Quantum contextuality of a single neutrino under interactions with matter

We study contextuality of the three neutrinos oscillations using the Klyachko–Can–Binicioglu–Shumovsky 5-observable test. We show how the interaction between neutrinos and normal matter affects contextuality—its loss and its possible revivals. We show that for open neutrino’s system, interacting with an environment, revivals of contextuality survive in a presence of decoherence included in the neutrino’s Lindblad–Kossakowski master equation in a simplest Markovian approximation.


Introduction
The most influential for quantum foundations triad-Einstein, Podolsky and Rosen [1]-claims that the physical theory should obey: (i) locality: the result of measurement carried out on the system A should not depend on the measurement performed on the spatially separated system B and (ii) objective realism: the properties of a system are set down before an act of measurement, which only reveals some pre-existing value. These requirements, if simultaneously satisfied, result in the celebrated EPR paradox. Quantum theory, objectively real and non-local, is recognized to be also contextual.
In our work we are interested in contextuality which, together with a temporal non-locality related to violation of the Leggett-Garg inequalities [2] and with the celebrated entanglement [3], is one of three strictly related [4,5] 'different types' of non-locality. Contextuality was first introduced (without using the term 'contextuality') by Bell in [6] and Kochen and Specker in [7]. It is said that theory is non-contextual if for a given set A of physical quantities: = { } A A A A , ,..., N 1 2 each yielding respective measurements' outcomes = ¼ a i N , 1, , i , the obtained values a i does not depend on the measurements' context, i.e. the set of another members of A measured along the quantity A i [8]. In other words, there exists a joint probability distribution and the probabilities of individual events can be calculated as the marginals of p joint [8,9]. An experimentally verifiable test for entanglement proposed by Bell [10] implemented in experiments performed by Alain Aspect and his collaborators [11] confirmed validity of quantum mechanics even though some physicists discerned the so-called 'loopholes' [12] opening the 'door' to the abolishment of quantum mechanics in favour of the hidden variable theories. These were taken into account in the future experiments [13,14] which still, despite of some earlier doubts confirmed the Bell non-locality [9] and possibility of entanglement [3]. In the recent years many advances were also made towards experimental verification of the quantum contextuality. The experiments were carried out in the systems of trapped ions [15], photons [16], fermions [17], deuterium nucleus [18] or neutron [19]. Moreover, the quantum contextuality gains in the popularity also in the quantum computation's community [20][21][22]. All these aspects indicate that the contextuality has become the active area of research. Roughly speaking the quantum contextuality tests verify if the expectation values of the set of observables can be described with the joint probability distribution [8]. There exist two classes of tests of the contextuality: state-dependent ones which provide an evidence for the non- The maximum violation of (2) admissible by the quantum mechanics occurs for Δ=−3.94. An example of the vectors ñ |v i and state yñ | for which the violation of inequality (1) or equivalently (2) is extremely large is given below [8]: sin , cos , sin , for the angles α=π/5 and β=2π/5. Quantum non-locality of the Bell type related to quantum entanglement [28,29] and the one of the Leggett-Garg type [30][31][32] have been studied recently in a context of oscillating neutrino systems. In particular, [33] reports violation of the LGI in neutrino systems based upon the Main Injector Neutrino Oscillation Search experiment's data. In our theoretical work, we want to study a complementary topic and examine how the system of 3 oscillating neutrinos exhibit the contextuality in the presence of matter. Furthermore, we want to investigate how the effect of this interaction becomes influenced by decoherence [34]. We show, although it may seem counter-intuitive, that the interaction with a normal matter modifying neutrino oscillations results in effective revivals of the quantum contextuality. Bearing in mind a high sophisticity and extreme difficulties present in any neutrino measurement [33], we would like to emphasize purely qualitative character of our predictions. We modestly attempt to present the calculations which, as we believe, indicate general properties of the oscillating neutrino systems.
The paper is organized as follows: in the next section we review the qutrit model of neutrino's oscillation and apply it to investigate revivals of contextuality in section 3 originating from the interaction with matter. In section 4 we investigate an influence of decoherence and the contextuality revival in decohering environment. Finally we conclude our work. where the matrix U PMNS in the case of Dirac neutrino (considered in this paper) is parametrized with three mixing angles q q q , , 12 23 13 and one CP-violating phase δ [35]: ) and δ being the CP-violating phase. As currently attainable sensitivity of experiments to estimate a value of δ in the following part of our work it is neglected and set to δ=0.

