# Testing Violation of the Leggett-Garg-type inequality in neutrino
oscillations of Daya-Bay experiment††thanks: This work is partly supported by
the Key Research Program of Frontier Sciences, CAS, under the Grants Number
NO. QYZDY-SSW-SLH006 of Chinese Academy of Sciences.

Qiang Fu1,2,3 Xurong Chen1<EMAIL_ADDRESS>1 Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou, 730000, China
2 Lanzhou University, Lanzhou 730000, China
3 University of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

The Leggett-Garg inequality (LGI), derived under the assumption of realism,
acts as the temporal Bell’s inequality. It is studied in electromagnetic and
strong interaction like photonics, superconducting qu-bits and nuclear spin.
Until the weak interaction two-state oscillations of neutrinos affirmed the
violation of Leggett-Garg-type inequalities (LGtI). We make an empirical test
for the deviation of experimental results with the classical limits by
analyzing the survival probability data of reactor neutrinos at a distinct
range of baseline dividing energies, as an analog to a single neutrino
detected at different time. A study of the updated data of Daya-Bay experiment
unambiguously depicts an obvious cluster of data over the classical bound of
LGtI and shows a $6.1\sigma$ significance of the violation of them.

###### pacs:

11.30.Er, 03.65.Ta, 13.25.Es, 14.60.Pq

To be published on European Physical Journal C.

##  I Introduction

Nonclassical features of the quantum system have experienced extensive study
since the inception of quantum mechanics. After a long debate between the
believer of local realism and quantum mechanics, a breakthrough study, Bell’s
inequality (BI), was provided by Bell Bell (1964). The unique feature of BI is
its testable formula from the consequence of the famous hypothesis called the
local realism (LR). The LR believers assume that any observable value of an
object, even not detected, must have a definite value and that results of any
individual measurement of the observables remain unaffected if they were in
space-like separation. Extensive experimental investigations Hensen  _et al._
(2015); Weihs  _et al._ (1998); Aspect  _et al._ (1982) over past several
decades tested the violation of BI. These studies conclude that any local
realism view of a microscopic object needs to be nonlocal. Based on these
studies of BI, Leggett and Garg furtherly derived a new series of inequalities
Leggett and Garg (1985) on the assumption of Macrorealism (MR), now known as
the Leggett–Garg inequalities (LGIs), that any system behaving as a
macroscopic realism must obey. From the structure of the LGIs, we can see them
as a analogue of Bell’s inequalities in temporal notion, which also makes it
possible to implement a rigorous test of quantum mechanics in a macroscopic
level which is usually very difficult in designing experiments in space-like
separation condition. By testing the LGtI, we can also perform a rigorously
loophole-free test of quantum mechanics Hensen  _et al._ (2015); Shalm  _et
al._ (2015); Giustina  _et al._ (2015); Gallicchio  _et al._ (2014).

Besides the nonlocal behavior and quantum correlation between different
particles, single particle states can also exist entanglement by the flavor
transitionBlasone  _et al._ (2009). Since the oscillation of neutrino flavors,
it offers us an ideal sources to test the quantum mechanics in weak
interaction and under a macroscopic view. For two-flavor neutrino oscillation,
a two level state’s matrix can be expressed in the form

$\displaystyle\begin{aligned} \rho=\frac{1}{4}\left[I\otimes
I+(\rm{r}\cdot\rm{\sigma})\otimes
I+I\otimes(\rm{s}\cdot\rm{\sigma})+\sum_{n,m=1}^{3}T_{mn}(\sigma_{m}\otimes\sigma_{n})\right].\end{aligned}$
(1)

Where the elements of matrix T are
$T_{mn}=Tr\left[\rho(\delta_{m}\otimes\delta_{n})\right]$. For two qu-bit
situation, many kinds of quantum correlations Hill and Wootters (1997);
Wootters (1998); Dakić  _et al._ (2010) like the entanglement, fidelity,
quantum discord and geometric discord have been derived to get their general
expressions. Mermin and Svetlichny Mermin (1990); Svetlichny (1987)
inequalities were derived for three or even many body system that have two
macroscopic distinct states. Using these results of two level states, a series
quantum-information theory calculation Gangopadhyay  _et al._ (2013); Banerjee
_et al._ (2015); Alok  _et al._ (2016); Gangopadhyay and Roy (2017) has been
applied on neutrino flavor oscillation in last several years. These quantum
correlations can be directly linked to the probabilities of flavor
oscillation, which lead to violation of classical boundary limits when
assuming the neutrino oscillation mixing angle is not vanishing.

