Quantum correlations and the neutrino mass degeneracy problem
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Abstract
Many facets of nonclassicality are probed in the context of three flavour neutrino oscillations including matter effects and CP violation. The analysis is carried out for parameters relevant to two ongoing experiments \(\hbox {NO}\nu \hbox {A}\) and T2K, and also for the upcoming experiment DUNE. The various quantum correlations turn out to be sensitive to the masshierarchy problem in neutrinos. This sensitivity is found to be more prominent in DUNE experiment as compared to \(\hbox {NO}\nu \hbox {A}\) and T2K experiments. This can be attributed to the large baseline and high energy of the DUNE experiment. Further, we find that to probe these correlations, the neutrino (antineutrino) beam should be preferred if the sign of mass square difference \(\varDelta _{31}\) turns out to be positive (negative).
1 Introduction
Quantum Mechanics has proved to be an incredibly successful theory. Not only have its predictions been verified with great accuracy, but it has also laid the foundation of new realms of technology, ready to revolutionize the information and communication sectors. Surprisingly, despite all of its success, the question of when does a system behave quantum mechanically rather than classically, still waits for a clear and unambiguous answer. This question becomes important while dealing with the nature of correlations between different subsystems of a composite system. These correlations can be spatial as well as temporal. Some of the widely studied spatial quantum correlations are entanglement [1], steering [2], nonlocality [3] and quantum discord [4]. The temporal correlations include Leggett–Garg (LG) [5] and LG type inequalities [6].
The quantum correlations have been studied mainly in the optical and electronic systems [7, 8, 9, 10]. Recently such studies have been extended to high energy physics owing to the advancement in various experimental facilities, see for example [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. The concept of single particle entanglement has been introduced in previous studies [25, 26, 27] which have also been demonstrated experimentally with single photon systems [28, 29, 30]. Later, the experimental schemes to probe nonlocality were generalized to include massive particles [31]. In [14], an experimental scheme is discussed for transferring this form of entanglement to spatially separated modes of stable leptonic particles. It allows to put mode entanglement in neutrino oscillations on equal footing with that in atomic and optical systems. Therefore, different flavour modes of neutrinos can be expressed as legitimate individual entities and entanglement in these flavour modes, i.e. modeentanglement, studied. An extensive study of quantum correlations in the context of two and three flavour neutrino oscillations is given in [17] and [18], respectively. However, in these works, matter and CP (charge conjugationparity) violating effects were not taken into account.
In this work, we study various facets of nonclassicality, quantified by spatial quantum correlations such as flavour entropy, geometric entanglement, Mermin and Svetlichny inequalities, in the context of three flavour neutrino oscillations, by taking into account the matter effects and CP violation. We discuss the behavior of these quantum correlations for the ongoing experiments like \(\hbox {NO}\nu \hbox {A}\) and T2K, and also for the upcoming experiment DUNE. We find that the various witnesses show sensitivity to the masshierarchy problem and CP violation in neutrino physics.
For a general and physically reliable study of the neutrino oscillation phenomena, one should look in terms of the wave packet approach, i.e. localization effects of production and detection processes should be considered. However, the plane wave approximation also holds good since the oscillation probability obtained with the wavepacket treatment is found to be in consonance with the planewave oscillation probability averaged over the Gaussian L / E distribution [32]. Here L and E represent the distance travelled by the neutrino and its energy.
The paper is organized as follows: In Sect. 2, we give a brief description of the dynamics of the neutrino oscillation in three flavour case in vacuum and constant matter density. Section 3 is devoted to a brief description of various quantum correlations studied in this work. Section 4 gives the results and their discussion. We finally summarize our work in Sect. 5.
2 Neutrino dynamics in vacuum and constant matter density

T2K (TokaitoKamioka) is an offaxis experiment [44, 45] using a \(\nu _{\mu }\) – neutrino beam originating at JPARC (Japan Proton Accelerator Complex) with energyrange of approximately 100 MeV to 1 GeV and the baseline of 295 km.