Three-flavour neutrino oscillations
The time evolution of the neutrino is governed by the Schrödinger equation: where Ψ F describes the flavour neutrino state for a given time t and   stands for the Hamiltonian [35,37,38]: kin pot which naturally consists of the kinetic H kin and H pot potential parts to be described below.
In the flavour basis-taking into account an ultrarelativistic limit for the neutrinos (with neutrino's and momentum p∼E )-the kinetic Hamiltonian reads: 3 eV 2 [36]. As for the potential part, the main contribution to that comes from neutrino's interaction via the coherent forward elastic scattering (non-coherent effects are negligible) with the matter's electrons generating chargedcurrent potential V CC . There exists also interaction with the neutrons, but since it is the same for all neutrinoʼs flavours, it does not affects the oscillations. Bearing all these facts in mind, the potential part takes the following form: Here the charged-current potential V CC is related to the electron's density in the matter with G F denoting the Fermi coupling constant and the electron number density n e .

Contextuality of neutrinos treated as a closed system
Neutrinos are the simple example of the three-dimensional system which may exhibit contextuality i.e. their time evolution can lead to violation of the KCBS test equation (2) for a given set of measurements. It is therefore natural to ask if and under which conditions neutrinos do exhibit contextuality indeed. In other words, one asks a natural question concerning physical parameters granting this very quantum property. For this purpose we apply the KCBS test briefly described in the Introduction. It is the simplest, most economic, state-dependent test requiring no more than five observables to be measured. For this purpose one sets the observables A i in such a way that initial electron neutrino state (δ=0): , 10 e 12 13 12 13 13 maximally violates the contextuality (Δ≈−3.94) i.e. it is the centre of the KCBS-pentagon see figure 1.
, , , , 1 2 3 4 5 creating pentagon around the state n ñ | e (equation (10)) indicate via the quantity Δ (equation (2)) how the contextuality of the oscillating neutrinos changes in time for a given set of parameters and the different initial states.
In order to obtain the set of vectors , , , , 1 2 3 4 5 , which maximally violate the KCBS inequality for the given initial electron state it is necessary to rotate all vectors specified in equation (3). First we map the vector yñ = | ( ) 1, 0, 0 into that in equation (3). To do this, it is necessary to find a rotation matrix fulfilling condition: In the first step to determine the axis of rotation R one calculates the normalized cross product of the vectors yñ | and n ñ | e : Afterwards, since : y n a n y á cos , 12 e e one can easily find an angle α of the rotation R(α). Then, one can notice that it is plausible to express any rotation as a transformed rotation about z-axis: where: a a a a a n y y n y n n y = - Then one can define the transformation matrix given in equation (13) as:

e
The simple calculations of equation (13) reveal that: Initially, except the 'optimal' electron neutrino ν e given in equation (10), also ν μ and ν τ are natural candidates for initial preparations which, however, to be KCBS-optimal would need to be accompanied by the different measuring operators obtained by an other rotation matrix being, however, analogous to the one given in equation (17). Since the results are in both cases qualitatively the same, we limit our considerations to the neutrinos initially prepared in the electron state and then analyse the KCBS test for the corresponding pentagon.
The quantifier of contextuality  D -3 is given by the KCBS inequality in equation (2). The time evolution of an initial state equation (10) is obtained via numerical solution of equation (6) which is then imposed into the KCBS test i.e. one calculates Δ for a given time instant. In figure 2 we plot Δ as a function of time t for different values of the coupling V CC indicating the strength of interaction with the normal matter. The set of observables i determined by the requirement of maximal initial violation of the KCBS test obviously remain fixed in time. Initially, see inset in figure 2, the electron neutrino ν e maximally violates the KCBS inequality as discussed above. Further, the contextuality is satisfied provided that the time-evolving state remains confined in a (relatively) small cone around the state ν e . The results presented in figure 2 indicate periodicity of Δ which becomes faster with an increase of V CC . In other words, the larger the effect of matter is, the faster neutrino loses its contextuality. On the other hand, an enhancement of V CC results in the quicker return of the neutrino to its initial contextuality and truncation of the time interval during which the neutrino violates the KCBS condition. To sum up, after a rapid contextuality loss indicated by Δ>−3, there occurs its effective revival with Δ<−3 solely due to interaction with normal matter i.e. ¹ V 0 CC followed further by another loss. We use here the adjective 'effective' since the effect is transient and for sufficiently long time there is no qualitative difference between the systems with the different values of V CC . The value of V CC describing the size of a matter effect is chosen very large V CC ∼1 meV. Although such a large value of the V CC coupling can in principle be achieved in astrophysics e.g. in neutron stars [36] or supernovae with a matter density of order 10 11 g cm −3 [39,40], we emphasize an essentially model character of our calculations. In particular the range of the parameter t is chosen suitable to express the qualitative rather than quantitative effects since it is comparable (or even larger) with a duration of coherence loss of neutrino in a Earth matter. Further increasing V CC would decrease the time needed for the first contextuality revival. In other words, the effect described here requires very restrictive and tailored conditions to occur.