Neutrino flavor oscillation is such a special process that it merely only
affected by their own properties like mass square differences, mixing angles
and the energies. Since neutrinos just interact with matters by weak
interaction with quite low cross-section. The influence of the environment act
on neutrinos’ propagation is much more negligible comparing with optical or
electrical system, which makes neutrinos an ideal particle on testing the
LGIs. As the mass eigenstate of a neutrino is not the same with its flavor
eigenstate, during propagation, neutrinos undergo flavor mixing among the
three flavored eigenstates. The MINOS experiment has been studied in Ref.
Formaggio  _et al._ (2016), which observed the violation of Leggett-Garg-type
inequalities, K3 and K4 terms, in a significance greater than $6\sigma$
Formaggio  _et al._ (2016). The MINOS experiment is an accelerator neutrino
experiment using decay in flight neutrinos with fixed baseline distance 735 km
and a large range of $\nu_{\mu}$ energy from 0.5 GeV to 50 GeV, which happens
to cover the largest violation of LGIs K3 and K4. The Daya-Bay Collaboration
reported an updated data analysis of electron anti-neutrino disappearance
channel An  _et al._ (2015), which gives a best fit of
$sin^{2}2\theta_{13}=0.084\pm 0.005$, we will investigate whether the Daya-Bay
Reactor Neutrino Experiment can observe the violation of LGIs.

##  II The Leggett-Garg-type inequalities

We focus on the simplest L-G-type inequality which is constructed as follows.
Consider a system with two absolutely distinguishable states correspond to an
observable quantity $Q(t)$ can be two different value +1 or -1. Assuming that
whenever the system was being measured, the observable quantity occupies a
value of either +1 or -1 for being in state 1 or 2 respectively. Then we can
define a macroscopic observable $Q(t)$ for the macroscopic system. And
$C_{t_{i},t_{j}}=\left<Q(t_{i})Q(t_{j})\right>$ as its two-time correlation
function, where $Q(t_{i})$ and $Q(t_{j})$ are observable quantity’s value when
being measured at time $t_{i}$ and time $t_{j}$. In this work we consider the
two state as the survival of electron anti-neutrino and the disappearance of
the electron anti-neutrino. When the neutrinos being created in the reactor by
beta decay process, they are totally in the state of the flavor eigenstate.
Since the PMNS matrix does not change with time, the two-flavor neutrino
oscillation obey the same survival probability. We shall introduce this
stationary assumption Huelga  _et al._ (1995) which requires the evolution of
the neutrino for different ordered time intervals are the same. And then the
$C(t_{i},t_{j})=C(t_{i}-t_{j})$ (if $t_{i}<t_{j}$). Next, consider a sequence
of times $t_{1},t_{2},t_{3}$ and $t_{4}$(here, $t_{1}<t_{2}<t_{3}<t_{4}$). If
we take a series of measurements for $Q(t)$ in these four times, it is
straightforward to determine four time correlations ($C_{12},C_{23},C_{34}$
and $C_{14}$). Then it is possible to adopt the stationary condition on
standard LGI procedure leading to K4 LG-type inequality involving four
correlation functions. For any sequence of measurements, any $Q(t_{i})$ has
the definite observable value, regardless of the choice of the pair
$Q(t_{i})Q(t_{j})$ it belongs to. So, combination
$Q(t_{1})Q(t_{2})+Q(t_{2})Q(t_{3})+Q(t_{3})Q(t_{4})-Q(t_{1})Q(t_{4})$ lies
always between -2 and +2. Similarly, $K_{3}$ inequality lies between -1 and
+1. If all the terms in the above formula are replaced by time correlations
(average), the Leggett-Garg-type inequalities are in form

$\displaystyle\begin{aligned} &K_{3}\equiv C_{12}+C_{23}-C_{13}\leq 1.\\\
&K_{4}\equiv C_{12}+C_{23}+C_{34}-C_{14}\leq 2.\end{aligned}$ (2)

The above inequalities imposes a constraint on macroscopic realism on temporal
separated joint probabilities in any two-state system.

