A quantuminformation theoretic analysis of threeflavor neutrino oscillations
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Abstract
Correlations exhibited by neutrino oscillations are studied via quantuminformation 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 Belltype 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 electronneutrino. Via these quantuminformation theoretic quantities, capturing different aspects of quantum correlations, we elucidate the differences between the flavor types, shedding light on the quantuminformation theoretic aspects of the weak force.
Keywords
Neutrino Oscillation Quantum Correlation Quantum Discord Oscillation Probability Tripartite Entanglement1 Introduction
The study of correlations in quantum systems has a vast literature and draws its practical importance from potential applications to quantum technologies such as quantum cryptography and teleportation [1]. Recently, there has been a move toward extending these studies to systems in the domain of particle physics [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. The neutrino is a particularly interesting candidate for such a study (see e.g. the review on flavor oscillation [22]). In Nature, neutrinos are available in three flavors, viz, the electronneutrino \(\nu _e\), muonneutrino \(\nu _\mu \), and tauneutrino \(\nu _\tau \). Owing to their nonzero mass, they oscillate from one flavor to another. This has been confirmed by a plethora of experiments, using both natural and “manmade” neutrinos.
Neutrino oscillations are fundamentally threeflavor oscillations. However, in some cases, they can be reduced to effective twoflavor oscillations [21]. These elementary particles interact only via weak interactions; consequently the effect of decoherence, as compared to other particles widely utilized for quantuminformation processing, is small. Numerous experiments have revealed interesting details of the physics of neutrinos [23, 24, 25, 26, 27, 28]. This paper asks what type of quantum correlations is persistent in the time evolution of an initial \(\nu _e\) or \(\nu _\mu \) or \(\nu _\tau \). It presents a systematic study of the manyfaceted 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.
Threeflavor neutrino oscillations can be studied by mapping the state of the neutrino, treating it as a threemode system, to that of a threequbit system [16, 17]. In particular, it was shown that the neutrino oscillations are related to the multimode entanglement of singleparticle states which can be expressed in terms of flavor transition probabilities. Here we take the study of such foundational issues further by characterizing threeflavor neutrino oscillations by quantum correlations. This is nontrivial as quantum correlations in threequbit systems are much more involved compared to their twoqubit counterparts.

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. [29].

Genuine multipartite nonlocality: Nonlocality—which 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 [30] and Svetlichny inequalities [31].

Dissension: A tripartite generalization of quantum discord which is a measure of nonclassicality of correlations [32].
2 Threeflavor neutrino oscillations
3 Study of quantuminformation theoretic properties in neutrino oscillations
Separability or the lack of separability, i.e., entanglement, is defined for a given state according to its possible factorization with respect to a given algebra [34]. The separability problem is in general a NPhard problem, and only necessary but not generally sufficient criteria exist to detect entanglement. For bipartite quantum systems it suffices to ask whether the state is entangled or not. In the multipartite 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 flavors 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 analyze the different facets of the correlations in the dynamics of the 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.
3.1 Study of the entanglement properties
Since the flavor entropy \(S_\mathrm{{flavor}}({\varPsi }(t)\rangle _e)\) is nonzero for almost all time instances, the state is entangled. For this and all the following plots, we use the oscillation period of an electron to a muonneutrino as a unit. Since there are tiny changes in the behavior in one period due to the existence of the third flavor, 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 flavor entropy becomes maximal. Then the \(\nu _\mu \) and \(\nu _\tau \) oscillation probabilities become greater than the survival probability of \(\nu _e\), 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 is repeated.
The entropy of all three neutrino flavors are compared in Fig. 2 showing that for the muon and tauneutrinos the entropy is nonzero for almost all time instances. Compared to the electronneutrino evolution, the flavor uncertainty of the other two flavors oscillates more rapidly and with higher amplitudes, reaching the maximal value more often.
Among the genuinely multipartite entangled states, there are two subclasses known for threequbit states, the GHZ and Wtype of states. In Ref. [29] 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 selfconsistent classification also in a relativistic framework [37]. 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, it remains in a certain entanglement class. We will therefore investigate this Lorentz invariant criterion, though let us emphasize that we do not take any relativistic effects of a boosted observer into account in this contribution.
3.2 Genuine multipartite modenonlocality
It is tempting to think that this is a failure of the method, in any case we can conclude that the full time evolution of a single neutrino cannot be described by a hybrid modenonlocal–local ensemble for all times. Since the violation of the Svetlichny inequality is only a sufficient witness of genuine tripartite nonlocality, but not a necessary condition, it is in principle possible that the timewindow where the inequality is satisfied may indeed contain this form of strong nonlocality. In any case, it seems safe to say that it should vanish close to the points where the neutrino state is characterized by a single flavor, and that genuine tripartite nonlocality is likely to be absent even in regions where genuine tripartite entanglement and absolute nonlocality may be present.
3.3 Dissension—a measure of nonclassicality
In Fig. 6 we plot the dissensions \(D_1,D_2\) minimized over all projective measurements for the time evolution of an initial electronneutrino. 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 nonzero 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 Wstate, 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 “twinhumped” 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 flavor (\(\tau \)).
4 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 analyzed 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 nonclassical correlations revealed by mutual information measures. In particular, dissension turned out to be larger than that for the perfect Wstate (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 electronneutrino 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 detail we have shown that even though a genuine modenonlocal correlation is usually present, there are specific time regions when it vanishes. This could be described as a possible failure of the method. In any case we have proven that for the full time evolution no hybrid modenonlocal–local theory can be constructed.
Summing up, we can conclude that foundational issues are more prominent in accelerator experiments (mainly producing muonneutrinos) than in reactor experiments (mainly producing electronneutrinos).
The weak force, being one of the four known fundamental forces in Nature, dominant in the flavor changing process of neutrinos, reveals strong genuine quantum features such as also shown for weakly decaying spinless Kmesons [42] or for the weakly decaying half integer spin hyperons [43]. 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.
Notes
Acknowledgments
B. C. gratefully thanks the Austrian Science Fund (FWF26783). The authors acknowledge Uma Sankar and Gerd Krizek for critically reading the manuscript and commenting on it.
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