Quantum correlations in terms of neutrino oscillation probabilities

Neutrino oscillations provide evidence for the mode entanglement of neutrino mass eigenstates in a given flavour eigenstate. Given this mode entanglement, it is pertinent to consider the relation between the oscillation probabilities and other quantum correlations. In this work, we show that all the well-known quantum correlations, such as the Bell's inequality, are directly related to the neutrino oscillation probabilities. The results of the neutrino oscillation experiments, which measure the neutrino survival probability to be less than unity, imply Bell's inequality violation.

Neutrino oscillations provide evidence for the mode entanglement of neutrino mass eigenstates in a given flavour eigenstate. Given this mode entanglement, it is pertinent to ask if other quantum correlations are present in neutrino evolution. In this study, we compute a number of such correlations for accelerator neutrinos in the approximation of two flavour νµ ↔ ντ oscillations. The point of minimum survival probability corresponds to the extremum point of all measures of quantum correlations. We find that Bell's inequality is always violated.

I. INTRODUCTION
The foundations of quantum mechanics are usually studied in optical or electronic systems. In such systems, the interplay between the various measures of quantum correlations is well known. Due to the technical advances in high energy physics experiments, in particular the meson factories and the long baseline neutrino experiments, it will be fruitful to test the foundations of quantum mechanics in such systems.
In [1], the interplay between various measures of quantum correlations were studied, for the first time, in the context of correlated unstable neutral meson systems. It was observed that the quantum correlations for these unstable systems can be non-trivially different from their stable counterparts. We now turn our attention to neutrinos, where such an interplay has not yet been studied, to the best of our knowledge. Neutrino is a particularly interesting candidate for such a study. Neutrino oscillations are experimentally well established [2][3][4][5][6][7]. Such oscillations are possible if both of the following conditions are satisfied: • The neutrino flavour state is a linear superposition of non-degenerate mass eigenstates.
• The time evolution of a flavour state is a coherent superposition of the time evolution of the corresponding mass eigenstates.
The coherent time evolution implies that there is mode entanglement between the mass eigenstates which make up a flavour state. Since neutrinos interact only through weak interactions, the effect of decoherence is minimal, when compared to other particles such as electrons and photons that are widely used in quantum information processing.
The quest for understanding quantum correlations could be thought to have begun with the efforts of * akalok@iitj.ac.in † subhashish@iitj.ac.in ‡ uma@phy.iitb.ac.in Einstein-Podolsky-Rosen (EPR) [8]. A better, quantitative understanding of EPR led to the development of Bell's inequality [9], with refinements leading to the Bell-CHSH (Clauser-Horn-Shimony-Holt) inequalities [10]. Violation of Bell's inequality quantifies the nonlocality inherent in the system. A weaker, though very popular and widely studied measure of quantum correlations, is entanglement [11]. This has been applied to understand the process of teleportation [12]. A still weaker measure is quantum discord [13,14] and was developed as the difference between the quantum generalizations of two classically equivalent formulations of mutual information. Since states with are separable and hence have no entanglement could still have non zero discord, our present understanding of quantum correlations is that it is a complex entity with many facets. There is now an abundance of measures of quantum correlations such as quantum work deficit [15], measurement induced disturbance [16] and dissonance [17].
In this paper we study several measures of quantum correlations in the context of two-flavour ν ν ↔ ν τ oscillations in long baseline accelerator neutrinos. Among the measures studied are Bell's inequality, concurrence, discord and teleportation fidelity. We find that the study of quantum correlations provides useful insights into the nature of two flavour neutrino oscillations. We find that these oscillations always violate Bell's inequality and that the teleportation fidelity is always greater than 2/3.
In this work, we first provide an introduction to the quantum mechanics of two flavour neutrino oscillations.
Here we see that the mode entanglement comes in a natural setting. We then make a brief discussion of the various measures of quantum correlations used here, to study neutrino oscillations. We then apply the correlation measures discussed, for ν ν ↔ ν τ oscillations in accelerator neutrinos. We finish with our conclusions.

