A quantum-information theoretic analysis of three-flavor neutrino oscillations

Correlations exhibited by neutrino oscillations are studied via quantum-information theoretic quantities. We show that the strongest type of entanglement, genuine multipartite entanglement, is persistent in the flavor changing states. We prove the existence of Bell-type nonlocal features, in both its absolute and genuine avatars. Finally, we show that a measure of nonclassicality, dissension, which is a generalization of quantum discord to the tripartite case, is nonzero for almost the entire range of time in the evolution of an initial electron-neutrino. Via these quantum-information theoretic quantities, capturing different aspects of quantum correlations, we elucidate the differences between the flavor types, shedding light on the quantum-information theoretic aspects of the weak force.

Neutrino oscillations are fundamentally three flavor oscillations.However, in some cases, it can be reduced to effective two flavor oscillations [21].These elementary particles interact only via weak interactions, consequently the effect of decoherence, as compared to other particles widely utilized for quantum information processing, is small.Numerous experiments have revealed interesting details of the physics of neutrinos [22][23][24][25][26][27].This paper asks what type of quantum correlations is persistent in the time evolution of an initial ν e or ν µ or ν τ .It presents a systematic study of the many-faceted aspect of quantum correlations.Herewith, it contributes to the understanding how Nature processes quantum information in the regime of elementary particles and, in particular, which aspect of quantum information is relevant in weak interaction processes.
Three-flavor neutrino oscillations can be studied by mapping the state of the neutrino, treating it as a three-mode system, to that of a three-qubit system [16,17].In particular, it was shown that the neutrino oscillations are related to the multi-mode entanglement of single-particle states which can be expressed in terms of flavor transition probabilities.Here we take the study of such foundational issues further by characterizing three-flavor neutrino oscillations by quantum correlations.This is nontrivial as quantum correlations in three-qubit systems are much more involved compared to their two-qubit counterparts.
The present study of quantum correlations in three flavor neutrino oscillations can be broadly classified into three categories: • Entanglement: We study various types of the inseparability properties of the dynamics of neutrino oscillations via the von Neumann entropy and in terms of a nonlinear witness of genuine multipartite entanglement introduced in Ref. [28].
• Genuine multipartite nonlocality: Nonlocalitythat is considered to be the strongest manifestation of quantum correlations -is studied in both its absolute and genuine tripartite facets, characterized by the Mermin inequalities [29] and Svetlichny inequalities [30].
• Dissension: A tripartite generalization of quantum discord which is a measure of nonclassicality of correlations [31].
The plan of the paper is as follows.In Sec.II, we provide a brief introduction to the phenomenology of neutrinos and introduce the three-flavour mode entangled state which will be analysed using information theoretic tools.The core of the paper is Sec.III, where we characterize three flavor neutrino oscillations in terms of various facets of quantum correlations.We then conclude by providing an outlook.

II. THREE FLAVOUR NEUTRINO OSCILLATIONS
The three flavours of neutrinos, ν e , ν µ and ν τ , mix to form three mass eigenstates ν 1 , ν 2 and ν 3 : where U is the 3×3 PMNS (Pontecorvo-Maki-Nakagawa-Sakata) mixing matrix parameterized by three mixing angles (θ Therefore, each flavor state is given by a linear superposition of the mass eigenstates, where α = e, µ, τ ; k = 1, 2, 3.As the massive neutrino states |ν k are eigenstates of the Hamiltonian with energy eigenvalues E k , the time evolution of the mass eigenstates |ν k is given by where |ν k are the mass eigenstates at time t = 0. Straightforwardly, the time evolution of flavor neutrino states computes to For example, if an electron-neutrino is produced at time t = 0 then its time evolution is given by |ν e (t) = a ee (t) |ν e + a eµ (t) |ν µ + a eτ (t) |ν τ , (7) where If we assume that the detected neutrinos have an energy of at least one MeV (the electron/positron mass), namely are in the ultrarelativistic regime, the flavor eigenstates are well defined in the context of quantum mechanics [16].In this approximation the survival probabilities take the form and the oscillation probabilities where ∆m As in the neutrino oscillation experiments, the known quantity is the distance L between the source and the detector and not the propagation time t, therefore the propagation time t is replaced by the source and detector distance L in the above equation.This is a valid approximation as all detected neutrinos in the oscillation experiments are ultrarelativistic.
The allowed ranges of the six oscillation parameters, three mixing angles and three mass squared differences, is obtained by a global fit to solar, atmospheric, reactor and accelerator neutrino data within the framework of three flavor neutrino oscillations.For normal ordering, the best fit values of three flavor oscillation parameters are [32] θ 12 = 33.48Following Ref. [16] we introduce the occupation number of neutrinos by making the following correspondence: Consequently, we can view the time evolution of a flavor eigenstate α = e, µ, τ as a three-qubit state, i.e., ) Therefore, flavor oscillations can be related to time variation of the tripartite entanglement of single particle states.