Decoherence effect: neutrino as an open system
The neutrino oscillations are mostly treated in the framework of the closed quantum systems and described according to the above mentioned unitary evolution. Nevertheless, every physical system-in particular that which is coupled to a matter, as it is the case of neutrinos-is subjected, at least to small extent, to decoherence effects originating from interaction with the environment and resulting in non-unitary corrections to its time evolution [41]. The system, treated as a whole, is assumed to be closed and evolves unitarily, while the reduced evolution of the (open) subsystem is determined by a non-unitary operator r r j j i i where r ( ) t i is the reduced density matrix of the time-evolving system [42]. Obviously the operator Γ cannot be arbitrary-the transformation must be completely positive [42][43][44] and satisfying the semi-group property i.e. obeying the composition law: , ,  (18) is generated by   defined in equation (7). The (non-Hamiltonian) Lindbladian part of the master equation equation (18): is responsible for the non-standard effects connected with dissipation and decoherence. Here N=3 denotes the dimension of the system, {· ·} , is an anti-commutator, the matrices F n stand for the generators of SU(N) and c ij are coefficients satisfying the set of following inequalities (in order to have all the properties, including complete positivity, of the map Γ retained [44,46]): Moreover, as the decoherence effects has been proposed as a possible explanation of certain experimental data [47][48][49][50], also the constraints on decoherence parameters are related to experimental data [51,52]-see [34] for a summary of a recent progress on that topic. Since in our case the dimension N=3, the matrices F n are generators of SU (3)  Let us emphasize that equation (18) is written in the flavour basis. The other option is to use the mass-states basis instead. However, the choice adapted and privileged here is related to a basis of measurable quantity-the flavour i.e. it is related to the pointer states [54] of the system and, as that, natural for open systems. Let us also notice that the Kossakowski-Lindblad form is invariant under changes of a basis but an interpretation of coefficients becomes different. Obviously, if one applies, instead of phenomenological, rigorous approach, one could-at least in principle-find a physically rather than mathematically (or formally) justified form of time evolution of the system. Let us notice that equation (18) can be, following [44], rearranged to the form which, not including any additional physical content, is of particular usefulness in numerical calculations: which allows to extract the 'Hamiltonian'  and 'dissipative'  (i.e. present in the open systems only) part of an effective time evolution generator  eff similarly as it was done in [30]. The notation r [ ] F v means that the density matrix is now represented by a 9-dimensional coherence-vector: The matrices  and  represent the 'ordinary' and dissipative (decohering) part of the Hamiltonian, respectively. Their explicit form is however too complicated (they are 9-dimensional matrices) to be reproduced here. At www.dropbox.com/sh/m678lxyn6fek2h6/AABM1Kg7wczX0JsRU-j2gcsHa?dl=0 one can find Mathematica code for their generation. In particular, following [44], after transforming the original, native problem equation (18)into the coherence-vector form equation (22) the numerical tool which is utilized is a Pythonbased exponentiation of finite dimensional matrices.
As one can expect, provided that it is sufficiently large, an effect of decoherence is lethal for contextuality as plotted in figure 3 for the simplest case of the constant Lindblad-Kossakowski matrix º " The larger the amplitude of c ij is, the faster the contextuality parameter Δ crosses the line Δ=−3 indicating contextuality loss. However, even in the presence of decoherence the contextuality revivals of contextuality are possible provided that the neutrinos are coupled to the normal matter with a sufficiently large V CC . This behaviour is depicted in figure 4. Initially one observes fast contextuality loss indicated by Δ>−3 but then, e.g. for V CC =0.3 meV due to its oscillatory behaviour again Δ<−3 i.e. neutrino regains its contextuality.
Both the decoherence and the neutrino-matter interaction are responsible for the above mentioned revivals of contextuality. One could expect the first of these two mechanisms to be rather destructive. It is clearly the case but the way how the decoherence affects contextuality of neutrino is highly related to V CC . The purity  r = Tr 2 [41] is a natural quantifier for a 'decoherence strength': it is maximal for pure states which are projectors satisfying the idempotence condition r r = 2 whereas minimal for maximally mixed states ρ proportional to an identity matrix. There is a relation between contextuality revivals of contextuality and the purity  r = ( ) t Tr 2 of the time-evolving neutrino. In the inset of the upper panel of figure 4 it is presented that the purity, due to environmental coupling, decreases (on average) with time but, for certain values of V CC oscillates. There is a relation between (decaying) oscillations of both the contextuality and the purity indicating a relation between decoherence and an interaction with matter leading to optimal parameters for transient revivals of contextuality. Let us emphasize, however, that in a presence of strong decoherence, as presented in the lower panel of figure 4 contextuality revivals are absent since the system purity loss is very rapid.
Let us indicate an apparent similarity between the contextuality decay and revivals of contextuality reported here and the entanglement sudden death [55] and rebirth [56]. This well known and celebrated effects peculiar for time-evolving open quantum systems [57] exemplify another analogy between different notions of quantum non-locality [4,5].