##  III Oscillating neutrinos

It is extensively verified that the flavor component of a neutrino oscillates
during its propagation. The oscillation properties of different neutrino
flavors are determined by there mixing angles ($\theta_{12},\theta_{23}$, and
$\theta_{13}$), a CP phase of the Pontecorvo-Maki-Nakagawa-Sakata matrix and
there mass-squared differences ($\Delta m_{32}^{2},\Delta m_{21}^{2}$)
Pontecorvo (1968); Maki  _et al._ (1962). Here, we will treat the Leggett-
Garg-type inequalities using the updated measurement of $\overline{\nu_{e}}$
survival channel in the results of the Daya-Bay experiment, where give the
newest best fit of mixing angle $\theta_{13}$ with large significance An  _et
al._ (2015). The measurement was updated later with full detector
configuration An  _et al._ (2015). For $\theta_{13}$ measurement, they used
baseline length dividing $\overline{\nu_{e}}$ energy as the variable to depict
the survival probability of $\overline{\nu_{e}}$ as

$\displaystyle\begin{aligned}
P_{\overline{\nu}_{e}\rightarrow\overline{\nu}_{e}}=1-cos^{4}\theta_{13}sin^{2}2\theta_{12}sin^{2}\frac{1.267\Delta
m_{21}^{2}L}{E}-sin^{2}2\theta_{13}sin^{2}\frac{1.267\Delta
m_{ee}^{2}L}{E}.\end{aligned}$ (3)

where $E$ is the energy of $\overline{\nu_{e}}$ in MeV, L is the propagation
distance between near and far point detector, $\theta_{12}$ is the solar
neutrino mixing angle and $\Delta m_{21}^{2}$ is their mass-squared difference
in $eV^{2}$. Notice that $\Delta m_{ee}^{2}$ is an effective mass-squared
difference Nunokawa  _et al._ (2005) in electron anti-neutrino disappearance
with the form of

$\displaystyle\begin{aligned} \Delta m_{ee}^{2}=cos^{2}\theta_{12}\Delta
m_{31}^{2}+sin^{2}\theta_{12}\Delta m_{32}^{2}.\end{aligned}$ (4)

Since the $m_{12}^{2}=(7.50\pm 0.20)\times 10^{-5}~{}\rm{eV^{2}}$, while the
$m_{ee}^{2}=(2.42\pm 0.11)\times 10^{-3}~{}\rm{eV^{2}}$ according to Ref. An
_et al._ (2015), we can choose a appropriate value of the ratio L/E to make
one of a term of $sin^{2}\frac{1.267\Delta m^{2}L}{E}$ vanishing. For Daya-Bay
experiment, the effect of parameter $\theta_{12}$ becomes far less enough than
that from $\theta_{13}$ to make $\theta_{12}$ regarded as negligibly small.
Given that there is an initial pure electron anti-neutrino source, after a
time t propagation, the survival probability of $\overline{\nu}_{e}$ will be:

$\displaystyle\begin{aligned}
P_{\overline{\nu}_{e}\rightarrow\overline{\nu}_{e}}=1-sin^{2}2\theta_{13}sin^{2}\frac{1.267\Delta
m_{ee}^{2}ct}{E}.\end{aligned}$ (5)

Although there is the MSW (Mikheyev-Smirnov-Wolfenstein) effect (usually
called matter effect) during the propagation of neutrino in matter. The effect
is only significant for high energy neutrinos and long range of matter, like
the solar neutrino experiment. The KamLAND and Super-K’s $P_{ee}$ day-night
discrepancy are only obvious for larger than 6 MeV neutrinosLisi and Montanino
(1997); Vissani (2017). Further more, the solar neutrino experiments involve
the matter effect caused by the electron in the solar, which electron density
$\epsilon_{\odot}$ is great larger than that in the Earth. Generally speaking,
a neutrino vector of state in flavor basis
$\left|\nu(t)\right>=(\nu_{e}(t)~{}\nu_{\mu}(t)~{}\nu_{\tau}(t))^{T}$ flows
the $\rm{Schr\ddot{o}dinger}$ equation:

