A quantum information theoretic quantity sensitive to the neutrino mass-hierarchy

In this work, we derive a quantum information theoretic quantity similar to the Leggett-Garg inequality, which can be defined in terms of neutrino transition probabilities. For the case of νμ → νe/ν̄μ → ν̄e transitions, this quantity is sensitive to CP violating effects as well as the neutrino mass-hierarchy, namely which neutrino mass eigenstate is heavier than the other ones. The violation of the inequality for this quantity shows an interesting dependence on mass-hierarchy. For normal (inverted) mass-hierarchy, it is significant for νμ → νe (ν̄μ → ν̄e) transitions. This is applied to the two ongoing accelerator experiments T2K and NOνA as well as the future experiment DUNE.


INTRODUCTION
An endeavour to extend the laws of quantum mechanics to the macroscopic level, leads to confrontation with the possibility of macroscopic coherence, viz. the superposition of macroscopically distinct states [1][2][3][4]. A famous example in this context would be the Schrödinger cat being in a superposition of alive and dead states at the same instance. This paradox can be explained by decoherence, where superposition of distinct states is destroyed by coupling to unwanted degrees of freedom [5]. Another approach are so called spontaneous collapse models [6] which propose an ontologically objective mechanism for a collapse in order to avoid the unobserved macroscopic superpositions. Recently, systems in high energy physics including neutrinos were studied to test their sensitivity to such dynamical reduction models [7][8][9].
Founded on the assumption of macrorealism (MR) and non-invasive measurability (NIM) the main goal of the Leggett-Garg Inequality tests (LGI) was to demonstrate macroscopic coherence [10,11]. Macro-realism means that the measurement of an observableQ of a macroscopic system reveals a well defined pre-existing value and NIM states that, in principle, we determine this value without disturbing the future dynamics of the system. These assumptions lead to bounds on certain combinations of two time correlation functions C(t i , t j ) = Q (t i )Q(t j ) , which may not be respected by quantum systems. Since the notion of "realism" is often linked to the existence of hidden variable theories, as a consequence, the violation of LGI nullifies the existence of such theories [12][13][14].
Bell's inequality could be considered as the spatial counterpart of Leggett-Garg inequality. Both capture the so called inherent non-locality of the system. Refs. [15] and [16] show the violation of Bell type inequality in the context of two and three flavor neutrino oscillations [17], respectively. In [10] a method was proposed for the determination of LG inequality and it was shown that LG inequality is violated in a MINOS experimental setup.
Similar observations were reported recently in the context of Daya-Bay experiment in [18]. This analysis was in the context of two flavor neutrino oscillations which does not capture the full dynamics. Moreover, it was based on survival probability, and hence not equipped to capture CP violation in the leptonic sector which has not been observed till date. On the other hand, CP violation has been observed and well established in the quark sector of the standard model [19]. The CP violation in mixing has also shown to be a crucial ingredient to violate a Bell inequality for entangled K-meson pairs [20]. An interplay between various facets of quantum correlations and CP violation in the quark sector has been a topic of numerous interests, see for example [21][22][23]. The complex Yukawa couplings can give rise to the complex phase of the quark mixing matrix. This can also be due to a relative CP violating phase in the vacuum expectation values of the Higgs fields. It is expected that an analogous mechanism would give rise to CP violation in the leptonic sector; several experiments have been put forward for electron-positron systems [24] as well for neutrinos.
In this work, we develop, invoking macrorealism and stationarity, LG type inequalities in the context of three flavor neutrino oscillations. Our findings are not only of pure foundational interest but as we show the welldesigned combination of correlation functions C(t i , t j ), witnessing the impossibility of assigning certain properties of realism to the neutrino system, suggests a resolution of the long standing mass hierarchy problem of neutrino physics. The mass hierarchy problem refers to the ordering of the neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) which is either ν 1 < ν 2 < ν 3 (normal hierarchy) or ν 3 < ν 1 < ν 2 (inverted hierarchy). The ordering of ν 1 < ν 2 is deduced from solar neutrino physics. Knowing the mass hierarchy is twofold interesting; it adds to our understanding of the mass origin in the leptonic sector and provides constraints to other important properties of the rich neutrino physics, such as Majorana versus Dirac particle [28,29], absolute neutrino masses, CP symmetry violation effects or the precision measurements of the flavour oscillations.
Moreover, the LG inequalities can be reduced to the basic observables in neutrino oscillation experiments which are the neutrino survival and transition probabilities. This makes our findings also interesting from the experimental point of view.
The paper is organized as follows. We show that the LG parameters, in various scenarios, can be expressed in terms of these experimentally measured quantities (survival and transition probabilities). We include matter as well as CP violating effects. We find that the LG type inequalities are sensitive to the mass hierarchy problem, i.e., by measuring the LG parameter, one can discern the degeneracy in ∆ 31 = m 2 3 − m 2 1 , where m 1 and m 3 are the masses of the neutrino mass eigenstates ν 1 and ν 3 , respectively. We further study the possibility of measurements of the LG function at currently running accelerator facilities, such as NOνA [27,30] and T2K [31].

