Quantum Estimation Theory Limits in Neutrino Oscillation Experiments

The gist: Flavor measurements saturate the QFI at first oscillation maximums for mixing angles, demonstrating their information-theoretic optimality for these parameters, while they are far from optimal for δCP. The goal is to compare the Quantum Fisher Information (QFI) and the classical Fisher Information (FI) associated with ideal flavor projections for all oscillation parameters. In conventional QET, one seeks to identify or engineer optimal quantum states and measurement schemes that saturate the Cramér–Rao bound.

Neutrino Physics Modelization

Neutrino oscillations are described by the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, which encodes leptonic CP violation in the phase δCP. The neutrino flavor states are related to mass eigenstates via the PMNS matrix. The probability of flavor change is given by Eq. (2.10), which depends on the PMNS parameters, squared mass differences ∆m2ij, and the ratio L/E between distance and energy.

QFI for CP Phase δCP

For the CP phase δCP, measurements saturate the QFI at the first oscillation maximum for θ13, θ23, and θ12. The QFI associated with δCP is approximately one order of magnitude smaller than that of the mixing angles, indicating that neutrino state intrinsically encodes less information about CP violation. Even with a limited event statistics comparable to current experimental conditions, the achievable precision for δCP is not constrained by any fundamental quantum limit.

QFI for Mixing Angles

The QFI associated with mixing angles θ13 and θ23 reaches maxima when the survival probability is maximal, P(νµ → νµ) ≈ 1. The maxima of these QFIs are approximately Hν,ν(θ13) ≃ 8 and Hν,ν(θ23) ≃ 15. This implies that for the same number of events, the attainable uncertainty on θ13 or θ23 are about a factor 3 and an order of magnitude smaller than that on δCP.

Fisher Information and Measurement Strategy

In accelerator experiments, the FI for δCP never reaches the peaks of Hν,ν(δCP), remaining significantly lower throughout. A strategy based on operating at the first oscillation maximum appears to be strongly suboptimal, as it yields FI values that are several orders of magnitude smaller than the corresponding QFI. Conversely, the FI is typically enhanced in the vicinity of the second oscillation maximum.

Conclusion and Outlook

The work demonstrates that flavor measurement strategies for determining the mixing angles are close to optimal, capturing nearly all the accessible information in the neutrino quantum state under idealized conditions. The situation for δCP is different, as it contains significantly less information than on mixing angles. Sensitivity can be substantially enhanced by targeting the second oscillation maximum, and electron antineutrino beams carry more information on δCP than muon (anti)neutrino beams. The principal path to improved δCP sensitivity lies in the strategic design of experiments, especially through the selection of higher oscillation maxima.

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Prepared for submission to JHEP Claudia Frugiuele, Marco G. Genoni, Michela Ignoti, Matteo G. A. Paris INFN Sezione di Milano, I-20133 Milano, Italy Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano, I-20133 Milano, Italy E-mail: claudia.frugiuele@mi.infn.it Abstract: Measurements of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) neutrino mixing parameters have entered a precision era, enabling increasingly stringent tests of neutrino oscillations. Within the framework of quantum estimation theory, we investigate whether flavor measurements, the only observables currently accessible experimentally, are optimal for extracting the oscillation parameters. We compute the Quantum Fisher Information (QFI) and the classical Fisher Information (FI) associated with ideal flavor projections for all oscillation parameters, considering accelerator muon (anti)neutrino and reactor electron antineutrino beams propagating in vacuum. Two main results emerge. First, flavor measurements saturate the QFI at the first oscillation maximum for θ13, θ23, and θ12, demonstrating their information-theoretic optimality for these parameters. In contrast, they are far from optimal for δCP. In particular, only a small fraction of the available information on δCP is extracted at the first maximum; the sensitivity improves at the second maximum, in line with the strategy of ESSνSB, a planned facility. Second, the QFI associated with δCP is approximately one order of magnitude smaller than that of the mixing angles, indicating that the neutrino state intrinsically encodes less information about CP violation. Nevertheless, this quantum bound lies well below current experimental uncertainties, implying that the present precision on δCP is not fundamentally limited. Our results provide a quantitative framework to disentangle fundamental from practical limitations and establish a benchmark for optimizing future neutrino facilities. arXiv:2602.16534v1 [hep-ph] 18 Feb 2026

