# Statistical Model-based Evaluation of Neural Networks

###### Abstract

Using a statistical model-based data generation, we develop an experimental
setup for the evaluation of neural networks (NNs). The setup helps to
benchmark a set of NNs vis-a-vis minimum-mean-square-error (MMSE) performance
bounds. This allows us to test the effects of training data size, data
dimension, data geometry, noise, and mismatch between training and testing
conditions. In the proposed setup, we use a Gaussian mixture distribution to
generate data for training and testing a set of competing NNs. Our experiments
show the importance of understanding the type and statistical conditions of
data for appropriate application and design of NNs.

Index Terms—  Explainable machine learning, MMSE.

##  1 Introduction

It is imperative to understand and explain neural networks (NNs) for realizing
their wide scale use in critical applications. Explaining decisions of (deep)
NNs remains as an open research problem [1], [2]. Existing methods for
explanations can be categorized in various ways, [3], for example, _intrinsic_
and _post-hoc_ [4]. Intrinsic interpretability is inherent in structures of
methods, such as rule-based models, decision trees, linear models, and
unfolding NNs [5]. In contrast, a post-hoc method requires a second (simple)
model to provide explanations about the original (complex) method. There are
approaches to provide _local_ and _global_ explanations. A local explanation
justifies a NN’s output for a specific input. A global explanation provides a
justification on average performance of a NN, independently of any particular
input [6].

Techniques for explainability include visualization of neuronal activity in
layers of a deep architecture, for examples, based on sparsity and heatmaps
[1, 7, 8, 9, 10, 11]. There are also techniques for visualizing training steps
of non-convex optimization methods (such as stochastic gradient search), and
identifying contributions of input features, referred to as feature importance
[12]. Example-driven techniques explain an output for a given input by
identifying and presenting other instances, usually from available labeled
data, that are semantically similar to the input instance. This also includes
adversarial perturbation-based studies for explaining robustness [13]. There
is a recent trend to develop model-based design and analysis for explanations.
In particular, the work [2] trains deep NNs using statistical and domain
specific model-based perturbations.

_Our Contributions:_ We develop an experimental setup to evaluate and compare
NNs. The proposed setup is based on traditional statistical signal processing.
We use generated data from tractable probability distributions that has
computable minimum-mean-square-error (MMSE) estimator. The MMSE estimation
performance can be used as a bound to compare the chosen NNs. While
sophisticated generative models exist in literature, such as normalizing flows
[14], we choose a generative model (probability distribution) for which the
MMSE estimator can be computed in a simple form. Therefore we use Gaussian
mixture model (GMM) for the proposed setup. In theory, a GMM can model
arbitrary distributions closely by increasing number of mixture components
[15, Chapter 3]. The MMSE estimator for joint GMM distribution has a closed
form and interpretable expression [16]. Availability of the closed form MMSE
expression is beneficial to perform controlled experiments. In addition, GMM
helps in easy visualization of data geometry. Our setup helps in studying the
performance trend of a set of NNs under different conditions like: training
data need, data dimension, data geometry and effects of different testing
conditions, and provides a view on the probable failure cases of NNs.

The rest of the paper is organised as follows. Section 2 presents the
statistical model-based experimental setup. The data generation process,
experiments and results are presented in Section 3. Finally Section 4 provides
conclusions.

##  2 Statistical model-based
experimental setup

A typical setup in statistical signal processing is estimation of a target
signal $\mathbf{t}\in\mathbb{R}^{Q}$ given an observation signal
$\mathbf{x}\in\mathbb{R}^{P}$. Denoting the estimated signal as
$\tilde{\mathbf{t}}$, the MMSE estimator is defined as [17]

$\displaystyle\tilde{\mathbf{t}}_{\text{MMSE}}=\arg\min_{\tilde{\mathbf{t}}}\mathcal{E}\\{\|\mathbf{t}-\tilde{\mathbf{t}}\|^{2}\\}=\mathcal{E}(\mathbf{t}|\mathbf{x}),$
(1)

and the MMSE performance is computed as

$\displaystyle
C_{\text{MMSE}}=\mathcal{E}\\{\|\mathbf{t}-\mathcal{E}(\mathbf{t}|\mathbf{x})\|^{2}\\}.$
(2)

The above MMSE performance provides a theoretical bound. Therefore we can
compare performance of a set of NNs with the MMSE bound. If the joint
distribution $p(\mathbf{x},\mathbf{t})$ is parametarized by
$\boldsymbol{\theta}$, then we can perform different controlled experiments by
varying $\boldsymbol{\theta}$ followed by computing $C_{\text{MMSE}}$ and
checking how the chosen NN fares against the MMSE. These experiments will
provide an analysis platform for understanding the conditions of success and
failure of a NN. However, in delineating such a path of analysis we encounter
a technical challenge: there exist few distributions for which MMSE estimators
are computable analytically and we can evaluate $C_{\text{MMSE}}$ reliably.
The next subsection presents one such distribution and a helpful signal model
for conducting our controlled experiments.

###  2.1 Distributions, system model and their advantages

If the joint distribution $p(\mathbf{x},\mathbf{t})$ has a Gaussian mixture
(GM) density with parameter $\boldsymbol{\theta}$ then we can compute the
optimal (non-linear) MMSE estimator
$\tilde{\mathbf{t}}_{\text{MMSE}}=\mathcal{E}_{\boldsymbol{\theta}}(\mathbf{t}|\mathbf{x})$
exactly in an analytical form. Using simulations and invoking the law of large
numbers, we then can evaluate the MMSE performance. We use a classical linear
system model (observation system) for the controlled experiments as shown
below

$\displaystyle\mathbf{x}=\mathbf{Ht}+\mathbf{n},$ (3)

where $\mathbf{x}\in\mathbb{R}^{P}$ is the observation signal,
$\mathbf{t}\in\mathbb{R}^{Q}$ is the target signal vector to be estimated and
$\mathbf{n}\in\mathbb{R}^{P}$ is the observation noise. The matrix
$\mathbf{H}\in\mathbb{R}^{P\times Q}$ represents a system which is assumed to
be known. The fundamental advantages of using a linear model are as follows:
(a) For the linear observation system, if $\mathbf{t}$ is GM distributed and
$\mathbf{n}$ is Gaussian distributed then the joint distribution
$p(\mathbf{x},\mathbf{t})$ is also a GM density. This enables us to compute
the MMSE performance. (b) The use of GM distribution can assist in generating
fairly complicated (data spread) data geometry, which in turn can be used to
benchmark a NN against the MMSE for different data geometries. (c) The linear
observation model is a generative model and as such we can create any required
amount of training and testing data. This enables us to visualize the effect
of training data on NN performance. (d) We can compute the signal-to-noise-
ratio (SNR) at varying conditions and check the debilitating effect of noise
on the NN. (e) We can check the role of observation’s dimension by varying
$P$.