III. STUDY OF QUANTUM INFORMATION THEORETIC PROPERTIES IN NEUTRINO OSCILLATIONS
Separability or the lack of separability, i.e., entanglement, is defined for a given state according to its possible FIG.1: Plot of the (normalized) flavor entropy (solid line, red) and the three probabilities (νe → νe(pink, dashed) (survival probability, Eq. ( 8)), νe → νµ(lightblue, dashed), νe → ντ (lightgreen, dashed) (oscillation probabilities, Eq. ( 9)) for an initial electron-neutrino state |Ψ(t = 0) e = νe(0) as a function of the distance traveled per energy L/E.factorization with respect to a given algebra [33].The separability problem is in general a NP-hard problem, and only necessary but not generally sufficient criteria exists to detect entanglement.For bipartite entangled quantum systems it suffices to ask whether the state is entangled or not.In the multi-partite case the problem is more involved since there exist different hierarchies of separability (defined later).We have defined the algebra by introducing the occupation number of the three flavours and our first goal is to understand the time evolution of neutrino oscillation in terms of tools for classifying and detecting different types of entanglement.
The next step would be to take potential measurement settings into account and analyse the different facets of correlations in the dynamic of neutrinos.In particular, we are interested whether there are correlations stronger than those predicted by any classical theory.The correlations are studied via two different approaches, one based on the dichotomy between predictions of quantum theory and different hidden parameter theories, and the other one quantifies the various information contents via entropies.

