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Neutrino Oscillations as a Probe of Macrorealism

This paper critiques previous claims of Leggett-Garg inequality violations in neutrino oscillations by identifying unsuitable macrorealistic modeling in statistical tests and proposes an improved methodology that yields revised, more modest evidence for such violations at the 2–3σ level using MINOS data.

Original authors: Kathrine Mørch Groth, Johann Ioannou-Nikolaides, D. Jason Koskinen, Markus Ahlers

Published 2026-01-30

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CC BY 4.0

Original authors: Kathrine Mørch Groth, Johann Ioannou-Nikolaides, D. Jason Koskinen, Markus Ahlers

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Is the World "Real" or "Quantum"?

Imagine you have a magic coin that can be Heads or Tails. In our everyday, "classical" world, this coin is always either Heads or Tails, even when you aren't looking at it. If you check it, it doesn't change what it was. This is called Macrorealism: the idea that things have a definite state at all times, and checking them doesn't disturb them.

But in the Quantum World, things are weird. A quantum coin might be spinning in a blur of both Heads and Tails at the same time. If you check it, it "chooses" a side, and that choice might change how it behaves later.

Scientists want to know: Do neutrinos (tiny, ghost-like particles) act like the magic spinning coin (Quantum) or the normal coin (Classical)?

To test this, they use a set of rules called Leggett-Garg Inequalities (LGIs). Think of these rules as a "lie detector test" for reality.

If the neutrino follows the rules, it behaves like a classical object (Macrorealism).
If it breaks the rules, it proves the neutrino is behaving in a truly quantum way.

The Problem with Previous Tests

In the past, scientists tried to run this "lie detector test" on neutrinos. They looked at data from big experiments (like MINOS) where neutrinos are shot through the Earth and detected later.

However, the authors of this paper argue that previous tests were flawed.

The Flaw: They used a "classical" model that was too simple. It was like trying to test if a car is an electric vehicle by checking if it has a gas tank. If the car didn't have a gas tank, the test said "It's electric!" But the test didn't account for the fact that the car might just be a toy car with no engine at all.
The Result: Previous studies claimed a very strong violation (6.2σ), essentially shouting, "We proved neutrinos are quantum!" The authors say, "Wait a minute, your test was rigged. You didn't account for measurement errors properly."

The New Method: A Better Way to Play the Game

The authors propose a new, more rigorous way to run the test. Here is how they did it, using an analogy:

The Analogy: The Music Playlist
Imagine you are trying to prove a song is a remix (Quantum) and not a standard track (Classical).

Old Way: You just listened to the song and guessed. If it sounded weird, you called it a remix.
New Way: You create thousands of fake "standard tracks" (simulated data) that include all the possible mistakes and static you'd expect from a normal recording. Then, you compare the real song against these thousands of fake ones.
- If the real song sounds way more different than almost all the fake ones, you can confidently say it's a remix.
- If the real song sounds similar to the fake ones, you can't be sure.

The authors did exactly this with neutrino data. They created "fake" neutrino data based on two different "Classical" scenarios:

Scenario A: The neutrino never changes its flavor (it stays the same).
Scenario B: The neutrino changes flavor randomly and fades away over time (like a signal losing strength).

They then compared the real MINOS/MINOS+ data against these thousands of simulated "Classical" scenarios.

What They Found

When they ran their new, stricter test, the results changed:

Previous Claim: "We are 99.9999% sure neutrinos violate the rules of classical reality!" (6.2σ significance).
New Finding: "We are 95% to 99% sure neutrinos violate the rules." (2 to 3σ significance).

The Translation:
The evidence is still there, but it's not as overwhelming as previously thought. It's like finding a fingerprint at a crime scene.

Old view: "This fingerprint is a perfect match! The suspect is definitely guilty!"
New view: "This fingerprint is a match, but it's a bit smudged. It's likely the suspect, but we can't be 100% certain without more evidence."

Why Does This Matter?

The paper doesn't claim this will lead to new technology or medical cures. Instead, it's about cleaning up the science.

Better Math: They fixed the statistical math so we aren't fooled by random noise or measurement errors.
Honesty: They showed that previous studies were too optimistic. The violation of "Macrorealism" is real, but it's a "soft" violation (2–3σ) rather than a "slam-dunk" one.
Future Proofing: By using a better method, future experiments will know exactly how to interpret their data without making the same mistakes.

Summary

The authors took a famous experiment that claimed to prove neutrinos are "quantum ghosts" and said, "Your proof is a bit shaky." They built a stronger, more realistic test that accounts for measurement errors. Their new test still says, "Yes, neutrinos are quantum," but the confidence level dropped from "Absolutely certain" to "Very likely." It's a correction that makes the scientific conclusion more robust and reliable.