$\displaystyle\begin{aligned}
i\frac{d}{dt}\left|\nu(t)\right>=\mathcal{H}\left|\nu(t)\right>\end{aligned}$
(6)

where the Hamiltonian can be replaced by an effective one as

$\displaystyle\begin{aligned}
\mathcal{H}\simeq\frac{1}{2E}\rm{U}~{}\rm{diag}(0,\Delta m_{21}^{2},\Delta
m_{31}^{2})U^{\dagger}+\rm{diag}(V,0,0).\end{aligned}$ (7)

where $V$ is the effective charged potential contribution to $\nu_{e}$
Wolfenstein (1978) given in the form

$\displaystyle\begin{aligned} V(x)\simeq 7.56\times
10^{-14}\left(\frac{\rho(x)}{\rm{g/cm^{3}}}\right)Y_{e}(x)~{}\rm{eV},\end{aligned}$
(8)

where $\rho(x)$ is the matter density along the track path of the
neutrino,$Y_{e}(x)$ (for the Earth $\simeq$ 0.5) is the number of electrons
normalized to the number of nucleons. For the matter of constant density, the
series expansion for three-flavor neutrino oscillation probabilities can be
derived from the Hamiltonian Eq.(7) Akhmedov  _et al._ (2004). For $\nu_{e}$
survival, the survival probability expansion to second order is

$\displaystyle\begin{aligned} P_{ee}=1-\alpha
sin^{2}2\theta_{12}\frac{sin^{2}A\Delta}{A^{2}}-4sin^{2}\theta_{13}\frac{sin^{2}(A-1)\Delta}{(A-1)^{2}},\end{aligned}$
(9)

where $\alpha=\frac{\Delta m_{21}^{2}}{\Delta m_{31}^{2}}\simeq 0.0297$, and
the abbreviation for $A$ and $\Delta$ is

$\displaystyle\begin{aligned} &\Delta\equiv\frac{1.27\Delta
m_{31}^{2}L}{E}~{}\rm{\frac{[ev^{2}][km]}{[GeV]}},\\\
&A\equiv\frac{2EV}{\Delta m_{31}^{2}\times
10^{-3}}~{}\rm{\frac{[GeV][eV]}{[eV^{2}]}}.\end{aligned}$ (10)

For this Daya-Bay analysis, we calculate the discrepancy of the $P_{ee}$
probability of a $6~{}MeV$ neutrino in the range of 0 to 10 km covering the
range of the experiment, about 2 km. From Fig. 1, we can draw a conclusion
that the matter effect is too small to be included in “short” baseline.

Figure 1: (color online). Illustration of the matter effect of the Earth in
the interval of 0 10 km. The plot is the discrepancy of vacuum $P_{ee}^{Vac}$
and the Earth $P_{ee}^{Earth}$. The neutrino is 6 MeV monoenergetic, and the
$\rho(x)=2.65~{}g/cm^{3}$ for the standard rock.

According to the expansion of the $P_{ee}$ with matter effect mentioned above,
we will use the oscillation probability just in the vacuum. Using the
stationary assumption, one can derive the four joint probabilities
$P_{\nu_{\alpha},\nu_{\beta}}(t_{i},t_{j})$, here $\alpha$ and $\beta$ stands
for $\overline{\nu}_{e}$ and another neutrino flavor, i and j are from 1 to 4
defined above. The two-time correlation function $C_{12}$ in this
$P_{\overline{\nu}_{e}\rightarrow\overline{\nu}_{e}}$ is given by

$\displaystyle\begin{aligned}
C_{12}=1-\left[sin2\theta_{13}sin\left(\frac{1.267\Delta
m_{ee}^{2}}{E}c(t_{2}-t_{1})\right)\right]^{2}=2P_{\overline{\nu}_{e}\rightarrow\overline{\nu}_{e}}(t_{2}-t_{1})-1.\end{aligned}$
(11)

Similarly, the correlation functions $C_{23}$, $C_{34}$, and $C_{14}$ can be
calculated in the same way. Using the Eq. 5, the quantity $K_{n}^{Q}$ can be
evaluated as defined in Eq. 2. By choosing the time intervals in a particular
way, we can achieve maximum value of $K_{4}$ when
$t_{4}-t_{3}=t_{3}-t_{2}=t_{2}-t_{1}=\delta t$. Under this condition, the
correlation functions depend on the baseline length $L$ and neutrino energy
$E_{\nu}$. We select the neutrinos measured $L_{eff}/E$ to make the
oscillation phase $\psi_{a}=\frac{1.267\Delta m^{2}}{E}c(\delta t)$ obey the
sum rule: $\psi_{12}+\psi_{23}+\psi_{34}=\psi_{14}$. With an experimental
arrangement in which measurements occur at some fixed distance from the
neutrino sources. Assuming the neutrino begins in the pure
$\left|\overline{\nu}_{e}\right>$

