension of residual entanglement and coherence measures to wave-packet neutrino oscillations, but quantitative predictions are hostage to an unconstrained free parameter (sigma_x) without physical motivation.

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CitationAbhishek Kumar Jha, Bhavna Yadav, Lekhashri Konwar. "Residual Entanglement vs. Residual Coherence: A Wave Packet Approach in Neutrino Oscillation with Matter." arXiv:2611.16709 (2026). https://pith.science/paper/TEQERXH6

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This paper does two things well. First, it derives closed-form expressions for three residual quantum correlation measures—Three-pi, Concurrence Fill, and residual l1-norm coherence—in terms of transition probabilities for an initial muon-flavor neutrino state, within the wave-packet formalism in constant matter. The derivations are careful and mathematically consistent. Second, it presents numerical results for DUNE and extended baselines, showing how these measures depend on wave-packet width, matter potential, baseline, and the CP phase. The plots are clear.

Now the soft spot, and it is load-bearing. The wave-packet width sigma_x is treated as a free parameter, and the paper uses two values (0.5e-9 m and 7e-9 m) with no physical justification. The coherence length scales linearly with sigma_x, so the damping pattern—and consequently the relative sizes of the correlation measures—changes dramatically with this parameter. The claim that genuine tripartite entanglement persists in the wave-packet case is true for these specific choices, but the central quantitative results (e.g., which measure dominates, CP-phase sensitivity) are parametric unless sigma_x is connected to actual production or detection processes. The stress-test concern is valid: the paper does not discuss how to determine sigma_x from the DUNE source, and does not even provide a plausible range from the neutrino beamline. The constant matter density approximation is a secondary simplification, common in such studies, but its impact at long baselines is not discussed.

Who is this for? Readers working on quantum information in neutrino oscillations who want the specific formulas or benchmark numerics for these three measures. The formalism is standard but the combination is new. I would not cite the numerical results in my own work without a justification of sigma_x, but the expressions themselves are useful.

The paper is worth sending to peer review because the derivations are solid and the topic has an audience. But a serious referee should push the authors to provide a physical basis for sigma_x, or at minimum frame the analysis as a parametric exploration with clear caveats. Deserves a major revision.

Referee report

Major objections (4)

[§V, Fig. 2, Fig. 3, Fig. 5, Fig. 6] The wave-packet width σ_x is a free parameter with no physical justification for the chosen values (0.5×10^{-9} m, 7×10^{-9} m, etc.). The paper does not connect these values to actual neutrino production/detection processes in DUNE (e.g., pion decay region, hadron production target). The quantitative results, including the relative sizes of correlation measures and the δ_CP sensitivity, depend sensitively on σ_x (see Fig. 2, Fig. 3). Without a justified range, the predictive power for DUNE is limited. The authors should either provide a physically motivated range for σ_x based on the DUNE source and detector characteristics, or clearly state that the results are illustrative and not predictions.
[§III, Eq. (28), Appendix A] The constant matter density approximation is used throughout, but the Earth's density varies significantly along the DUNE baseline (1300 km). The paper does not discuss the impact of this approximation on the results. For a quantitative analysis relevant to DUNE, the authors should either justify that constant density is sufficient (e.g., by comparing with an adiabatic approximation) or acknowledge this as a limitation and estimate the error.
[§IV, Eqs. (40), (41), (44)] The derivation of the correlation measures in terms of transition probabilities relies on the specific form of the density matrix in Eq. (38), which assumes that the neutrino state is pure and that the Gaussian averaging over time yields the given form. The authors should verify that the density matrix in Eq. (38) is indeed correct for the wave-packet formalism with matter, particularly the off-diagonal elements. The expressions for Three-π and Concurrence Fill are quoted from Ref. [29] (Li et al., 2022), which used the plane-wave approximation. The authors should confirm that these expressions remain valid for the wave-packet case, given that the density matrix structure is the same (only the coefficients F^m_{αγ} change). This is likely fine, but a brief justification would strengthen the paper.
[§V, Fig. 3, Fig. 6] The oscillogram plots (L-δ_CP plane) are presented for a single energy (E=2.5 GeV for DUNE, E=45 GeV for extended baseline). The energy dependence is crucial for DUNE, which has a broad energy spectrum (0.5-8 GeV). The paper should discuss how the results would change when integrated over the DUNE flux, or at least show results for a few representative energies. The current single-energy analysis may not reflect the experimental sensitivity.