LEGGETT-GARG-TYPE INEQUALITIES IN THREE-FLAVOUR NEUTRINO OSCILLATIONS
The first Subsection is devoted to a brief introduction to the main quantities which will be later characterized in the case of three-flavour neutrinos, namely the so-called Leggett-Garg-type Inequalities (LGtI), Subsection 3. In the second Subsection we describe the oscillation formalism.

Leggett-Garg-type Inequalities
Consider a generic finite-dimensional quantum system which is assumed to have a finite transition probability P i→j (t) to change from state |i to state |j during its evolution up to time t. At that time t, imagine that one tries to ascertain whether the system is in a certain target state. This corresponds to performing a measurement of an observableQ which is dichotomic, i.e. whose possible outcomes are ±1 depending on whether the system is or is not found in the target state. The degree of coher-ence of the dynamics is then captured by the autocorrelation function C(t i , t j ) ≡ Q (t i )Q(t j ) [1,2]. According to any realistic description, any measurement would have revealed nothing more than a pre-existing state of the system, and thus the dynamical random variable corresponding to its possible outcome represents a definite value at any instant of time [2]. This would imply a bound on the following linear combination of autocorrelation functions where we have assumed by construction t 1 < t 2 < t 3 .
It is worth stressing that Eq. (1) can be derived from realism alone, without any assumption of non-invasive measurement (NIM) [26]. As stressed already in the Introduction, the aim of the present work is to investigate the coherence of three-flavour neutrinos' dynamics. The latter however do not have a clear macroscopic counterpart, and moreover any insofar measurement of neutrinos requires destroying them, thus rendering the assumption of NIM ill-posed. For these reasons, we decide to go along the line of [13] and characterize the eventual coherence of neutrino's dynamics by testing the predictions of a fully quantum-mechanical description against a sub-class of realistic theories, namely the stationary ones. According to the latter condition, the autocorrelation function C(t i , t j ) actually only depends on the timedifference t j − t i , this leading to the following simplification of Eq. (1) [13,25], known as Leggett-Garg-type Inequalities: where we have assumed that t 1 = 0 and that It is finally worth stressing that, according to [1], a proper derivation of Eq.(2) can be derived provided that: (i) the system is prepared in a given target state |j at initial time t = 0; (ii) the system undergoes a Markovian evolution; (iii) the conditional probabilities P (j, t + τ |j, τ ) are time-translation invariant, i.e., P (j, t + τ |j, τ ) = P(j, t|j, 0). The validity of (i) will be granted by the assumption to start from a well-defined flavour eigenstate of neutrinos and (ii) descends automatically from the fact that, as discussed below in the following Subsection, the dynamics is unitary (and thus Markovian). It is not difficult to show that the timetranslation invariance of conditional probabilities holds here.
Neutrino evolution in vacuum and in constant matter density.
Here we give the fundamental elements of the quantum mechanical description of neutrino dynamics. In the fol-lowing Section, we will employ them to calculate the autocorrelation function C(t i , t j ) of a particular dichotomic observable and compare the result with that of a stationary and realistic theory through the test of violations of LGtI (2).