A. Study of the entanglement properties
Entanglement measures quantify how much a quantum state ρ fails to be separable.Axiomatically, it must be a nonnegative real function of a state which cannot increase under local operations and classical communication (LOCC), and is zero for separable states.An entropic function generally quantifies the average information gain by learning about the outcome obtained by measuring a system.The von Neumann entropy, a quantum mechanical analogue of Shannon entropy, is defined by S(ρ) = −ρ log ρ and is zero for pure states and log(d) for the totally mixed state, where d is system dimension, and the log function usually refers to base 2. The entan-glement content can be computed by the entropy of the subsystems since the full system is pure.
Considering the three possible partial traces of the three-qubit state under investigation, we obtain a concave function of the single-mode probabilities |a αβ (t)| 2 , i.e., with ρ j := Tr all but not subsystemj |Ψ(t (14) which we call the flavor entropy.This function is plotted in Fig. 1 together with the survival probabilities ν e → ν e , Eq. ( 8), and the oscillation probabilities ν e → ν µ , ν e → ν τ , Eq. ( 9).
Since the flavor entropy S flavour (|Ψ(t) e ) is nonzero for almost all time instances, the state is entangled.For this and all the following plots, we use as units, the oscillation period of an electron to a muon-neutrino.Since there are tiny changes in the behaviour in one period due to the existence of the third flavour, we always plot two periods.When the amount of all three probabilities, both the survival as well as the oscillation probabilities, become nearly equal, the flavour entropy becomes maximal.Then the ν µ and ν τ oscillation probabilities become greater than the survival probability of ν e resulting in a decrease in the uncertainty of the total state followed by an increase, when the probabilities get closer.Next, the uncertainty in the total state drops again and the pattern gets repeated.
The entropy of all the three possible flavour entropies are compared in Fig. 2 showing that for the muon-and tau-neutrinos the entropy is non-zero for almost all time instances.Compared to the electron-neutrino evolution, the flavor uncertainty of the other two flavours oscillates more rapidly and with higher amplitudes, reaching the maximal value more often.
Let us now refine the picture by investigating the type of entanglement in neutrino oscillations.A tripartite pure state can be, for example written as where k gives the number of partitions dubbed the kseparability.If k equals the number of involved states, in our case k = 3, the joint state is called fully separable, else it is partially separable.An important class of states are those that are not separable within any bi-partition, and are called genuinely multi-partite entangled.In general they allow for applications that outperform their classical counterparts, such as secret sharing [34,35].It should be noted that since a k = 3-separable state is necessarily also k = 2-separable, k-separable states have a nested-convex structure.Among the genuinely multi-partite entangled states, there are two subclasses known for three qubit states, the GHZ-and W-type of states.In Ref. [28] a general framework was introduced to detect and define different relevant multipartite entanglement subclasses and refined in several follow ups.In particular it has been shown to allow for a self-consistent classification also in a relativistic framework [36].Generally, one would expect from a proper classification of different types of entanglement that for a relativistically boosted observer, which causes a change of the observed state, but not of the expectation value, that it remains in a certain entanglement class.We will therefore investigate this Lorentz invariant criterion, though let us emphasise that we do not take any relativistic effects of a boosted observer into account in this contribution.
The necessary criterion for a tripartite qubit state with one excitation ("1") to be bipartite reads If this criterion is violated the state ρ has no bipartite decompositions, i.e., it is genuinely multipartite entangled.
The positive terms are exactly the only non-zero offdiagonal terms of the W -state, |W = 1 √ 3 {|100 + |010 + |001 }, with one excitation in the computational basis, whereas the negative terms are only diagonal terms.Note that these negative terms are all zero for the W -state in the given basis such that only this state obtains the maximum value.
Obviously, this criterion depends on the basis representation of the state ρ and has therefore to be optimized over all local unitary operations.Indeed, taking the "flavour basis" as the computational basis, Eq. ( 13), the unoptimized criterion becomes which is not violated for all times.Consequently, optimization over all local unitaries has to be taken into account for each time point and is plotted for an initial electron-, muon-and tau-neutrino in Fig. 3.We find that the states at each time point are always genuine multipartite entangled if at least two amplitudes of the state, Eq. ( 13), are non-zero, i.e., for almost all time instances.The results depicted in Fig. 3 also prove that in the course of the time evolution the genuine multipartite W state (all amplitudes equal to 1 √ 3 ) is reached.Hence, Nature exploits the maximum genuine multipartite entanglement in the occupation number basis.