$\displaystyle\begin{aligned}
K_{n}^{Q}=-2+2\sum_{a=1}^{n-1}P_{ee}(\psi_{a})-2P_{ee}(\sum_{a=1}^{n-1}\psi_{a}).\end{aligned}$
(12)

Where the n can be 3 or 4 in this paper, which corresponds to $K_{3}$ or
$K_{4}$ LGI. In quantum mechanical, the commutators of operators can be
nonvanishing. While in a classical system, operators observable values must
commute, then the macrorealism derived $K_{n}$ will become:

$\displaystyle\begin{aligned}
K_{n}^{C}=\sum_{a=1}^{n-1}C_{i,i+1}-\prod_{a=1}^{n-1}C_{i,i+1}.\end{aligned}$
(13)

The Daya-Bay Collaboration released updated oscillation results as a function
of effective baseline distance $L_{eff}$ over the average energy
$\left<E_{\nu}\right>$ in bins An  _et al._ (2015). For their six anti-
neutrino detectors (ADs) placed in three separate experimental halls (EHs) and
three nuclear reactors neutrino sources, the effective baseline varies for
each detected anti-neutrino. The Daya-Bay experiment covers energy between 1
to 8 MeV. The range of effective baseline and energy correspond to a phase
range of $\left(0,3/4\pi\right)$, within which the violations of LGI will be
observed near the minimum point of the anti-neutrino survival probability.

To test the violations of the $K_{3}$ and $K_{4}$ inequalities, we make an
operation on the data from the Daya-Bay neutrino experiment. Daya-Bay
experiment extracted the survival probabilities of neutrinos using Daya-Bay
and Ling-Ao nuclear power stations’ reactors. We use all the measurement
positions including EH1, EH2 and EH3 of Daya-Bay. The reactors provide
different sources of neutrinos with several fixed baseline and an energy
spectrum with peaks. We make a $\theta_{13}$ fit over the Daya-Bay updated
data and get the fit error band and center value of $P_{ee}$ shown in Fig. 2.
With the best fit of $sin^{2}2\theta_{13}$ and the $1\sigma$ error band of it,
we generated a large sets of pseudodata. Then we select all sets of data
points in Fig 3 which obey the sum rule of phase with the precision of $0.5\%$
($\psi_{1}+\psi_{2}\in\psi_{3}$) and $0.1\%$
($\psi_{1}+\psi_{2}+\psi_{3}\in\psi_{4}$) for the $K_{3}$ and $K_{4}$
respectively. For $K_{3}$ ($K_{4}$) situation, 48 (56) correlation triples
(quadruples) satisfy the sum rule. While the updated measurement only includes
the static errors and we simply assumed that the errors at small phase of the
oscillation probability are the fitting error.

Figure 2: (color online). The electron anti-neutrino survival probability
versus effective propagation distance $L_{eff}$ over anti-neutrino energy
$E_{\nu}$. The dashed blue curve indicates the prediction fitting directly to
the measured Daya-Bay values of $P_{ee}$. The red band indicates a $1\sigma$
confidence interval around the fitted prediction. The blue triangles, green
rectangles and black dots are the binned data of Daya-Bay EH1, EH2 and EH3
from Ref. An  _et al._ (2015) respectively.

The violation of Leggett-Garg-type inequalities has been tested and confirmed
by the MINOS experiment with the $K_{3}$ and $K_{4}$ being inconsistent with
realism prediction over $5\sigma$ Formaggio  _et al._ (2016). Since the
violation of Leggett-Garg-type inequalities happens when the mixing angle of
two flavors is not zero, we suppose that the violation could be observed in
the $\overline{\nu_{e}}$ survival channel at Daya-Bay. In order to estimate
the significance from events number of the violations, we simulated the
statistical quantity by creating large sample of pseudodata based on the
fitting result of observed $P_{ee}$ values. The pseudodata are generated by a
Gaussian distribution model with the means and variances matched to the center
values and deviations of the best fit. Each set of simulated data gives an
artificial number of LGI violations for $K_{3}$ and $K_{4}$, from which we can
calculate the level of inconsistency of the predictions between quantum and
classical $K_{n}$.