A general neutrino state in flavor basis (|ν α , α = e, µ, τ ) is given by the superposition |Ψ(t) = ν e (t) |ν e + ν µ (t) |ν µ + ν τ (t) |ν τ . The same state can be expressed in terms of mass eigen-basis (|ν i , i = 1, 2, 3), |Ψ(t) = ν 1 (t) |ν 1 + ν 2 (t) |ν 2 + ν 3 (t) |ν 3 . The coefficients in two representations are connected by a unitary matrix where U αi are the elements of a 3 × 3 unitary PMNS (Pontecorvo-Maki-Nakagawa-Sakata) mixing matrix U parametrized by three mixing angles (θ 12 , θ 23 , θ 32 ) and a CP − violating phase δ. Experimental values for the PMNS mixing matrix can be taken from the particle data group [19]. The coefficients in flavor basis at different times, i.e., ν α (t) in terms of ν α (0), is given by In the absence of matter effects, the elements of the flavor evolution matrix U f are functions of the PMNS matrix parameters. When neutrinos propagate through matter with constant (electron) density N e , because of a feeble interaction with electrons, the Hamiltonian, which is diagonal in the mass-eigen basis, H m = diag[E 1 , E 2 , E 3 ], picks up an interaction term, diagonal in flavor basis, V f = diag[A, 0, 0]. Here A = ± √ 2G F N e , the matter density parameter and G F is the Fermi coupling constant. The flavor evolution operator for constant matter density, takes the form [32] where φ = e −i T rHm 3 L , c 1 = det(T )tr(T −1 ) and the Hamiltonian in mass basis is The matrixT = U T U −1 , where U is the PMNS mixing matrix. T is a hermitian matrix, form given in [32], with eigenvalues λ n (n = 1, 2, 3). When neutrinos travel through a series of matter densities with matter density parameters A 1 , A 2 , . . , A n with thicknesses L 1 , L 2 , . . . , L n , the total evolution operator is given by where L = n i=1 L i and U f (L i ) is calculated for density parameter, A i . This operator, for example, can be used to study neutrino oscillations in a mantle-core-mantle step function model simulating the Earths matter density profile [33].
Leggett-Garg type inequality for neutrinos.
Here we derive an analytic expression for the LGtI (2) in the case of three-flavoured neutrinos, when the latter are prepared in a specific flavour state |ν α (α = e/µ/τ ) and where the dichotomic observable selected iŝ Q = 2 |ν α ν α | − 1. The latter therefore makes quantitative the inquiry as to whether the neutrino is still in the initial flavour eigenstate.
Under the stationary assumption, the autocorrelation functions C(t i , t j ) ≡ C(0, t) are straightforwardly found and can be compactly expressed in terms of the probability P α→β (t) = | ν α (t)|ν β | 2 as Note that both the survival probability P α→α (t) and the transition probability P α→β (t) actually depend on many physical parameters such as the neutrino energy E, the mass square differences ∆ ij = m 2 j − m 2 i , the matter density parameter A, the mixing angles θ ij and the CP − violating phase δ. For clarity of notation however, we will keep the dependence on all these parameters implicit except for the energy E, for reasons that will be clear in a short-while. Care however should be taken when moving from neutrinos to anti-neutrinos since both the CPviolating phase δ and the interaction potential A reverse their sign [34]. The LGtI defined in Eq. (2) thus becomes which shows its experimental feasibility, being clearly expressed only in terms of measurable quantities, i.e., survival and transition probabilities.
In light of its physical grasp, the following considerations on Eq. (9) naturally follow. Neutrino oscillation experiments typically operate in the ultra-relativistic regime and therefore the time-dependence in the probabilities P α→β (t, E) can be equivalently replaced by the length L travelled by neutrinos [37]. An experimental verification of Eq. (9) would thus require two detectors placed at L and 2L, respectively. The current experimental facilities however do not allow for such a setup. It is nevertheless possible to bypass such obstacle by looking for matching energiesẼ satisfying the implicit equation P α→β (2L, E) = P α→β (L,Ẽ). (10)

APPLICATIONS AND DISCUSSION
We will now exploit our result Eq. (9) to explicitly evaluate the LGtI for neutrinos and antineutrinos using realistic parameters from two existing experimental facilities, namely NOνA and T 2K. Since both of them employ ν µ sources and are appearance experiments, i.e., they study the transition probabilities P (µ → e)(t, E), we will concentrate on the evaluation of for ∆ 31 , where the considerations made at the end of the previous Section have been taken into account.