II. QUANTUM MECHANICS OF TWO FLAVOUR NEUTRINO OSCILLATIONS
It is well known that there are three flavour states of neutrinos, ν e , ν µ and ν τ [18,19]. In the oscillation formalism, it is assumed that they mix via a 3×3 unitary matrix to form the three mass eigenstates ν 1 , ν 2 and ν 3 . Neutrino oscillations occur only if the three corresponding masses, m 1 , m 2 and m 3 , are non-degenerate. Of the three masssquared differences ∆ ij = m 2 i − m 2 j (where i, j = 1, 2, 3 with i > j), only two are independent. Oscillation data tells us that ∆ 21 ≈ 0.03 × ∆ 32 , hence ∆ 31 ≈ ∆ 32 . One of the three mixing angles parametrizing the mixing matrix, θ 13 , is measured to be quite small (about 0.14 radians) [20][21][22][23].
In considering neutrino oscillations, in general, one should use the full three flavour oscillation formulae. A number of studies do this, fitting all the available neutrino oscillation data to the three flavour formulae [24][25][26]. In a number of cases, the three flavour formula reduces to an effective two flavour formula, if one or both of the small parameters, ∆ 21 /∆ 32 and θ 13 , are set equal to zero. In the present work, we consider ν µ ↔ ν τ oscillations in long baseline accelerator neutrinos. For such a system, both the above parameters can be neglected in doing leading order calculations. Then the problem reduces to that of two flavour mixing of ν µ and ν τ to form two mass eigenstates ν 2 and ν 3 . The corresponding oscillations are described by one mixing angle θ (≡ θ 23 in three flavour mixing) and one mass-squared difference ∆ (≡ ∆ 32 in three flavour mixing).
Neutrino oscillations arise due to the mixing between flavour states (eigenstates of weak interaction, which are detectable in lab) leading to mass eigenstates (which are the propagation eigenstates). In the case of two flavour mixing, the relation between the flavour and the mass eigenstates is described by a 2 × 2 rotation matrix, U (θ), Therefore, each flavour state is given by a superposition of mass eigenstates, where α = µ or τ and j = 2, 3. The time evolution of the mass eigenstates |ν j is given by where |ν j are the mass states at time t = 0. Thus, we can write The evolving flavour neutrino state |ν α can also be projected on to the flavour basis in the form where |ν α is the flavour state at time t = 0 and |Ũ αα (t)| 2 +|Ũ αβ (t)| 2 = 1. We introduce occupation number states as [27,28] Eq. (5) can therefore be rewritten as where,Ũ Now the state in Eq. (7) has the form of a mode entangled single particle state [29][30][31][32][33]. Such mode entangled states have been the subject of intense discussions over the last two decades, resulting in the general consensus of subspace entanglement as a generalized feature of inter particle entanglement [34]. It has been the subject of many theoretical and experimental proposals as well as successful experimental realizations [32,33] in atom-photon systems. In the case of electron neutrinos, it is not possible to define an entangled two flavour state as given in Eq. (7). One must necessarily involve all three flavours which makes the algebra very complicated. Hence we limit our attention to accelerator neutrinos and do not consider reactor neutrinos, which areν e . Eq. (7) can be used to relate flavour oscillations to bipartite entanglement of single particle states. The corresponding density matrix is given by where |Ũ αα (t)| 2 = c 4 + s 4 + 2s 2 c 2 cos ∆t 2E , with c ≡ cos θ and s ≡ sin θ. Since accelerator neutrinos have energy of order of a GeV, they are ultra relativistic.