Minor objections (8)

[§II, Eq. (10)] The definition of Three-π in Eq. (10) uses N^2_{AB} etc., but the text earlier defines N_{AB} as negativity. The notation is consistent, but it would be helpful to explicitly state that N^2_{AB} is the squared negativity.
[§IV, Eq. (43)] The expressions for C_{l1}(ρ_{AB}) and C_{l1}(ρ_{AC}) are both given as 2√(P^m_{µe} P^m_{µµ}). This seems to assume that the reduced density matrices have the same form. Please verify that this is correct for the specific density matrix in Eq. (38).
[§V, Fig. 1 caption] The caption for Fig. 1 says 'upper panels present a comparison between PW and wave-packet formalisms' but the panels are labelled (a)-(c) for probabilities. The description in the text is clear, but the caption could be more explicit about which panel corresponds to which probability.
[§V, Fig. 2] The curves in Fig. 2 are described in the text but the legend in the figure itself is small and may be hard to read. Consider using larger fonts or more distinct line styles.
[§V, Fig. 6] The color scales in the oscillogram plots (Fig. 6) are not uniform across panels, making direct comparison difficult. The authors should use consistent color bars for each measure across different parameter choices.
[§VI, Conclusion] The conclusion states that 'genuine tripartite entanglement in neutrino oscillations is not restricted to the PW approximation but also persists within the wave-packet approximation.' This is a qualitative claim that is supported by the non-zero values of Three-π and Concurrence Fill in the wave-packet case. However, the paper does not provide a quantitative threshold for 'persistence' (e.g., how small can the measures be and still be considered non-zero?). This is a minor point, but clarifying would strengthen the claim.
[General] The paper uses the notation P^m_{αγ} for transition probabilities in matter, but sometimes refers to them as P(ν_μ → ν_e) etc. Consistency would be improved by using the same notation throughout.
[References] Reference [41] (Ravari et al., 2022) is cited for the wave-packet formalism in matter. It would be helpful to also cite the original wave-packet papers (e.g., Giunti and Kim, 1998) more prominently in the derivation.

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The paper

The paper claims

The paper derives expressions for residual entanglement and coherence in three-flavor neutrino oscillations using a wave-packet approach in matter and analyzes their behavior for DUNE and extended baselines.

The reading

Neutrinos oscillate between flavors as they travel, and this can be described using quantum mechanics. Usually, physicists treat neutrinos as plane waves, but a more realistic model treats them as wave packets with a finite size. This paper uses that wave-packet model to study quantum correlations like entanglement and coherence in three-flavor neutrino oscillations, including the effect of matter (the Earth's crust) on the DUNE experiment's baseline (1300 km) and beyond. The authors find that the wave-packet width (size) and matter potential significantly affect the oscillation probabilities and the quantum correlations. Smaller wave packets cause stronger damping of oscillations. The residual l1-norm coherence (a measure of quantum superposition) dominates over entanglement measures. At very long distances, all correlations approach constant values, independent of distance and the CP-violating phase. The paper provides formulas connecting these quantum correlation measures to the measurable oscillation probabilities.

Extended reading notes

Core claim

The paper claims that genuine tripartite entanglement (measured by Three-pi and Concurrence Fill) and residual l1-norm coherence persist in three-flavor neutrino oscillations when using a wave-packet formalism with constant matter density, and that these measures can be expressed in terms of transition probabilities for an initial muon-flavor state. The central quantitative results are the expressions in Eqs. (40), (41), and (44) and the numerical analysis showing how these measures depend on wave-packet width, matter potential, baseline, and CP phase for DUNE and beyond.

Load-bearing premise

The weakest assumption is the use of a Gaussian wave-packet with a fixed width sigma_x, and the assumption that the momentum distribution is sharply peaked (sigma_p << E^2/m_i). The wave-packet width is treated as a free parameter, and the results depend sensitively on its value. The paper does not provide a physical justification for the chosen values (0.5e-9 m, 7e-9 m, etc.) or connect them to actual production/detection processes in DUNE. The constant matter density approximation is also a simplification, as the Earth's density varies along the baseline.