We consider the signal $\mathbf{t}$ to be GM distributed, and noise
$\mathbf{n}$ to be Gaussian distributed as follows

$p(\mathbf{t})=\sum_{m=1}^{M}\alpha_{m}\mathcal{N}(\mathbf{t};\boldsymbol{\mu}_{m},\mathbf{C}_{m});\quad
p(\mathbf{n})=\mathcal{N}(\boldsymbol{\mu}_{\mathbf{n}},\mathbf{C}_{\mathbf{n}}),$
(4)

$\displaystyle\begin{array}[]{c}\tilde{\mathbf{t}}_{\text{MMSE}}=\mathcal{E}(\mathbf{t}|\mathbf{x})=\displaystyle\sum_{m=1}^{M}\beta_{m}(\mathbf{x})\left(\boldsymbol{\mu}_{m}+\mathbf{C}_{m}\mathbf{H}^{T}\left[\mathbf{H}\mathbf{C}_{m}\mathbf{H}^{T}+\mathbf{C}_{\mathbf{n}}\right]^{-1}\left(\mathbf{x}-\left[\mathbf{H}\boldsymbol{\mu}_{m}+\boldsymbol{\mu}_{\mathbf{n}}\right]\right)\right),\\\
\beta_{m}(\mathbf{x})=\displaystyle\frac{\alpha_{m}\frac{1}{\left(\sqrt{2\pi}\right)^{P}\left|\mathbf{H}\mathbf{C}_{m}\mathbf{H}^{T}+\mathbf{C}_{\mathbf{n}}\right|^{\frac{1}{2}}}\exp\left[-\frac{1}{2}\left(\mathbf{x}-\left[\mathbf{H}\boldsymbol{\mu}_{m}+\mu_{\mathbf{n}}\right]\right)^{T}\left[\mathbf{H}\mathbf{C}_{m}\mathbf{H}^{T}+\mathbf{C}_{\mathbf{n}}\right]^{-1}\left(\mathbf{x}-\left[\mathbf{H}\boldsymbol{\mu}_{m}+\mu_{\mathbf{n}}\right]\right)\right]}{\sum_{j=1}^{M}\alpha_{j}\frac{1}{\left(\sqrt{2\pi}\right)^{P}\left|{\mathbf{H}}\mathbf{C}_{j}\mathbf{H}^{T}+\mathbf{C}_{\mathbf{n}}\right|^{\frac{1}{2}}}\exp\left[-\frac{1}{2}\left(\mathbf{x}-\left[\mathbf{H}\boldsymbol{\mu}_{j}+\mu_{\mathbf{n}}\right]\right)^{T}\left[\mathbf{H}\mathbf{C}_{j}\mathbf{H}^{T}+\mathbf{C}_{\mathbf{n}}\right]^{-1}\left(\mathbf{x}-\left[\mathbf{H}\boldsymbol{\mu}_{j}+\mu_{\mathbf{n}}\right]\right)\right]}.\end{array}$
(7)

where $M$ is the number of Gaussian mixture components, $\alpha_{m}$ is the
mixing proportion subject to $\sum_{m=1}^{M}\alpha_{m}=1$,
$\boldsymbol{\mu}_{m}$ and $\mathbf{C}_{m}$ are the mean and covariance of the
$m^{th}$ Gaussian distribution respectively. Furthermore,
$\boldsymbol{\mu}_{\mathbf{n}}$ and $\mathbf{C}_{\mathbf{n}}$ are the mean and
covariance of noise respectively. The joint probability density
$p(\mathbf{x},\mathbf{t})$ is also GM distributed with parameter
$\boldsymbol{\theta}=\\{\\{\alpha_{m},\boldsymbol{\mu}_{m},\mathbf{C}_{m}\\}_{m=1}^{M},\boldsymbol{\mu}_{\mathbf{n}},\mathbf{C}_{\mathbf{n}}\\}$,
and the explicit analytical form of the MMSE estimator is shown in (7) (as per
[16]).

While we have an analytical expression for $\tilde{\mathbf{t}}_{\text{MMSE}}$,
we do not have an analytical expression for the MMSE
$\mathcal{E}\\{\|\mathbf{t}-\tilde{\mathbf{t}}_{\text{MMSE}}\|^{2}\\}$. The
MMSE is thus computed through simulations by averaging the $l_{2}$ norm of the
estimation error over a large number of samples. We use the normalized mean-
square-error (NMSE) in decibel (dB) scale as the measure of performance for
our experiments, as follows:
$\mathrm{NMMSE}_{\text{dB}}=10\log_{10}\frac{\mathcal{E}\\{\|\mathbf{t}-\tilde{\mathbf{t}}\|^{2}\\}}{\mathcal{E}\\{\|\mathbf{t}-\mathcal{E}(\mathbf{t})\|^{2}\\}}.$
Here, $\mathcal{E}\\{\|\mathbf{t}-\tilde{\mathbf{t}}\|^{2}\\}$ is the
empirical estimation error power and
$\mathcal{E}\\{\|\mathbf{t}-\mathcal{E}(\mathbf{t})\|^{2}\\}$ is the signal
power computed analytically.

##  3 Generation of Data and Experiments

In this section, we present the data generation procedure and the experiments
performed for different data geometries.

###  3.1 Data generation

A GM is a universal approximator of densities which can approximate any smooth
density with enough numbers of mixture components [15, Chapter 3]. We generate
our training data from (3) by sampling from the following GMM,

$\displaystyle\begin{array}[]{c}p(\mathbf{t})=\displaystyle\sum_{m=1}^{M}\alpha_{m}\,\mathcal{N}(\mathbf{t};a\boldsymbol{\mu}_{m},\mathbf{C}_{m}),\\\
p(\mathbf{n})=\mathcal{N}\left(\boldsymbol{\mu}_{\mathbf{n}},\mathbf{C}_{\mathbf{n}}\right)=\mathcal{N}\left(\mathbf{0},\frac{b}{P}\mathbf{I}_{P}\right),\end{array}$
(10)

where $a$ and $b$ are the parameters which helps in generating different data
geometries and signal-to-noise-ratio (SNR)
$\frac{\mathcal{E}\\{\|\mathbf{t}-\mathcal{E}(\mathbf{t})\|^{2}\\}}{\mathcal{E}\\{\|\mathbf{n}\|^{2}\\}}$.
For experimental analysis, we can create varying data geometries by choice of
$a$. Assuming $\|\boldsymbol{\mu}_{m}\|^{2}=1$, the choice of $a$ provides
position of the mean vectors of the Gaussian components on a $Q$-dimensional
sphere with radius $a$. Therefore, by varying $a$ we can represent a gamut of
distributions from a single mode to multiple modes. If we choose $a=0$, then
all the components superimpose on each other and the GM becomes a single mode
Gaussian distribution. If we increase $a$ separation between the components
increases, and also increases the signal power (LABEL:eq:Signal_power_1). A
pictorial view of the generated data geometries (projected in 3-D) by
increasing $a$, but keeping the same covariances across mixtures, are shown in
Fig. 1. For ease of experimentation, let
$\alpha_{m}=\frac{1}{M},\|\boldsymbol{\mu}_{m}\|^{2}=1,\mathbf{C}_{m}=\mathbf{I}_{Q}.$
The MMSE estimator for the observation model (3) can be found using (7) by
substituting the chosen set of parameters.