\(\hbox {NO}\nu \hbox {A}\) (NuMI OffAxis \(\nu _e\) Appearance), the long baseline experiment, uses neutrinos from NuMI (Neutrinos at the Main Injector) beamline at Fermilab optimized to observe \(\nu _\mu \rightarrow \nu _e\) oscillations. This experiment uses two detectors, both located at 14 mrad off the axis of the NuMI beamline, the near and far detectors are located at 1 km and 810 km from the source, respectively. The flavour composition of the beam is \(92.9\%\) of \(\nu _\mu \) and \(5.8\%\) of \(\bar{\nu }_\mu \) and \(1.3\%\) of \(\nu _e\) and \(\bar{\nu }_e\); the energy of the neutrino beam varies from 1.5 GeV to 4 GeV. The spectrum for NuMI beamline for various offaxis locations is given in [46, 47, 48].

DUNE is an experimental facility which uses NuMI neutrino beam with energy range of 1–10 GeV from Fermilab and has a long baseline of 1300 km. This enables L/E, of about \(10^3\) km/GeV, to reach good sensitivity for CP measurement and determination of mass hierarchy [49].
In the next section we analyze the behavior of various quantum correlations in the context of the experiments described above.
3 Measures of quantum correlations
 1.Flavour entropy For the pure states (15), the standard measure of entanglement is given as [18]This measure serves as a tool to probe the nonclassicality of the system. In the context of neutrino oscillation, the flavour entropy parameter \(\mathscr {S} = 0\) for an initially prepared neutrino state \(\nu _\alpha \) (\(\alpha =e, \mu ,\tau \)), and reaches its upper bound \(\mathscr {S}=1\) for the maximally nonclassical state in the W class \(\frac{1}{\sqrt{3}} (100\rangle + 010\rangle + 001\rangle )\) [50].$$\begin{aligned} \mathscr {S}(\xi _{i}^2) =&\sum _{i=1}^{3} \xi _i ^2 \log _2(\xi _i^2) \nonumber \\&\sum _{i=1}^{3} (1\xi _i ^2) \log _2(1\xi _i^2). \end{aligned}$$(16)
 2.Tripartite geometric entanglement Tripartite geometric entanglement G for the pure states, given in Eq. (15), is defined as the cube of the geometric mean of Shannon entropy over every bipartite section.where \( H(p)\equiv  p ~log_2(p)(1p)~log_2(1p)\) is the bipartite entropy. This is a weaker condition than genuine tripartite nonlocality discussed below. The genuine tripartite entanglement does not exist if \(G=0\).$$\begin{aligned} G = H(\xi _1(t)^2)H(\xi _2(t)^2)H(\xi _3(t)^2), \end{aligned}$$(17)
 3.Absolute and genuine tripartite nonlocality (Mermin and Svetlichny inequalities) The violation of a Bell type inequality (viz., CHSH) for a two qubit state is said to imply nonlocality. A generalization to three party system is not straightforward. Mermin inequality is based on the assumptions that all the three qubits are locally and realistically correlated; hence a violation would be a signature of the tripartite nonlocality shared among the qubits. It was shown in [51, 52] that the biseparable states also violate the Mermin inequality. This motivated Svetlichny to formulate a hybrid nonlocallocal realism based inequality, the Svetlichny inequality. A three qubit system may be nonlocal if nonclassical correlations exist between two of the three qubits. Such a state would be absolute nonlocal and will violate Mermin inequality [53] for a particular set of detector setting (A,B,C) and (\(A^\prime \),\(B^\prime \),\(C^\prime \)). The two Mermin inequalities are:However, a violation of Mermin inequality does not necessarily imply genuine tripartite nonlocality. A state violating a Mermin inequality may fail to violate a Svetlichny inequality, which provides a sufficient condition for genuine tripartite nonlocality [54] and is given by$$\begin{aligned} \begin{aligned} M_1 \equiv \left\langle ABC'+ AB'C + A'BC  A'B'C'\right\rangle \le 2,&\\ M_2 \equiv \left\langle ABC  A'B'C  A'BC'  AB'C'\right\rangle \le 2. \end{aligned} \end{aligned}$$(18)$$\begin{aligned} \sigma \equiv M_1 + M_2 \le 4. \end{aligned}$$(19)
4 Results and discussion
For DUNE experiment, Fig. 1 depicts the variation of the maximum of various quantum witnesses like flavour entropy, geometric entanglement, Mermin parameters (\(M_1\), \(M_2\)) and Svetlichny parameter (\(\sigma \)) with respect to the CP violating phase \(\delta \), for the case of neutrino and antineutrino, respectively. It can be seen that all the witnesses show different characteristics for the positive and negative signs of large mass square difference \(\varDelta _{31}\). Figures 2 and 3 depict the same for ongoing \(\hbox {NO}\nu \hbox {A}\) and T2K experiments, for neutrino beam. The corresponding antineutrino plots show similar features, such as inversion of masshierarchy, as in the DUNE plots and hence are not depicted here.