Original abstract

In this work, we investigate quantum correlations in three-flavor neutrino oscillations beyond the standard plane-wave (PW) description by employing a wave-packet formalism in the presence of constant matter density. We study three quantum correlation measures, namely the residual $l_1$-norm coherence, Concurrence Fill, and Three-$\pi$ entanglement, and express them in terms of transition probabilities for an initial muon-flavor state. The analysis is performed for the DUNE setup and extended beyond its baseline to study the interplay between wave-packet decoherence and matter effects over long distances. For the DUNE baseline, finite wave-packet widths suppress and smooth the oscillations relative to the PW case, with smaller widths causing stronger damping, while matter effects modify the amplitudes and positions of the oscillation features. Among the considered measures, residual $l_1$-norm coherence dominates over the entanglement measures. At very long propagation distances, wave-packet decoherence progressively washes out the oscillatory behavior, driving the oscillation probabilities and quantum correlation measures toward averaged values, with smaller wave-packet widths leading to earlier damping. Further, in the $L-\delta_{\rm CP}$ plane, residual $l_1$-norm coherence exhibits weak $\delta_{\rm CP}$ dependence, while Concurrence Fill and Three-$\pi$ entanglement show greater sensitivity at the DUNE baseline. Beyond a characteristic baseline in the extended propagation regime, all three measures approach constant values, independent of $L$ and $\delta_{\rm CP}$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-13: 1 request, 6,544 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.19 · human participants 0.02 · controlled comparison 0.55 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.88 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this work, we investigate quantum correlations in three-flavor neutrino oscillations beyond the standard plane-wave (PW) description by employing a wave-packet formalism in the presence of constant matter density. We study three quantum correlation measures, namely the residual $l_1$-norm coherence, Concurrence Fill, and Three-$\pi$ entanglement, and express them in terms of transition probabilities for an initial muon-flavor state. The analysis is performed for the DUNE setup and extended beyond its baseline to study the interplay between wave-packet decoherence and matter effects over long distances. For the DUNE baseline, finite wave-packet widths suppress and smooth the oscillations relative to the PW case, with smaller widths causing stronger damping, while matter effects modify the amplitudes and positions of the oscillation features. Among the considered measures, residual $l_1$-norm coherence dominates over the entanglement measures. At very long propagation distances, wave-packet decoherence progressively washes out the oscillatory behavior, driving the oscillation probabilities and quantum correlation measures toward averaged values, with smaller wave-packet widths leading to earlier damping. Further, in the $L-\delta_{\rm CP}$ plane, residual $l_1$-norm coherence exhibits weak $\delta_{\rm CP}$ dependence, while Concurrence Fill and Three-$\pi$ entanglement show greater sensitivity at the DUNE baseline. Beyond a characteristic baseline in the extended propagation regime, all three measures approach constant values, independent of $L$ and $\delta_{\rm CP}$.

background contribution method result interpretation main claim

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s0	In this work, we investigate quantum correlations in three-flavor neutrino oscillations beyond the standard plane-wave (PW) description by employing a wave-packet formalism in the presence of constant matter density.	contribution 0.98	0.90	0.05	asserted 0.89
s1	We study three quantum correlation measures, namely the residual $l_1$-norm coherence, Concurrence Fill, and Three-$\pi$ entanglement, and express them in terms of transition probabilities for an initial muon-flavor state.	contribution 0.86	0.94	0.05	asserted 0.99
s2	The analysis is performed for the DUNE setup and extended beyond its baseline to study the interplay between wave-packet decoherence and matter effects over long distances.	method 0.61	0.91	0.05	asserted 0.96
s3	For the DUNE baseline, finite wave-packet widths suppress and smooth the oscillations relative to the PW case, with smaller widths causing stronger damping, while matter effects modify the amplitudes and positions of the oscillation features.	result 1.00	0.93	0.13	asserted 1.00
s4	Among the considered measures, residual $l_1$-norm coherence dominates over the entanglement measures.	result 1.00	0.94	0.28	asserted 1.00
s5	At very long propagation distances, wave-packet decoherence progressively washes out the oscillatory behavior, driving the oscillation probabilities and quantum correlation measures toward averaged values, with smaller wave-packet widths leading to earlier damping.	result 0.98	0.91	0.21	asserted 1.00
s6	Further, in the $L-\delta_{\rm CP}$ plane, residual $l_1$-norm coherence exhibits weak $\delta_{\rm CP}$ dependence, while Concurrence Fill and Three-$\pi$ entanglement show greater sensitivity at the DUNE baseline.	result 1.00	0.93	0.10	asserted 1.00
s7	Beyond a characteristic baseline in the extended propagation regime, all three measures approach constant values, independent of $L$ and $\delta_{\rm CP}$.	result 1.00	0.92	0.24	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.69 · evidence from simulation or computation 0.68.

What was examined, and what was not

sources read

arXiv listing abstract v1

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full text
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references and citing sentences
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P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** presents the neutrino oscillation (P(νµ → νe) [1(a)] & P(νµ → ντ ) [1(b)]) and survival (P(νµ → νµ))[1(c)] probabilities of the initial state |νµ⟩ as functions of neutrino energy [GeV] for the DUNE baseline (L = 1300 km) within the wave-packet formalism, incorporating matter effects. For the DUNE configuration, the matter potential is fixed at V = 1.01 × 10−13 eV,

**Figure 2.** displays the dependence of the residual quantum correlation measures on the propagation distance L (km) for the DUNE configuration, considering a fixed neutrino energy of E = 2.5 GeV. The analysis considers three residual quantum correlation measures: the residual entanglement measures, Three-π and Concurrence Fill, along with the residual l1-norm coherence (Rl1-norm). The left panel shown in

**Figure 3.** FIG. 3: Oscillogram plots of the

**Figure 4.** shows the neutrino oscillation and survival probabilities with the propagation distance L, considering the range beyond the DUNE baseline and E = 45 GeV [41]. The analysis is performed within the wave-packet formalism, taking matter effects into account. In the upper

**Figure 5.** FIG. 5: For an initial state

**Figure 6.** FIG. 6: Oscillogram plots of the

Figures reproduced from arXiv:2611.16709 by the authors.

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