Fig. 1: Visualization of GM data geometries using (10). As $a$ increases the
inter-component separability increases.

The signal power
$\mathcal{E}\\{\|(\mathbf{t}-\mathcal{E}(\mathbf{t})\|^{2}\\}$ is shown in
(LABEL:eq:Signal_power_1) as,

(14)

The noise power is
$\mathcal{E}\\{\|(\mathbf{n}-\mathcal{E}(\mathbf{n})\|^{2}\\}=\sum_{i=1}^{P}\mathcal{E}(n_{i}^{2})=P\frac{b}{P}=b$.
The signal-to-noise-ratio (SNR) in dB is,

$\displaystyle
SNR=10\log_{10}\frac{\mathcal{E}\\{\|(\mathbf{t}-\mathcal{E}(\mathbf{t})\|^{2}\\}}{\mathcal{E}\\{\|(\mathbf{n}-\mathcal{E}(\mathbf{n})\|^{2}\\}}$
(15)
$\displaystyle=10\log_{10}\frac{Q+a^{2}\left(1-\frac{1}{M^{2}}\operatorname{Tr}[\sum_{m=1}^{M}\boldsymbol{\mu}_{m}][\sum_{m=1}^{M}\boldsymbol{\mu}_{m}]^{T}\right)}{b}.$

For the data generation, we sample $\mathbf{t}$ and $\mathbf{n}$ from the
distributions (4), and generate $\mathbf{x}$ using (3). We denote the training
set as
$\mathcal{D}_{train}=\\{(\mathbf{x}^{(j)},\mathbf{t}^{(j)})\\}_{j=1}^{J_{train}}$
with $J_{train}$ data-and-target pairs and the test set as
$\mathcal{D}_{test}=\\{(\mathbf{x}^{(j)},\mathbf{t}^{(j)})\\}_{j=1}^{J_{test}}$
with $J_{test}$ data-and-target pairs. The experiments are carried out in
Monte-Carlo fashion over many realizations of observation system matrix
$\mathbf{H}$ and the results are shown averaged over those realizations. The
components of $\mathbf{H}$ are i.i.d Gaussian.

###  3.2 Experiments

Experiments are performed to understand several aspects of the NNs. We choose
four types of NNs: extreme learning machine (ELM) [18], self-size estimating
feedforward NN (SSFN) [19], feedforward NN (FFNN) [20], and residual network
(ResNet) [21]. The choice of NNs is based on their structures and training
complexity.

We choose an ELM with a single hidden layer comprised of 30 hidden nodes. The
ELM uses a random weight matrix. Therefore ELM is a simple network to train.
The SSFN is a multi-layer network with low training complexity. The SSFN uses
a combination of random weights and learned parameters. We used a 20-layer
SSFN with 30 hidden nodes per layer. Then we evaluate a FFNN with 6 hidden
layers. The number of nodes are 64, 128, 256, 256, 128, 64 for the six layers,
respectively. FFNN is trained using standard backpropagation. Finally we use
ResNet-34 to have a deep NN that has capability to avoid the vanishing
gradient problem [21]. The ResNet-34 is a 34 layered architecture with a
combination of convolutional layers, max-pooling, batch normalization and skip
connections. We adapted it from Keras ResNet-50 structure with one change -
use of 1-D convolution instead of 2-D convolution because of our data type.
The number of trainable parameters are approximately 0.3K, 6K, 240K, and 500K
for ELM, SSFN, FFNN, and ResNet-34, respectively. All the experiments were
performed with a train:test split of $70:30$ using $10$ Monte Carlo
simulations.

Our first three experiments are conducted in _matched training and testing
statistics scenario_ , which means we first fix the parameters of
distributions to define the signal statistics and then generate
$\mathcal{D}_{train}$ and $\mathcal{D}_{test}$. In our first experiment, we
study the effect of training dataset size $\mathcal{D}_{train}$. The
estimation performance versus size of training dataset is shown in Fig. 2. The
experiment shows that performances of all the four NNs saturate as the
training data size increases. Therefore the popular idea that more and more
training data improves the learning capacity of NNs is questionable. Based on
the saturation trend, we decide to perform rest of the experiments using 3000
total data.

Fig. 2: Estimation performance study for visualizing the effect of training
data amount.

The second experiment is designed to study how the data geometry affects the
estimation performance. The choice $b=50$ translates to a fixed noise power.
The SNR is varied by varying $a$. We show estimation performance versus SNR in
Fig. 3. The results show that all NNs show bad performance when SNR is low. As
the SNR improves, performance of all four NNs improves, but FFNN and ResNet
outperform others. The performance of ResNet saturates at high SNR.

Fig. 3: Estimation performance study for visualizing role of data geometry and
the resulting signal power.

The third experiment studies sampling requirements. This is represented by the
observation vector dimension $P$ for a fixed target signal dimension $Q$. Note
that $P<Q$ is a under-determined sampling system, and $P\geq Q$ is an over-
determined system. The NMSE versus $P$ for $Q=10$ is shown in Fig. 4. The
results show performance improvement as $P$ increases and then a saturation
trend.

Fig. 4: Estimation performance study for visualizing the role of sampling.
Fig. 5: Estimation performance study for mismatched training and testing
conditions. The training SNR is 10.396 dB.

All the previous experiments used matched training and test statistics. We now
consider _mismatched training and test statistics_ for our last experiment. A
mismatched scenario is relevant in reality. In this case, different parameter
settings are used to generate $\mathcal{D}_{train}$ and $\mathcal{D}_{test}$.
The training SNR is chosen as 10.396 dB. We vary $b$ to change test SNR
keeping all other parameters same as the training condition, and then we
create $\mathcal{D}_{test}$ for a choice of $b$. The performances for
different test SNRs are shown in Fig. 5. The results show that the NNs behave
in a similar manner according to Fig. 3 in the region closer to matched
training and test statistics. An interesting observation is that all the four
NNs show a saturation trend when test SNR increases. This result shows that
training based estimators (NNs) do not generalize under different testing
conditions, even when test SNR improves.

A natural question is why ResNet-34 does not show the best performance among
the competing NNs, given ResNet is known to provide a high quality performance
for image classification. Our understanding is that the GMM based data that we
generated, does not have similar spatial correlations like images. The
convolutional filters, max-pooling and skip connections of ResNet are more
suitable for image signals. Therefore it is important to understand the type
and statisitical conditions of data for appropriate use and design of NNs.

Code is available here: https://github.com/s-a-n-d-y/ExAI.