B. Genuine multipartite nonlocality
We now ask the question whether in the course of the flavor oscillations, Bell-type nonlocality is persistent, i.e., there are correlations stronger than those predicted by any classical hidden variable theory.For that we investigate the Svetlichny inequalities [30] which are a sufficient criteria for proving genuine tripartite nonlocality.In short, the idea is whether by measuring three observables A, B, C and obtaining the results a, b, c, the probability P (a, b, c) can be assumed to be factorizable as where f, h are probabilities conditioned to the hidden variable λ with the probability measure dω.The factorization, here chosen between the partitions A, B versus C, corresponds to Bell's locality assumption in his original derivation if considered for two systems.(The requirement of a full factorization, i.e., the additional factorization f(ab|λ) = q(aλ)•r(bλ), which correspond to absolute locality, is explored later by the inequalities ( 20)).
Then the necessary criteria for such a factorization of the conditioned probabilities, qubit 1 and 2 versus 3, are given by with D = B + B and D = B − B .Since we are not interested in a particular hidden parameter model, we also consider the two other bipartitions, namely 2|13 and 3|12, and take the maximum over them.In Fig. 4 we have plotted the maximum of I a and I b , over all bipartitions, for the time evolution of an initial electron-, muon-and tau-neutrino.In addition, each data point corresponds to the maximum of the optimization over all possible observables A, B, C. In the case of an initial electron-neutrino we find regions in the time evolution when the criterion does not detect genuine tripartite nonlocality, whereas for the two other neutrino flavours we observe a stronger oscillating behaviour.Summing up, whereas genuine nonlocal correlation is largely present in the time evolution, there are specific time regions when it vanishes.
Requiring that for all three measurements a hidden parameter model should exist can be revealed by the following set of inequalities [29] which are connected to the Svetlichny inequality by I a 12|3 = M a + M b (see Refs. [30,37]).These are the Mermin inequalities and their violation is an indicator of absolute nonlocality.Again we are interested in finding a contradiction to any hidden parameter model, thus we consider all bipartitions and take the maximum.The results are plotted in Fig. 5 (including an optimization over all four arbitrary operators A, D, C, D ).For all times (except when the state is separable) the two inequalities are violated when optimized over all measurement settings.This shows that assuming that the mode correlations can be simulated by an ensemble where all three subsystems are correlated to each other for all time instances is not possible.In contrast, correlations simulated by a hybrid nonlocal-local ensemble, captured by inequalities (19), may exist for time instances close to the separable state, however, only for the electron-neutrino dynamics.
To sum up, except for small time regions neutrino oscillations exhibit all the strong correlations, entanglement and Bell-type nonlocality, that are considered to give an advantage to quantum theory over classical theories for a number of information processing tasks.For completeness, in the next section we investigate the behavior of a measure of nonclassicality weaker than entanglement.

C. Dissension-a measure of nonclassicality
Classical mutual information, quantifying the information between two random variables A and B, can be defined by I(A:B) = H(A) − H(A|B), where H(A) = − i p i log p i is the Shannon entropy of the probabilities p of the outcomes of A and H(A|B) := H(A) − H(A, B) represents the classical conditional entropy and H(A, B) is the joint entropy of the pair of random variables (A, B) (see, e.g., Ref. [1]).Mutual information can be generalized for three random variables A, B, C by any of the following three equivalent expressions [38] While the second of these expressions suggests a straightforward quantum generalization, by replacing the Shannon entropy by the corresponding von Neumann entropy S(ρ) ≡ −Tr(ρ log ρ), the first and third expressions lead to complications since the average conditioned entropy depends on the basis chosen and on the choice of the random variables A, B, C. Let us point out that in strong contrast to the bipartite mutual information the tripartite mutual information may also be negative.This is the case if for instance knowing the random variable C enhances the correlation between A and B. Following the concept of quantum discord [39,40], which quantifies nonclassical correlations, in Ref. [31] two measures for nonclassicality, called dissension, were introduced: where J i are the quantum analogs of classical tripartite mutual information I i , Eq. ( 21), namely Here k is chosen such as to minimize the uncertainty.A given state is denoted to be nonclassical for any departure of D 1 or D 2 from 0.Here D 1 and D 2 deviations from zero can be associated to nonclassicality Dicke detecting genuine multipartite entanglement, Eq. ( 16), optimized over local unities for the three initial flavour states (a) νe, (b) νµ, (c)ντ with respect to the distance traveled per energy L/E in units of the oscillation period of the two lightest neutrinos (300 data points).The criterion detects genuine multipartite entanglement if greater than zero and is maximal (= 1) only for the W -state. accessed by only one-mode or two-mode measurements, respectively.We find that J 2 is always zero, since S(A, B, C) is zero because the total state is pure and since S(A) = S(B, C), S(B) = S(A, C), S(C) = S(A, B), this being a particularity of the W class of states.Furthermore, any permutation of the entropy S(A|Π B,C ) = 0 is zero, since any projection onto the two-mode subspace gives a pure state which has zero uncertainty.Consequently, the relevant measures reduce in our case to Here the three terms in the bracket of D 1 refer to the single electron-neutrino, single muon-neutrino and single tau-neutrino mode measurements, respectively.The three terms in the bracket of D 2 refer to joint bipartite measurements in the muon-tau, electron-tau, muonelectron mode subspace, respectively (which are minimized to zero).In Fig. 6 we plot the dissensions D 1 , D 2 minimized over all projective measurements for the time evolution of an initial electron-neutrino.The first notable point is that both measures are very sensitive to whether the nonclassicality is accessed by single or bipartite measurements and both measures are non-zero for almost all times.Interestingly, we find that for both measures min D 1 , min D 2 and all measurement types there are time regions for which the value exceeds the corresponding value for the W -state, which has (min D 1 , min D 2 ) = (−1.738,0.918).For single measurements dissension D 1 is still considerably smaller than the values for the GHZ state (min D 1 = −3), in contrast to D 2 where min D 2 = 1.Moreover, a strong "twin-humped" pattern of D 2 in the time evolution is found for joint measurements in the subspace of the two heavier neutrinos showing the existence of the third neutrino flavour (τ ).