NOνA
With a long baseline of 810 km, this experiment uses Fermi NUMI beamline as its source. The neutrino energy range in NOνA varies within 1 to 10 GeV. The neutrinos pass through Earth's mantle, whose density parameter is A = 1.7 × 10 −13 eV (ρ ≈ 4.5 gm/cm 3 ). In Fig. 1 we plot the maximum of K 3 as given by Eq. (11) with respect to neutrino energy, as function of δ. More explicitly, the maximization is performed for ∆ 31 at fixed values of the CP −violating phase {δ k } and within the energy window of the experimental setup. It is also worth stressing that the value ofẼ is found by solving Eq. (10) after imposing the above constraints on the mass square difference ∆ 31 and δ = δ k .  It is immediately evident from Fig. 1 that the computation of the LGtI K 3 remarkably allows also to remove the well-known degeneracy in the neutrino masses ∆ 31 for all values of CP violating phase δ. This result alone provides a benchmark achievement in the field of neutrino physics and is another clear evidence, alongside with [15,16], of how the crossbreed with quantum information can help shedding unexpected light on neutrino physics.  For the sake of completeness, we also give in Fig. 2 the result of K 3 , for a fixed value of energy corresponding to the neutrino flux at 4.3 GeV and for both signs of mass square difference, with respect to CP −violating phase. It is worth noticing that here the LGtI still allows for the discernment of the mass degeneracy. One can infer that the maximal violation of K 3 can be at the energies different from that at which the experiment is designed to have the maximum energy flux.

T2K
This facility's aim is to measure the last unknown lepton sector mixing angle θ 13 by observing ν e appear-ance in a ν µ beam. The neutrino source is in this case represented by the J-PARC accelerator in Tokai, Japan. T 2K uses Super-Kamiokande as the far detector to measure neutrino rates at a distance of 295 km from the accelerator, and near detector to sample the beam just after production. The rock density parameter is A = 1.01 × 10 −13 eV (ρ ≈ 2.8 gm/cm 3 ).  Dotted and crossed curves correspond to the positive and negative signs of ∆31, respectively. Length L is taken to be 295 km and energy E is varied between 1 GeV to 2 GeV. The top and bottom panels correspond to neutrinos (A positive) and anti-neutrinos ( A negative), respectively. The horizontal grid-lines in above plots are drawn to highlight the regions in which degeneracy in ∆31 can be removed.
As for the NOνA experiment, we give in Figs. 3 and 4 the results of the calculation of the maximum of K 3 and K 3 as given by Eq. (11), respectively for energy varying in the experimentally allowed range and for the value of energy equal to 1.6 GeV.
Analogous conclusions as those stemming from the analysis of the data taken from NOνA can be drawn, namely the LGtI appear to be violated both in the case of neutrinos and antineutrinos and for almost every value of the CP −violating phase. In particular, we recover also here that the degeneracy in the mass square can be lifted for all values of δ. This shows the robustness and the repeatability of the proposed scheme to remove the degeneracy on ∆ 31 put forward in the present work.  It is finally important to stress that an important factor determining such a neat distinction of the two masses ±∆ 31 is played by the considerable length L of the two baselines considered, i.e., NOνA and T 2K. The latter in fact, as explained above, directly relates with the flying time of the neutrinos analyzed and thus directly affects the figure of merit considered in Eq. (11).

CONCLUSION
In this work we have evaluated LGtI in the context of three-flavor neutrino oscillations and found a closed expression for the K 3 function in terms of simple and measurable quantities such as the neutrino survival and oscillation probabilities. Having taken into account also for matter, we have shown that the LGtI is violated for both neutrinos and anti-neutrinos for almost all the values of the CP − violating phase δ. In addition, the explicit calculation of the maximum of K 3 using inputs from two important experiments, namely NOνA and T 2K, have shown how this information-theoretical figure of merit allows to lift the degeneracy on the square masses ∆ 31 for all the values of the CP −violating phase. The present work therefore paves the way for interesting cross fertilizations of ideas where foundational concepts of quantum mechanics are used fruitfully to address fundamental problems in particle physics such as CP violation and the mass hierarchy problem.