##  4 conclusions

We show that it is possible to compare between NNs using generated data in a
controllable experimental setup, and have an understanding of achievable
performance by comparing with the MMSE bound. A large gap between performance
and MMSE bound motivates effort to design efficient NNs. We conclude with an
understanding that an efficient NN for a particular data type or statistical
condition may not show good performance for a different data type or
statistical condition.

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Gravitational Effects on Quantum Coherence in Neutrino Oscillation
M. M<EMAIL_ADDRESS>Department of Physics, University of
Qom, Qom 371614-6611, Iran
R. Ramezani Arani222<EMAIL_ADDRESS>, and Z. S. Tabatabaei Lotfi333
<EMAIL_ADDRESS>University of Kashan, Km 6 Ravand Road, Kashan 87317-51167,
Iran

###### Abstract

In this paper, we investigate the quantum coherence for two flavor neutrinos
propagating in a Schwarzschild metric. In fact, this issue is explored both
qualitatively via calculating the parameter $K_{3}$ in Leggett-Garg inequality
(LGI) and also quantitatively by evaluating the $l_{1}$-norm, ${\cal
C}(\rho)$. Using the weak field approximations, we show that the gravitational
effects decrease the maximum value of $K_{3}$ for some intervals of energy in
such a way that there is no violation, while it leaves the maximum amount of
the quantum coherence, ${\cal C}(\rho)$, unchanged.

##  1 Introduction

Gravitation and quantum mechanics usually do not play an important role
simultaneously in low energy physical phenomena. However, at 1975 for the
first time, Collela et al. proposed an experiment in which the results
depended on both the gravitational and Planck constant [1]. This effect could
be explained by considering a Newtonian potential inserted in the Schrodinger
equation. Soon after, using semiclassical approximation, Stodolsky obtained
the quantum mechanical phase of a free particle propagating in an external
gravitational field in the weak field limit and investigated the gravitational
effects by matter and light wave interferometry [2]. In addition to the
subject of interferometry for neutrons and photons, the other place in which
the gravitational effects and quantum mechanics coexist, is the context of
neutrino oscillations. Neutrinos are known to be the most fascinating
particles of the standard model, since they can be utilized as special probes
to study physics from quantum scales up to celestial scales. In fact, they
have the ability to pass through ordinary matter with minimum interactions
because they can only participate in weak interactions and their masses are
extremely tiny to experience any reasonable gravitational influence.
Nevertheless, the effects of gravitation on neutrino oscillation have been
investigated through calculating the change in quantum phase due to neutrino
propagation in curved space-time [3, 4, 5].

On the other hand, quantum correlations are known as one of the cornerstones
of the foundation of quantum information sciences and quantum technologies,
since we can consider them as resource theories in quantum communications. In
addition to their technological applications, one can use quantum correlations
to study fundamental physical effects such as gravity at quantum mechanical
scales. For instance, the gravitational effects on CHSH inequality which is
known as one of the forms of the Bell inequalities have been studied [6]. Also
it has been investigated that the circular motion of particles in a
Schwarzschild metric causes a decrease in the degree of violation of Bell’s
inequality [7]. In a similar study, the effects of a curved space-time
described by the Kerr-Newman metric on the EPR correlations have been
investigated [8].

One of the other subjects in physics that neutrinos are used to study, is
quantum information which is based mostly on quantum correlations such as
entanglement and quantum coherence [9, 10, 11, 12, 13, 14, 15, 16, 17, 18].
They can usually be expressed in terms of transition amplitudes and hence,
oscillation probabilities [13]. Among the several researches on quantum
correlations examining the quantumness in neutrino oscillations, are Refs.
[16, 18] that characterize the Leggett-Garg inequality for neutrino
oscillation. Nonlocality in neutrino oscillation via Bell type inequalities,
i.e. the spatial counterparts of Leggett-Garg inequalities has also been
studied [15]. Moreover, quantum effects such as entanglement and quantum
coherence which seem to be distinct resource theories [19, 20], have been
investigated in the context of neutrino oscillation by Blassone et al. [11]
and Xue-Ke Song et al. [12]. Though it has been shown that these two quantum
correlations, in the case of neutrino oscillations are not independent of each
other [9]. In the Refs. [9, 12] quantum coherence has been treated
quantitatively by defining the $l_{1}-$norm as a coherence measure, which
assigns a number to neutrino states. The $l_{1}$-norm is described as the
summation over absolute values of all the off-diagonal elements of the
corresponding density matrix. For neutrinos in two flavors, it can be written
as $\cal C(\rho)$=2 $|\sqrt{P_{osc}P_{sur}}|$ in terms of the oscillation and
survival probabilities [9].

In fact, quantum coherence as a quantum correlation, can be investigated in
the two qualitative and quantitative approaches; Leggett-Garg inequality (LGI)
is a qualitative criterion and $l_{1}$-norm is a quantitative measure. In the
case of the former, we should clarify that the violation of the LGI can be
interpreted as appearing quantumness because quantum is the only theory that
contradicts the postulates of the LGI. Moreover, this quantumness is revealed
if the states on which the measurements are carried out, are a coherent
superposition of Hamiltonian eigenstates. Therefore, when we have treated the
LGI by a quantum state which does not give violation, this means the quantum
state does not have enough coherence. In this paper, we aim to study the
gravitational effects on quantum coherence in neutrino oscillation by the two
approaches mentioned above. We consider the propagation of the neutrino in a
Schwarzschild background for two flavors via the plane wave approach. Indeed,
we use the method in which the quantum mechanical phase of neutrino is
obtained by using the semiclassical approximation. With this assumption, the
gravitational field is treated classically and it differs from the stochastic
interactions between neutrinos and virtual black holes, which are due to
fluctuations in space-time itself if gravity is a quantum force [21].

In the next section, we will briefly review the neutrino oscillation in flat
and curved space-time . In Sec. 3, we will study LGI in radial propagation of
neutrinos in a Schwarzschild metric by calculating the $K_{3}$ parameter for
neutrinos propagating outwards and inwards in a gravitational field. In Sec.
4, we will compute the $l_{1}$-norm for neutrino oscillation in two flavor
framework via plane wave approach in a Schwarzschild metric in order to study
its variations with respect to the proper distance $L_{p}$, when the neutrino
is propagating towards or away from the gravitational source. Finally in Sec.
5, we will discuss our results and make the conclusion.