III. INTRODUCTION TO SOME CORRELATION MEASURES
In this section, we introduce various quantum correlations inherent in the state given in Eq. (7) and briefly outline the procedure used to compute them. In all our subsequent calculations, the states considered are represented by 4 × 4 density matrices.
Bell's inequality: Bell's inequality is used to study the non-locality of a given system. Its physical content is that a system that can be described by a local realistic theory will satisfy this inequality. However, quantum mechanics provides many examples where this inequality gets violated [35]. It is worth testing for such a violation in the time evolution of neutrinos. The density matrix ρ, in general, can be expanded in the form T mn (σ m ⊗ σ n )] .
(11) The elements of the correlation matrix T are given by T mn = T r [ρ(σ m ⊗ σ n )]. Let u i (i = 1, 2, 3) be the eigenvalues of the matrix T † T . Then the Bell-CHSH inequality can be written as M (ρ) < 1, where M (ρ) = max(u i + u j ) (i = j) [36].
Concurrence: Non-locality is the strongest measure of quantum correlations. A weaker, though popular and extensively studied measure, is entanglement. For the case of entangled two-level systems it is synonymous with concurrence. For a state with density matrix ρ, the concurrence is [11] where λ i are the square roots of the eigenvalues of ρρ in decreasing order. Hereρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ) and is obtained by applying the spin flip operation on ρ.
Geometric discord: A still weaker measure of quantum correlations is quantum discord which points out that classicality and separability are not synonymous. To obtain an analytical formula for quantum discord is a very difficult task as it involves an optimization over local measurements, requiring numerical methods. To overcome this difficulty, another measure of quantum correlation called geometric discord was introduced in [37] which quantifies the amount of non-classical correlation, of an arbitrary quantum composite system, in terms of its minimal distance from the set of classical states. For ρ, geometric discord can be shown to be where T is the correlation matrix defined above, y is the vector whose components are y m = Tr(ρ(σ m ⊗ I 2 )), and λ max is the maximum eigenvalue of the matrix ( y y † + T T † ) [37]. Teleportation fidelity: Apart from the above foundational measures of various aspects of quantum correlations, a need was felt to have a measure that defines the practical use of quantum correlations. This was supplied by teleportation. The classical fidelity of teleportation in the absence of entanglement is 2/3. Whenever the maximum teleportation fidelity, F max > 2/3, quantum teleportation is possible. F max , is easily computed in terms of the eigenvalues {u i } of T † T mentioned above and is given by [36]. This expression allows for a useful interplay between teleportation fidelity and Bell's inequality. This is so because N (ρ) ≥ M (ρ).
Hence a Bell's inequality violation automatically implies F max > 2/3. Interestingly, this does not rule out the possibility of entangled states that, while they do not violate Bell's inequality, can nevertheless be useful for teleportation.

IV. RESULTS AND DISCUSSIONS
We find that all the quantum correlations defined above are time dependent. We introduce the dimensionless time variable φ = ∆ t/2E to study this time dependence. In Fig. 1, all the correlations are plotted for the range φ = [0, 2π]. These values are easily realizable for long baseline accelerator neutrinos. It can be seen from the figure that the Bell-CHSH inequality, the strongest measure of non-locality, is always ≥ 1. This implies that the time evolution of neutrinos is highly non-local and cannot be explained by any local realistic theory. Also, the teleportation fidelity is always ≥ 2/3 and it exceeds the threshold of 2/3 whenever Bell's inequality is violated. This is in agreement with the cases of stable optical and electronic systems and is in contrast with unstable oscillating neutral meson systems [1]. The weaker measures of quantum correlations, such as discord, are always bounded from above by concurrence, as expected. Also, all measures of quantum correlations are symmetric about the half-period.
Bell's inequality violation is maximal at φ = π/2, 3π/2. At φ = π, Bell's inequality becomes 1 and concurrence becomes zero, coinciding with the value at φ = 0, meaning that the neutrino modes are not entangled at this instant. An explanation for this is provided in the right panel of Fig. 1, where the time evolution of the magnitude of off-diagonal elements of the density matrix (9) has been plotted. The off-diagonal elements of the density matrix are an indication of the quantum coherence in the system. At φ = 0, as can be seen from the right panel of Fig. 1, these off diagonal elements are zero. This means the neutrino modes are not entangled. At this instant of time, Bell's inequality and teleportation fidelity are 1 and 2/3, respectively. As φ increases, the off-diagonal elements become non zero and the states become entangled. The magnitude of the off-diagonal elements depend upon the mixing angle θ and the phase φ. Whenever φ = nπ, where n is an integer, the offdiagonal elements become zero and the entanglement is lost. When φ = (2n + 1)π/2, the off-diagonal elements are maximum as is the entanglement.

V. CONCLUSIONS
In this work we have computed four measures of quantum correlations for oscillations of accelerator neutrinos. We find that Bell's inequality is always ≥ 1 and hence the evolution of neutrinos is highly non local in nature. Teleportation fidelity is always greater than 2/3 thus obeying the usual relation between Bell's inequality and teleportaion fidelity, as seen in electronic and photonic systems. This is in contrast to the unstable oscillating neutral mesons. Geometric discord is always bounded from above by concurrence which is a satisfactory feature as discord is a weaker measure of quantum correlations compared to concurrence. All the measures of quantum correlations are directly proportional to the off diagonal order present in the system.