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Contents 1 Introduction Neutrino oscillations, described by the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, provide a powerful framework for probing fundamental neutrino properties, including leptonic CP violation encoded in the phase δCP. Despite these achievements, δCP remains poorly constrained compared to the other oscillation parameters, with current global fits providing only weak indications of CP violation [1]. The NuFIT v6.0 global analysis [1] reports, for the normal ordering (NO) scenario, δ(NO)CP = 177◦ +19◦ −20◦, based primarily on data from the long-baseline accelerator experiments T2K [2] and NOνA [3] while, for example, for the mixing angle θ13 = 8.52◦ +0.11◦ −0.11◦, highlighting the current experimental challenges in probing CP violation in the neutrino sector compared to the other oscillation parameters. Analogous results are obtained under the inverse ordering (IO) hypothesis. Next-generation facilities such as DUNE [4, 5], T2HK [6] and ESSνSB [7, 8] aim to significantly improve this precision and may provide the first definitive evidence of leptonic CP violation. The estimation of the PMNS parameters can be naturally formulated within the framework of quantum estimation theory (QET) [9–12] by treating the neutrino flavor states να(t)⟩ as parameter-dependent quantum states whose evolution, governed by the PMNS matrix, encodes information about the oscillation parameters. The sensitivity to these parameters can then be quantified in terms of the classical Fisher Information (FI) and the Quantum–1–Fisher Information (QFI), which set the ultimate bounds on their attainable precision via the Cramér–Rao bound. In conventional QET, the main goal is to identify or engineer optimal quantum states and measurement schemes that saturate the Cramér–Rao bound, thereby achieving the ultimate precision allowed by quantum mechanics. In neutrino physics, however, the situation is fundamentally different: the quantum state and the measurement basis are fixed by nature. The neutrino flavor states are uniquely determined by the PMNS matrix, and the detection process effectively performs a flavor measurement. Consequently, the central question is not only what is the ultimate precision limit imposed by the neutrino flavor state itself, but also whether the unavoidable restriction to flavor measurements constitutes an additional intrinsic limitation on the achievable precision for specific oscillation parameters. This is precisely the question we address in this work by comparing the QFI and the FI associated with flavor measurements across all neutrino oscillation parameters. This study extends the analysis of Ref. [13], which focused exclusively on δCP, to all neutrino oscillation parameters, enabling a systematic comparison of their achievable estimation precision. The present paper is structured as follows. We first give a brief overview on QET and describe how this formalism can be adapted to neutrino oscillations and specify the assumptions underlying our analysis in Section 2. We then compute the QFI in Section 3 and the FI in Section 4 for all oscillation parameters, providing a systematic comparison of their sensitivities.

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2 Problem Modelization In this section, we present the basic framework of QET [9–12] and describe how we can describe the estimation of the parameters involved in a neutrino oscillations’ experiment as a quantum metrology protocol, outlining all the assumptions made to simplify our study. 2.1 Quantum estimation theory in a nutshell Let us consider a physical parameter λ we want to estimate and a (classical) measurement that is mathematically described by the conditional probability p(xλ) of observing the classical outcome x given the value of the parameter λ.

1. No longer rely solely on statistical fluctuations for parameter estimation; AI systems can utilize Quantum Fisher Information (QFI) to establish fundamental limits, as the quantum state intrinsically encodes less information about CP violation when compared to mixing angles.

2. Implement optimal measurement strategies based on oscillation maxima; the paper shows that the second peak typically yields larger FI than the first for certain parameter spaces, suggesting AI can dynamically tune effective baselines or beam energy to enhance sensitivity beyond current fixed-beam setups.

3. Develop tailored analysis pipelines for different neutrino beams; since electron antineutrino beams carry more information on δCP than muon (anti)neutrino beams, AI models can prioritize electron antineutrino data when seeking CP violation insights.

4. Benchmark experimental design using the comparison between QFI and FI; AI can assess whether flavor measurements saturate the QFI at the first oscillation maximum for θ13, θ23, and θ12 to confirm if current strategies are optimal or require strategic tuning.