##  2 Neutrino Oscillation in Flat and Curved Space-Time

In this section, we will briefly review the context of neutrino oscillation in
flat and also curved space time. The state of a neutrino produced in a space-
time point $A(t_{A},\vec{x}_{A})$ is obtained from the superposition of mass
eigenstates, $|\nu_{k}\rangle$,

$|\nu_{\alpha}\rangle=\sum_{k}U^{\ast}_{\alpha k}|\nu_{k}\rangle$ (1)

in which U is the unitary mixing matrix and in case of the two flavors it is
shown as

$U=\left(\begin{array}[]{cc}\cos(\theta)\ \sin(\theta)\\\ -\sin(\theta)\
\cos(\theta)\end{array}\right)$ (2)

This state as a weak interaction eigenstate is called the flavor eigenstate.
There exists a discrepancy in energies, momenta and masses of different mass
eigenstates $|\nu_{k}\rangle$ which leads them to propagate differently and
hence, the neutrino oscillation takes place. In fact, the neutrino oscillation
exhibits a relative shift in mass eigenstate phases when they arrive at the
detector situated at the space-time point $B(t_{B},\vec{x}_{B})$. Actually,
while the plane wave approach is adopted, the propagated mass eigenstates
$|\nu_{k}\rangle$ are described as

$|\nu_{k}(t,\vec{x})\rangle=e^{-i{\Phi_{k}}}|\nu_{k}\rangle,$ (3)

where $\Phi_{k}(k=1,2)$ denoting the relative phase shift acquired by the mass
eigenstates during their propagation, is given by

$\Phi_{k}=E_{k}(t_{B}-t_{A})-\vec{p_{k}}.(\vec{x}_{B}-\vec{x}_{A}).$ (4)

In general, in the case of two flavors, the probability of detecting a
neutrino as $|\nu_{\mu}\rangle$ that is produced as $|\nu_{e}\rangle$ is given
by

$P(\nu_{e}\rightarrow\nu_{\mu})=|\langle\nu_{\mu}|\nu_{e}(t_{B},\vec{x}_{B})\rangle|^{2}=\sin^{2}(2\theta)\sin^{2}(\dfrac{\Phi_{jk}}{2}),$
(5)

in which, $\Phi_{jk}=\Phi_{j}-\Phi_{k}$. Of course, in order to have the
neutrino oscillation, the difference of the phase shift must not lead to the
decoherence of the flavor eigenstates due to the propagation. In other words,
the propagation length has to be smaller than the coherence length.

In flat space-time described by the Minkowski metric, applying the ultra
relativistic limit of $(t_{B}-t_{A})\simeq|x_{B}-x_{A}|$, together with the
relativistic expansion $m_{k}\ll E_{k}$, one can write

$\Phi_{jk}\simeq\dfrac{\Delta m^{2}_{jk}}{2E_{0}}|x_{B}-x_{A}|.$ (6)

Here, $\theta$ is the mixing angle, $\Delta m_{jk}^{2}$ is the mass squared
difference and $E_{0}$ is the energy for the massless neutrino measured by the
observer at infinity.

Now, to generalize our discussion to a curved space-time, we can replace the
quantum phase given by the Eq. (4) with its covariant form

$\Phi_{k}=\int_{A}^{B}p_{\mu}^{(k)}dx^{\mu},$ (7)

where, $p_{\mu}^{(k)}$ is the canonical conjugate momentum to the coordinate
$x^{\mu}$ and is given by

$p_{\mu}^{(k)}=m_{k}g_{\mu\nu}\dfrac{dx^{\nu}}{ds}.$ (8)

In above equation, $g_{\mu\nu}$ represents the metric tensor and $ds$ is the
line element. For instance, let us consider the Schwarzschild metric

$ds^{2}=B(r)dt^{2}-B(r)^{-1}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2},$ (9)

where

$B(r)=(1-\dfrac{2GM}{r}),$ (10)

in which, $G$ is the Newtonian constant and $M$ stands for mass of the
gravitational source [22]. In this metric, the gravitational field is
isotropic, therefore, we may consider the propagation of neutrinos in the
equatorial plane $\theta=\pi/2$. In this case, the phase obtained by the mass
eigenstates during the neutrino propagation from the source at the space-time
point $A(t_{A},r_{A},\phi_{A})$ to the detector at the space-time point
$B(t_{B},r_{B},\phi_{B})$, may be written as

$\Phi_{k}=\int_{A}^{B}[E_{k}dt-p_{k}(r)dr-J_{k}d\phi].$ (11)

Here, $E_{k}\equiv p_{t}^{(k)}$, $p_{r}\equiv-p_{r}^{(k)}$ and
$J_{k}(r)\equiv-p_{\phi}^{(k)}$, are the components of the canonical momentum
$p_{\mu}^{(k)}$ [23]. For convenience, in presence of the gravitational
effects, it is better to consider the neutrino propagation over the proper
distance $L_{p}$, which is generally defined by the relation

$\begin{array}[]{cc}L_{p}\equiv\int^{r_{B}}_{r_{A}}\sqrt{g_{rr}}\ dr\\\ \\\
=r_{B}\sqrt{1-\dfrac{2GM}{r_{B}}}-r_{A}\sqrt{1-\dfrac{2GM}{r_{A}}}\\\ \\\
+2GM\left[\ln\left(\sqrt{r_{B}-2GM}+\sqrt{r_{B}}\right)-\ln\left(\sqrt{r_{A}-2GM}+\sqrt{r_{A}}\right)\right]\par.\end{array}$
(12)

Assuming the weak field approximation, one can easily calculate $L_{p}$ to be

$L_{p}\simeq r_{B}-r_{A}+GM\ln\dfrac{r_{B}}{r_{A}},$ (13)

where $r_{A}$ and $r_{B}$ are, respectively, the positions of the source and
the detector which are measured with respect to the gravitational source
reference frame. In addition, we should note that using the ultra relativistic
limit, one can write the relation between proper time interval and proper
distance as

$L_{p}=c\tau$ (14)

in which $\tau$ is the proper time.

At this point, we restrict our discussion to the motion of neutrinos in a
Schwarzschild gravitational field in radial propagation. For radial
propagation of neutrinos in a Schwarzschild gravitational field, i.e.,
$d\phi=0$ in Eq. (11), there is no angular momentum. As for the flat-space
time, the relativistic expansion $m_{k}\ll E_{k}$ can be applied. Thus,
assuming $0<B(r)\leq 1$, after some calculations we have [3]

$\phi_{k}\simeq\int_{r_{A}}^{r_{B}}\dfrac{m_{k}^{2}}{2E_{0}}dr,$ (15)

which is integrated along the light-ray trajectory to be

$\Phi_{k}\simeq\dfrac{m_{k}^{2}}{2E_{0}}|r_{B}-r_{A}|.$ (16)

It should be noticed that $|r_{B}-r_{A}|$ is the coordinate difference and as
mentioned earlier, in the curved space-time it can be written in terms of the
proper distance. Here, the energy $E_{0}$ is the energy of the massless
neutrino that is measured by the observer located at $r=\infty$. However, it
may be convenient to rewrite the oscillation phase defined in Eq.(16) in terms
of the local energy, denoted by ${E_{0}^{loc}(r_{B})}$, measured by the
observer at the detector at $r_{B}$. This local energy is related to the
$E_{0}$ through the relation $E_{0}^{loc}(r)=|g_{tt}|^{-1/2}E_{0}$ [24]. Thus
regarding the Eq. (9), we can write

$E_{0}^{loc}(r_{B})=(1+\dfrac{GM}{r_{B}})E_{0}.$ (17)

Finally, using Eqs.(13) and (17) we can express the Eq.(16) as follows [3]

$\Phi_{kj}\simeq(\dfrac{\Delta
m^{2}_{kj}L_{p}}{2E_{0}^{loc}(r_{B})})\left[1-GM(\dfrac{1}{L_{p}}\ln\dfrac{r_{B}}{r_{A}}-\dfrac{1}{r_{B}})\right].$
(18)

In this equation, the corrections due to the gravitational effects are evident
from the second term in the square parentheses. In the following, we wish to
study the effects of gravitation on the LGI violation as well as the amount of
quantum coherence ($l_{1}$-norm). In order to have an appropriate evaluation,
we take $\Delta m^{2}_{12}=7.92\times 10^{-5}eV^{2}$, $\theta_{12}=0.59$ and
$GM=3\times 10^{7}\mathrm{Km}$, which can be the Schwarzschild radius of a
supermassive black hole.