IV. CONCLUSIONS AND OUTLOOK
To sum up, we have computed several information theoretic quantities detecting and classifying correlations for the time evolution of an initial electron-, muon-or tauneutrino.We find that for almost all time instances the neutrino states exhibit genuine quantum features.
We have analysed in detail the dynamics of initial neutrino states via various types of entanglement properties, correlations that cannot be simulated by realistic hidden variable theories and non-classical correlations revealed by mutual information measures.In particular, dissension turned out to be larger than that for the perfect W -state (Dicke state), for some time values, in strong contrast to the measures not involving measurements, i.e., the flavor entropy and the criterion detecting genuine multipartite entanglement.What physical significance this carries, if any, remains to be seen.
Qualitatively, there are differences between an initial electron-neutrino and the other two neutrinos, i.e., with the former showing less nonclassical features when compared to its heavier counterparts, a point that may merit further scrutiny.In any case, we can conclude that foundational issues are more prominent in accelerator experiments (mainly producing muon-neutrinos) than in reactor experiments (mainly producing electron-neutrinos).
The weak force, being one of the four known fundamental forces in Nature, dominant in the flavour changing process of neutrinos, reveals strong genuine quantum features such as also shown for weakly decaying spinless K-mesons [41] or for the weakly decaying half integer spin hyperons [42].The next step would be to understand how and whether Nature takes advantage of these strong quantum correlations for information processing in a natural setting.24), minimized over all projective measurements for the time evolution of an initial νe as a function of the distance per energy L/E: (a) single-mode measure D1 and (b) two-mode measure D2.The colors encode the dependence on the reference mode: (red, νe), (blue, νµ), (green,ντ ).The horizontal lines corresponds to the optimized values of the W -state, respectively.Curiously, the measures are always non-zero, detecting non-classical correlations, and exceed the value of the W -state in both cases.

FIG. 2 : 1 FIG. 3 :
FIG.2: Plot the (not-normalized) flavour entropy S flavor , Eq. (14), for the three initial flavour states (a) νe, (b) νµ, (c)ντ in terms of the distance traveled per energy L/E in units of the oscillation period of the two lightest neutrinos.The horizontal (dotted) line corresponds to the value of the W -state.

35 FIG. 4 : 3 FIG. 5 :
FIG.4: Plot of the Svetlichny criteria detecting genuine multipartite nonlocality, Eq. (19), optimized over possible bipartitions and optimized over all six different observables for the three initial flavour states (a) νe, (b) νµ, (c)ντ as a function of the distance traveled per energy L/E in units of the oscillation period of the two lightest neutrinos (300 data points).The criterion detects genuine multipartite nonlocality if the value is above 4.

1 FIG. 6 :
FIG.6: Plots of the dissensions, Eq.(24), minimized over all projective measurements for the time evolution of an initial νe as a function of the distance per energy L/E: (a) single-mode measure D1 and (b) two-mode measure D2.The colors encode the dependence on the reference mode: (red, νe), (blue, νµ), (green,ντ ).The horizontal lines corresponds to the optimized values of the W -state, respectively.Curiously, the measures are always non-zero, detecting non-classical correlations, and exceed the value of the W -state in both cases.