Technical Summary: Neutrino Oscillations as a Probe of Macrorealism

Problem Statement
Leggett-Garg Inequalities (LGIs) provide a framework to test the hypothesis of macrorealism (MR)—the view that a system possesses a definite state at all times and can be measured without disturbance. While violations of LGIs have been proposed as evidence for quantum behavior in neutrino flavor oscillations, previous analyses (e.g., Refs. [7–10]) have relied on statistical methodologies that the authors argue are flawed. Specifically, prior studies often utilized a "classical" Leggett-Garg string ( $K^C_n$ ) or resampled oscillation data without simulating a proper macrorealistic background. The authors contend that these approaches yield unreliable significance levels because the classical string only violates LGIs when experimental uncertainties produce unphysical correlations (e.g., survival probabilities $P > 1$ ), rather than reflecting genuine macrorealistic behavior.

Methodology
The paper proposes a revised statistical framework to probe LGIs in neutrino data, illustrated using muon-neutrino survival data from the MINOS and MINOS+ experiments. The methodology involves three key improvements:

Generalized Leggett-Garg Strings: The authors generalize the standard LG string ( $K_n$ ) to allow for arbitrary sequences of measurements and sign assignments ( $\sigma_i = \pm 1$ ). They define an optimal LG string, $K_{max}^n$ , which maximizes the violation potential for a given set of correlators, subject to the condition that the product of signs is $-1$. This allows for the construction of LG strings from non-time-ordered sequences and different phase combinations (e.g., "1+3" vs. "2+2" phase splits) that were previously ignored.
Macrorealistic Background Models: Instead of using a classical string or resampling data, the authors generate pseudo-data based on explicit macrorealistic background hypotheses ( $H_0$ $H_{0}$ ). They consider two models:
- $H_0^a$ : A fully correlated classical system with constant correlators $C(\tau) = 1$ .
- $H_0^b$ : A decorrelating classical system following a Markovian process with $C(\tau) = e^{-\Gamma\tau}$ , where $\Gamma$ is fitted to the data.
  These models generate pseudo-samples with the same variance as the experimental data to establish a baseline for "chance" violations.
Robust Test Statistic: To address the issue that small fluctuations in macrorealistic pseudo-data can frequently combine to violate LGIs (especially near the boundary $C=1$ ), the authors introduce a test statistic based on the root-mean-square (RMS) of one-sided $z$ -scores ( $z_{RMS}$ ). This metric quantifies the quality of the violation relative to the uncertainty, rather than just the quantity of violations.

Key Results
Applying this methodology to MINOS/MINOS+ data (39 energy bins, $L \approx 735$ km):

Optimization of Sequences: The authors identified a significantly larger number of unique phase combinations satisfying the matching condition (Eq. 9) compared to previous studies, including combinations with $n=4$ and $n=5$ measurements.
Significance of Violation: When comparing the observed data against the macrorealistic background models:
- Against the fully correlated model ( $H_0^a$ ), the post-trial significance of LGI violation is estimated at 2.1 $\sigma$ .
- Against the decorrelating Markovian model ( $H_0^b$ ), the significance is 3.7 $\sigma$ .
Comparison to Previous Claims: These results represent a reduction in significance compared to the 6.2 $\sigma$ claim in Ref. [7]. The authors attribute this reduction to the more rigorous treatment of background hypotheses and the avoidance of unphysical correlations in the statistical test.
Model Consistency: The analysis confirms that the constant correlation model ( $C=1$ ) is a poor description of the MINOS data ( $\chi^2/dof \approx 31$ ) compared to the best-fit oscillation model ( $\chi^2/dof \approx 1.8$ ). However, the statistical test for macrorealism specifically isolates the quantum nature of the correlations from the general goodness-of-fit.

Significance and Claims
The paper claims that its primary contribution is a more robust statistical methodology for testing macrorealism in neutrino oscillations. By moving away from the "classical LG string" approach and instead simulating explicit macrorealistic background models with a refined test statistic, the authors argue that previous claims of LGI violations were overstated.

The authors conclude that while MINOS/MINOS+ data still shows evidence for LGI violation (at the 2–3 $\sigma$ level), the statistical evidence is less definitive than previously reported. They assert that their findings necessitate a critical re-examination of similar results in other accelerator and reactor neutrino datasets (such as NO $\nu$ A, Daya Bay, and RENO) that utilized similar, less rigorous background modeling. The work establishes that testing LGI violation is distinct from simply confirming neutrino oscillations; it requires a specific statistical test of the correlations against a macrorealistic null hypothesis that accounts for measurement uncertainties without inducing spurious violations.