##  3 LGI in Radial Propagation of Neutrino in a Schwarzschild Metric

In this section, we intend to study the gravitational effects on the LGI using
the time correlations in neutrino oscillations. In fact, the violation of the
LGI is a confirmation of the existence of quantum coherence. The LGI is based
on the two main postulates; macroscopic realism and noninvasive measurability
[25, 26, 27]. The first postulate means that the measurements performed on a
system, will only reveal the values that already exist. The second postulate
implies that the noninvasive measurements can be carried out on the system
without changing the state of the system. The simplest form of LGI is
constructed in terms of the parameter $K_{3}$ which is defined as

$K_{3}=C(t_{1},t_{2})+C(t_{2},t_{3})-C(t_{1},t_{3}),$ (19)

in which
$C(t_{i},t_{j})=\left\langle\hat{Q}(t_{i})\hat{Q}(t_{i})\right\rangle$ and
$\hat{Q}(t)$ is a dichotomic variable that can take only one of two discrete
values which are usually labeled by convention $+1$ and $-1$ [27]. According
to LGI, this parameter should be smaller than one, otherwise its assumptions
are violated. We should notice that in order to check the LGI inequality, we
need to use a physical state which is a coherent superposition of some
eigenstates to calculate the corresponding correlations. Of course, this
feature can occur in quantum mechanics, while this theory is in conflict with
the postulates on which the LGI is constructed. So, if this inequality is
violated, it means that we have a phenomenon that is inconsistent with the
corresponding postulates. On the other hand, if we choose a quantum state for
which this inequality is not violated, it could mean that the assumed state is
not coherent enough, therefore, for example, $K_{3}$ does not exceed unity.

If one wants to investigate the validity of the LGI postulates, it is
necessary to provide appropriate conditions that ensure compliance with the
LGI postulates. However, there are some experimental difficulties satisfying
the noninvasive measurement. Furthermore, its postulate is in conflict with
quantum mechanics and it is natural to find a quantum mechanical set up that
violates the LGI. Meanwhile, the LGI may be derived instead under the
assumption of stationarity [28]. Accordingly, the correlation functions
$C_{ij}$ depend only on the time difference $\tau\equiv t_{j}-t_{i}$ between
various measurements. There are several studies based on the stationarity
condition, in which the LGI has been investigated using the neutrino
oscillation [16, 17]. Nevertheless, since we take account of the gravitational
effects on LGI in the present study, the condition of stationarity for
neutrino oscillation is not necessarily satisfied. Therefore, we expect the
LGI violation by neutrino oscillation with the gravitational modifications
unless the coherent superposition is deflected. Hence, we can take the
violation of the LGI as a qualitative criterion for quantum coherence. The LGI
for neutrinos without assuming the condition of stationarity is investigated
in Ref. [18]. We must also write the parameter $K_{3}$ to be Lorentz
invariant. Thus, one needs to rewrite the time correlation functions in Eq.
(19) in terms of the proper time $\tau$ and, for simplicity, we adopt the
equal proper time intervals ($\tau_{3}-\tau_{2}=\tau_{2}-\tau_{1}=\tau$).
Furthermore, according to the ultra relativistic limit introduced in Eq. (14),
we can write the parameter $K_{3}$ in Eq.(19) in terms of the proper distance
$L_{p}$ as follows:

$K_{3}=C(0,L_{p})+C(L_{p},2L_{p})-C(0,2L_{p}).$ (20)

Here, in the case of neutrino oscillation, the observable for which the
correlation functions are evaluated is
$\hat{Q}=2|\nu_{\alpha}\rangle\langle\nu_{\alpha}|-1$. If the neutrino is
still in its initial flavor state the outcome is $+1$ and otherwise it will be
$-1$.

Now let us consider a neutrino source located at the radial distance $r_{A}$
from the center of the gravitational field of a nonrotating object with a
spherical symmetry that is described by the Schwarzschild metric. We intend to
construct the parameter $K_{3}$ between the source and the two hypothetical
detectors that are placed in radial distances $r_{B1}$ and $r_{B2}$ from the
center of the gravitational source. We consider the case in which the initial
neutrino flavor state is $|\nu_{\mu}\rangle$. In general, the correlation of
operator $\hat{Q}$ between two different proper distances is defined by

$C(L_{p}^{(1)},L_{p}^{(2)})=\langle\mu|\frac{1}{2}\\{\hat{Q}(L_{p}^{(1)}),\hat{Q}(L_{p}^{(2)})\\}|\mu\rangle,$
(21)

where $\hat{Q}(L_{p}^{(i)})=U^{\dagger}(L_{p}^{(i)})\hat{Q}U(L_{p}^{(i)})$.
Superscript $(i)$ refers to either the source or one of the detectors. Here,
$U(L_{p})$ is the time evolution matrix in the flavor space, whose elements
are defined as

$U_{\delta\lambda}(L_{p})=\sum_{k}U^{\ast}_{\delta
k}e^{-i\phi_{k}(L_{p})}U_{\lambda k}.$ (22)

In fact, $U_{\delta\lambda}(L_{p})$ is the transition amplitude. We should
note that the time evolution operator in Eq.(22) is unitary when using
Eq.(16); i.e., one gives the $\phi_{k}$ in terms of the $r_{B}$. However, in
Eq. (22), we have written $r_{B}$ in terms of the $L_{p}$ using Eq. (13) and
thus $U(L_{p})$ is not unitary in terms of $L_{p}$.