Summary
Neutrino oscillations are one of most fundamental realization of the quantum three-level systems [35] exhibiting all the quantum 'unusual' features with the contextuality as a particular example. In our work, we investigated the relation between contextuality quantified by the KCBS 5-observable test and the effect of matter interacting with neutrinos and showed that this interaction results in effective revivals of contextuality in time. This counter-intuitive property is of a transient character and hence is related to a time period indicating duration of evolution. To be precise: one cannot globally enhance contextuality solely due to matter interaction but, as we showed, one can expect contextuality loss and revivals of contextuality occurring more frequently in a certain time interval in a presence of matter affecting neutrino's oscillation in comparison to the case where the matter is absent. Effective revivals of contextuality remain also present in neutrinos' oscillation affected by Markovian decoherence phenomenologically modelled via Kossakowski-Lindblad master equation equation (18). The Kossakowski-Lindblad master equations describing decoherence (with complete positivity as an underlying guideline for their construction) are fairly general despite of their Markovian limitation. They allow for credible predictions of time-evolving quantum systems weakly coupled to environment. Clearly, for large amplitudes of Kossakowski-Lindblad dissipatiors in equation (18) contextuality is lost as the system became effectively classical. Nevertheless, we showed that for weak and intermediate decoherence revivals of contextuality remain present. We also indicated that there is a relation between contextuality revivals and the purity of the system calculated for the neutrino state solving equation (18). For large decoherence, when the purity is monotonically lost one cannot expect contextuality revivals which do occur together with a (damped) oscillations of purity indicating time intervals of faster and slower purity loss.
Three-dimensional systems-qutrits-are the simplest objects exhibiting contextuality yet rich enough to attract attention [26,58]. A 'natural qutrits'-neutrinos-exhibiting often unexpected properties and attracting researchers from very different sub-disciplines of physics are a stage of the most fundamental properties of a quantum world. Contextuality, one of such properties, is one more test for our ability of acceptance counterintuitivity of Nature. Our studies are based on two main model assumptions: (i) the first, concerning neutrinos, is formalized in equation (6); (ii) the second is about the utilized decoherence approximation given in equation (18) with the Kossakowski-Lindblad underlying guidelines and the complete positivity of time evolution among them. We hope that our theoretical investigations performed from a quantum mechanical perspective can contribute, yet modestly, to a better understanding quantum properties of neutrinos.