A general feature observed in these results is that the different measures of nonclassicality are sensitive to the sign of \(\varDelta _{31}\). The distinction being more prominent in DUNE experiment compared to the \(\hbox {NO}\nu \hbox {A}\) and T2K experiments. This can be attributed to the high energy and long baseline of the DUNE experiment.
The quantum correlation measures studied in this work can attain their upper bounds for some specific values of L / E [18]. In the present study, however, by taking into account the matter effects and CP violation, we are restricting L / E within the experimentally allowed range; consequently the various nonclassical measures do not reach their maximum allowed values. Mermin inequalities are violated for all values of \(\delta \) which means that if one of the three parties is traced out, still there will be residual nonlocality in the system. Violation of the Svetlichny inequality reflects the nonlocal correlation between every subsystem of the tripartite system. To achieve significant violation of correlation measures one should use neutrinobeam if the sign of \(\varDelta _{31}\) is positive (normal mass hierarchy), while antineutrinobeam should be used in case of negative sign of \(\varDelta _{31}\) (inverted mass hierarchy).
From the definitions of flavour entropy (Eq. 16) and geometric entanglement (Eq. 17), it is clear that these are measurable quantities since these are written in terms of survival and oscillation probabilities making them suitable for experimental verification. Expressing the Mermin and Svetlichny parameters in terms of measurable quantities is nontrivial here. However, guided by the previous work [17], the measures of quantum correlations viz. BellCHSH inequality, teleportation fidelity and geometric discord have been expressed in terms of survival and transition probabilities for two flavour neutrinosystem. It could be envisaged that such an exercise, though complicated, could be carried out for the three flavour case.
5 Conclusion
Different facets of nonclassicality have been investigated for the neutrino system by considering the three flavour scenario of neutrino oscillation. The matter effects are included in order to carry out the analysis in the context of the ongoing neutrino experiment \(\hbox {NO}\nu \hbox {A}\) and T2K and also for the future experiment DUNE. The analysis is carried out by considering both neutrino and antineutrino beams for the experiments. The quantum correlations show sensitivity to the neutrino mass hierarchy, i.e. the sign of \(\varDelta _{31}\). It is a general feature displayed by all the correlations that the sensitivity to the mass hierarchy becomes more prominent for the high energy and long baseline experiment like DUNE compared to \(\hbox {NO}\nu \hbox {A}\) and T2K experiments. The results also suggest that in order to probe the various measures of nonclassicality in neutrino sector, one must use neutrino beam for the positive sign of \(\varDelta _{31}\) and an antineutrino beam otherwise.
Notes
Acknowledgements
We acknowledge useful discussions with B.C. Hiesmayr and G. Guarnieri. This work is partially supported by DST IndiaBMWfW Austria Project Based Personnel Exchange Programme for 20172018. SB acknowledges partial financial support from Project no. 03 (1369)/16/EMRII, funded by the Council of Scientific & Industrial Research, New Delhi.
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