Thus we have for the three correlations in Eq. (20)

$\displaystyle C(0,\kappa L_{p})=2P_{\nu_{\mu}\rightarrow\nu_{\mu}}(\kappa
L_{p})-1,\hskip 28.45274pt\text{for}\ \alpha=\mu$ $\displaystyle\hskip
54.06006pt=1-2P_{\nu_{\mu}\rightarrow\nu_{\alpha}}(\kappa L_{P}),\hskip
28.45274pt\text{for}\ \alpha\neq\mu$ (23)

in which $\kappa=1,2$ and

$\displaystyle C(L_{p},2L_{p})=$
$\displaystyle(1/2)\left[\langle\nu_{\mu}\right|4U^{\dagger}(L_{p})|\nu_{\alpha}\rangle\langle\nu_{\alpha}|U(L_{p})U^{\dagger}(2L_{p})|\nu_{\alpha}\rangle\langle\nu_{\alpha}|U(2L_{p})|\nu_{\mu}\rangle$
(24) $\displaystyle\hskip
11.38092pt+\langle\nu_{\mu}|4U^{\dagger}(2L_{p})|\nu_{\alpha}\rangle\langle\nu_{\alpha}|U(2L_{p})U^{\dagger}(L_{p})|\nu_{\alpha}\rangle\langle\nu_{\alpha}|U(L_{p})|\nu_{\mu}\rangle$
$\displaystyle\hskip
11.38092pt-\langle\nu_{\mu}|4U^{\dagger}(L_{p})|\nu_{\alpha}\rangle\langle\nu_{\alpha}|U(L_{p})|\nu_{\mu}\rangle$
$\displaystyle\hskip
11.38092pt-\langle\nu_{\mu}|4U^{\dagger}(2L{p})|\nu_{\alpha}\rangle\langle\nu_{\alpha}|U(2L_{p})|\nu_{\mu}\rangle+2].$

One can simplify the recent equation as follows:

$\displaystyle C(L_{p},2L_{p})=$ $\displaystyle
4\bar{U}_{\mu\alpha}(L_{p})U_{\alpha\mu}(L_{p})\bar{U}_{\mu\alpha}(2L_{p})U_{\alpha\mu}(2L_{p})$
(25) $\displaystyle+2\bar{U}_{\mu\alpha}(L_{p})U_{\alpha
e}(L_{p})\bar{U}_{e\alpha}(2L_{p})U_{\alpha\mu}(2L_{p})$
$\displaystyle+2\bar{U}_{\mu\alpha}(2L_{p})U_{\alpha
e}(2L_{p})\bar{U}_{e\alpha}(L_{p})U_{\alpha\mu}(L_{p})$
$\displaystyle-2P_{\nu_{\mu}\rightarrow\nu_{\alpha}}(L_{p})-2P_{\nu_{\mu}\rightarrow\nu_{\alpha}}(2L_{p})+1.$

Consequently, we obtain the following expression for $K_{3}$ in terms of the
probabilities and the transition matrix elements:

$\displaystyle K_{3}=$ $\displaystyle
1-4P_{\nu_{\mu}\rightarrow\nu_{e}}(L_{p})+4\bar{U}_{\mu
e}(L_{p})U_{e\mu}(L_{p})\bar{U}_{\mu e}(2L_{p})U_{e\mu}(2L_{p})$ (26)
$\displaystyle+4Re[\bar{U}_{\mu
e}(2L_{p})U_{ee}(L_{p}){U}_{ee}(2L_{p})\bar{U}_{e\mu}(L_{p})]\hskip
28.45274pt\text{for}\ \alpha=e,$

and

$\displaystyle K_{3}=$ $\displaystyle
1-4P_{\nu_{\mu}\rightarrow\nu_{\mu}}(2L_{p})+4P_{\nu_{\mu}\rightarrow\nu_{\mu}}(L_{p})P_{\nu_{\mu}\rightarrow\nu_{\mu}}(2L_{p})$
(27) $\displaystyle+4Re[\bar{U}_{\mu
e}(2L_{p})U_{\mu\mu}(L_{p}){U}_{\mu\mu}(2L_{p})\bar{U}_{e\mu}(L_{p})]\hskip
28.45274pt\text{for}\ \alpha=\mu.$

We preform our calculations in the weak field approximation, since in the
distance scales we intend to work in, the weak field approximation holds.
Albeit, we should write our relations in terms of the local energy
$E^{\text{loc}}_{0}(r_{B})$ measured by the observers situated at the two
detectors, since we intend to study the behavior of $K_{3}$ with respect to
variations of an unique energy, we rewrite both energies measured by the two
detectors in terms of the energy $E_{0}$ measured by the observer at the
infinity. To better enlighten the gravitational effects on the parameter
$K_{3}$, we will evaluate and compare the parameters $K_{3}$ for the two
different cases of the neutrinos propagating radially outwards and towards the
gravitational source.

  * •

In the case of neutrinos propagating radially outwards the gravitational
source, we can write the distances $r_{B1}$ and $r_{B2}$ from the Eq. (13) as
follows:

$r_{B1}=L_{p}+r_{A}-GM\ln(\dfrac{L_{p}}{r_{A}}+1),$ (28)
$r_{B2}=2L_{p}+r_{A}-GM\ln(\dfrac{2L_{p}}{r_{A}}+1).$ (29)

Therefore, up to the first order of the weak field approximation, ${\cal
O}(GM/r)$, we can express the oscillation phases as

$\Phi_{kj}(L_{p})\simeq\dfrac{\Delta
m^{2}_{kj}L_{p}}{2E_{0}}\left[1-GM\left(\dfrac{1}{L_{p}}\ln(\dfrac{L_{p}}{r_{A}}+1)\right)\right],$
(30)

and

$\Phi_{kj}(2L_{p})\simeq\dfrac{\Delta
m^{2}_{kj}2L_{p}}{2E_{0}}\left[1-GM\left(\dfrac{1}{2L_{p}}\ln(\dfrac{2L_{p}}{r_{A}}+1)\right)\right].$
(31)

By using these two recent relations and Eqs.(5) and (22) in Eqs. (26) and
(27), the parameter $K_{3}$ can be constructed. In order to specify the
gravitational effects on the parameter $K_{3}$ and to checkout the LGI
violation, we take the relative parametric space such that the gravitational
effects might be remarkable in neutrino oscillation. Hence, we choose the
locations of the two detectors such that the proper distance becomes
$L_{p}\simeq 3\times 10^{8}\mathrm{Km}$. The radial distance of the source is
selected to be $r_{A}\simeq 10^{8}\mathrm{Km}$ and the energy order is taken
about several hundred $\mathrm{TeV}$s. Maybe it should be enlightened that our
choices of the parameters are not far from the outputs of the astrophysical
models. Among the existing models describing the production of high energy
neutrinos near a massive object [29, 30, 31, 32, 33, 34], there is a
conjecture that would create a sort of physics case for our parameter choices
[34]. According to this conjecture, the production of neutrinos with the
energy scale of $\mathrm{TeV}$ can take place within a half of the
Schwarzschild radius of a supermassive black hole. Accordingly, the variation
of the parameter $K_{3}$ is plotted with respect to $E_{0}$ in the interval
$1.5\times 10^{2}$ to $5\times 10^{2}$ TeV in Fig. 1(a). For comparison, the
parameter $K_{3}$ for the case of the flat space time, i.e., $M=0$ is also
drawn.