To estimate the confidence level of these results being inconsistent with a
realism expected prediction, we make a fit of histogram filled by predicted
LGI violations number under realism model 7 to a beta-binomial distribution,
thus to estimate the deviation of classical predictions from the actually
observed number of LGI violations. For the actual number of LGI violations (41
in 48 data points), there exists a $6.1\sigma$ deviation from the expected
distribution of the classical prediction.

A similar statistical test is made for LGI $K_{4}$. Using the filter of the
phase sum rule described above, we get a number of 30 (in total of 56 data
points) exceed the classical limits. As Fig. 4 shows, there are obvious
clusters of points over the classical bound of $K_{3}$ and $K_{4}$. The
discrepancy between the observed events number and the classical predicted
events originating from the fluctuation is very clear. Our $K_{4}$ data also
possesses $6\sigma$ deviation from the classical prediction.

Figure 3: (color online). The histograms of number of K3 (upper) and K4
(lower) values that violate the LGI bound. The left curves with red filling
indicate the expected classical distributions, while the right cures with blue
filling indicate the quantum corresponding quantity.

Figure 4: (color online). The distribution of K3 (upper) and K4 (lower) versus
the effective propagation length divided by neutrino energy reconstructed from
$P_{ee}$. The black dot data show a cluster over the LGI bound. We also show
the expected distributions of classical (red circles) and quantum (blue
circle) predictions. Note that the $K_{3}$ and $K_{4}$ can be multiple values,
since there are many triples and quadruple satisfy the phase sum rule.

##  IV Discussion

The results mentioned above clearly constrain the validity of quantum
mechanics in such a macroscopic area. Values of LGtI $K_{3}$ and $K_{4}$ are
violated with the QM’s prediction at the confidence level of over $6\sigma$
comparing with the classical bound for the neutrino $\theta_{13}$ mixing in
our estimation. We get to present that anti-electron neutrino oscillations
also violate the limits of Leggett-Garg inequality. The detected violations
act as a new affirmation of quantum nonlocality existing in neutrino system
during its long range propagation. These violations were observed over the
near and far detectors placed at three experimental halls (EHs) with the
baseline long enough to make the test not being a Bell-like inequality test.
Besides, it should be worthwhile to make a detailed data analysis on the Daya-
Bay experiment involving three-flavor neutrino oscillation, in order to
achieve more data points of LGI $K_{3}$ and $K_{4}$. It could be worth to test
the quantum mechanics in such a weak interaction context. Although tests of
incompatible of LGI and QM have been achieved by photonics and electronic
experiments Emary (2012); Kofler and Brukner (2013), nuclear spin qubits
Waldherr  _et al._ (2011) and even condensed states Zhou  _et al._ (2015),
there are few reports of LGI violation in particle physics. Even though the
MINOS and Daya-Bay experimental setup show the LGtI violations, these two
experiments are all in the context of two flavor neutrino oscillation, which
can not reveal the CP violation. Since the entanglement exists between a pair
of neutral meson and anti-meson, which will violate the Bell inequality
Hiesmayr  _et al._ (2012), three-flavor oscillation analysis involving
neutrinos and anti-neutrinos may shed light on the study of CP-violating
phase. Naikoo  _et al._ (2017)

Acknowledgments: The authors thank Jarah Evslin for helpful discussions and
thank J. A. Formaggio for the illuminating suggestions and answers for our
questions. This work was supported by the Key Research Program of Frontier
Sciences, CAS, under the Grants Number NO. QYZDY-SSW-SLH006 of Chinese Academy
of Sciences.

## References

  * Bell (1964) J. S. Bell, Physics 1, 195 (1964).
  * Hensen  _et al._ (2015) B. Hensen _et al._ , Nature 526, 682 (2015), arXiv:1508.05949 [quant-ph] .
  * Weihs  _et al._ (1998) G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81, 5039 (1998), arXiv:quant-ph/9810080 [quant-ph] .
  * Aspect  _et al._ (1982) A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, 91 (1982).
  * Leggett and Garg (1985) A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985).
  * Shalm  _et al._ (2015) L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, Phys. Rev. Lett. 115, 250402 (2015).
  * Giustina  _et al._ (2015) M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-A. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, Phys. Rev. Lett. 115, 250401 (2015).
  * Gallicchio  _et al._ (2014) J. Gallicchio, A. S. Friedman, and D. I. Kaiser, Phys. Rev. Lett. 112, 110405 (2014), arXiv:1310.3288 [quant-ph] .
  * Blasone  _et al._ (2009) M. Blasone, F. Dell’Anno, S. De Siena, and F. Illuminati, EPL 85, 0002 (2009), arXiv:0707.4476 [hep-ph] .
  * Hill and Wootters (1997) S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997).
  * Wootters (1998) W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
  * Dakić  _et al._ (2010) B. Dakić, V. Vedral, and i. c. v. Brukner, Phys. Rev. Lett. 105, 190502 (2010).
  * Mermin (1990) N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).
  * Svetlichny (1987) G. Svetlichny, Phys. Rev. D 35, 3066 (1987).
  * Gangopadhyay  _et al._ (2013) D. Gangopadhyay, D. Home, and A. S. Roy, Phys. Rev. A 88, 022115 (2013).
  * Banerjee  _et al._ (2015) S. Banerjee, A. K. Alok, R. Srikanth, and B. C. Hiesmayr, The European Physical Journal C 75, 487 (2015).
  * Alok  _et al._ (2016) A. K. Alok, S. Banerjee, and S. U. Sankar, Nucl. Phys. B909, 65 (2016), arXiv:1411.5536 [hep-ph] .
  * Gangopadhyay and Roy (2017) D. Gangopadhyay and A. S. Roy, The European Physical Journal C 77, 260 (2017).
  * Formaggio  _et al._ (2016) J. A. Formaggio, D. I. Kaiser, M. M. Murskyj, and T. E. Weiss, Phys. Rev. Lett. 117, 050402 (2016).
  * An  _et al._ (2015) F. P. An, A. B. Balantekin, and H. R. e. Band (Daya Bay Collaboration), Phys. Rev. Lett. 115, 111802 (2015).
  * Huelga  _et al._ (1995) S. F. Huelga, T. W. Marshall, and E. Santos, Phys. Rev. A 52, R2497 (1995).
  * Pontecorvo (1968) B. Pontecorvo, Sov. Phys. JETP 26, 984 (1968), [Zh. Eksp. Teor. Fiz.53,1717(1967)].
  * Maki  _et al._ (1962) Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).
  * Nunokawa  _et al._ (2005) H. Nunokawa, S. Parke, and R. Z. Funchal, Phys. Rev. D 72, 013009 (2005).
  * Lisi and Montanino (1997) E. Lisi and D. Montanino, Phys. Rev. D 56, 1792 (1997).
  * Vissani (2017) F. Vissani, (2017), arXiv:1706.05435 [nucl-th] .
  * Wolfenstein (1978) L. Wolfenstein, Phys. Rev. D 17, 2369 (1978).
  * Akhmedov  _et al._ (2004) E. K. Akhmedov, R. Johansson, M. Lindner, T. Ohlsson, and T. Schwetz, JHEP 04, 078 (2004), arXiv:hep-ph/0402175 [hep-ph] .
  * Emary (2012) C. Emary, Phys. Rev. B 86, 085418 (2012).
  * Kofler and Brukner (2013) J. Kofler and i. c. v. Brukner, Phys. Rev. A 87, 052115 (2013).
  * Waldherr  _et al._ (2011) G. Waldherr, P. Neumann, S. F. Huelga, F. Jelezko, and J. Wrachtrup, Phys. Rev. Lett. 107, 090401 (2011).
  * Zhou  _et al._ (2015) Z.-Q. Zhou, S. F. Huelga, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 115, 113002 (2015).
  * Hiesmayr  _et al._ (2012) B. C. Hiesmayr, A. Di Domenico, C. Curceanu, A. Gabriel, M. Huber, J.-Å. Larsson, and P. Moskal, The European Physical Journal C 72, 1856 (2012).
  * Naikoo  _et al._ (2017) J. Naikoo, A. K. Alok, S. Banerjee, S. U. Sankar, G. Guarnieri, and B. C. Hiesmayr, (2017), arXiv:1710.05562 [hep-ph] .