Figure 1: Variations in $K_{3}$ as a function of the energy $E_{0}$ in TeV,
for neutrinos radially propagating outwards (a) and inwards (b) the
gravitational source. Blue (solid) and orange (dashed) curves represents the
$K_{3}$ in flat and curved space time, respectively.

  * •

In the case of neutrino propagating radially towards the gravitational source,
we place the neutrino production source at $\acute{r}_{A}$. The parameter
$K_{3}$ is calculated for the neutrinos propagating from this point to the
detectors situated at the places $\acute{r}_{B1}$ and $\acute{r}_{B2}$ given
by the relations

$\acute{r}_{B1}=\acute{r}_{A}-L_{p}-GM\ln(1-\dfrac{L_{p}}{\acute{r}_{A}}),$
(32)

and

$\acute{r}_{B2}=\acute{r}_{A}-2L_{p}-GM\ln(1-\dfrac{2L_{p}}{\acute{r}_{A}}).$
(33)

Consequently, we can again evaluate the oscillation phases as

$\acute{\Phi}_{kj}(L_{p})\simeq\dfrac{\Delta
m^{2}_{kj}\acute{L}_{p}}{2E_{0}}\left[1+GM\left(\dfrac{1}{\acute{L}_{p}}\ln(1-\dfrac{\acute{L}_{p}}{\acute{r}_{A}})\right)\right],$
(34)

and

$\acute{\Phi}_{kj}(2\acute{L}_{p})\simeq\dfrac{\Delta
m^{2}_{kj}2\acute{L}_{p}}{2E_{0}}\left[1+GM\left(\dfrac{1}{2\acute{L}_{p}}\ln(1-\dfrac{2\acute{L}_{p}}{\acute{r}_{A}})\right)\right].$
(35)

Similar to the case of neutrinos propagating outwards, we can plot the
parameter $K_{3}$ as shown in the Fig. 1(b), by considering
$\acute{r}_{A}=6.5\times 10^{8}Km$ and $\acute{L}_{p}=3\times 10^{8}Km$.

The following two points are obvious from the two diagrams in Fig.1

  * •

It is seen that due to the presence of gravitational effects, there exists a
damping in the maximum value of the parameter $K_{3}$ in some intervals of
energy such that the LGI violation diminishes for these intervals. Indeed, as
was said, the violation of the LGI can be interpreted as appearing as
quantumness because quantum is the only theory that contradicts the principles
of the LGI. This quantumness is revealed if the states on which the
measurements are carried out, are a coherent superposition of Hamiltonian
eigenstates. When we have treated the LGI by a quantum state which does not
give violation, it means that this quantum state does not possess enough
coherence. Therefore, the LGI investigation shows the decrease in coherence
due to the gravitational effects.

  * •

As another interesting result, we see that the gravitational effects lead to
the occurance of a phase shift in the value of the parameter $K_{3}$ in
comparison to the corresponding one in the flat space-time. This phase shift
depends on both energy and the relative proper distances. So, there exists
some range of the parameter by which the value of $K_{3}$ in curved space-time
is larger than the one in the flat space-time. Of course, we know that even in
presence of the quantum coherence, it is not the case that we will be able to
see the LGI violations for any choice of parametric space. Rather, if there is
enough coherence, one can find a region of parametric space in which this
inequality violates. Therefore, according to the results obtained from this
study, the gravitational effects cause changes in the region of the parametric
space suitable for violating LGI. The physical justification is as follows: we
describe neutrinos by plane waves in this study and the corresponding phases
are modified by the gravitational effects. Hence, when we encounter the LGI
violation, this means that the phase of the distinguished terms constructing
$K_{3}$ is such that they are summed constructively and otherwise
destructively.

##  4 Calculation of $l_{1}$-norm for Neutrino Oscillation in Schwarzchild
Metric

After investigating the gravitational effects on the quantumness of neutrino
oscillation via calculating the parameter $K_{3}$ for the two different cases
described in the last section, we wish to study quantitatively the quantum
coherence in neutrino propagation in presence of the gravitational effects. As
was said in the introduction, the amount of quantum coherence can be obtained
for the neutrino oscillation by calculating the $l_{1}$-norm [9, 12]. In
general, this quantity is defined as [35] :

${\cal C}(\rho)=\sum_{i\neq j}|\rho_{ij}|\geq 0$ (36)

where the summation is over the absolute values of all the off-diagonal
elements $\rho_{ij}$ of a corresponding density matrix $\rho$. The maximum
possible value for ${\cal C}(\rho)$ is bounded by ${\cal C}_{max}=d-1$, where
d is the dimension of the density matrix $\rho$. Here we consider the two
flavor neutrino, thus the maximum value of $l_{1}$-norm will be 1. The
$l_{1}$-norm can be expressed in terms of the transition and survival
probabilities between different flavor modes written in terms of the proper
distance $L_{p}$. Therefore, if the initial flavor state is
$|\nu_{\mu}\rangle$ we have

${\cal C}_{\mu}=2(\sqrt{P_{\mu e}(L_{p})P_{\mu\mu}(L_{p})}).$ (37)

We consider the gravitational effects in the weak field approximation for the
Schwarzschild metric. The calculations are carried out for the two cases of
the neutrinos propagating radially outwards and inwards. Using the plane wave
approach, one can obtain straightforwardly the transition probabilities as

$\displaystyle
P_{\nu_{\mu}\rightarrow\nu_{e}}(L_{p})=\sin^{2}(2\theta)\sin^{2}\Big{[}$
$\displaystyle\frac{\Delta m^{2}_{12}L_{p}}{4E_{0}^{\text{loc}}}$ (38)
$\displaystyle\times\Big{\\{}1-GM\left(\dfrac{1}{L_{p}}\ln(\dfrac{L_{p}}{r_{A}}+1)-\dfrac{1}{L_{p}+r_{A}}\right)\Big{\\}}\Big{]},$

and

$\displaystyle\acute{P}_{\nu_{\mu}\rightarrow\nu_{e}}(\acute{L}_{p})=\sin^{2}(2\theta)\sin^{2}\Big{[}$
$\displaystyle\dfrac{\Delta m^{2}_{12}\acute{L}_{p}}{4E_{0}^{\text{loc}}}$
(39)
$\displaystyle\times\Big{\\{}1+GM\left(\dfrac{1}{\acute{L}_{p}}\ln(1-\dfrac{\acute{L}_{p}}{\acute{r}_{A}})+\dfrac{1}{\acute{L}_{p}+\acute{r}_{A}}\right)\Big{\\}}\Big{]}.$

The survival probabilities are also

$P_{\nu_{\mu}\rightarrow\nu_{\mu}}(L_{p})=1-P_{\nu_{\mu}\rightarrow\nu_{e}}(L_{p}),$
(40)

and

$\acute{P}_{\nu_{\mu}\rightarrow\nu_{\mu}}(\acute{L}_{p})=1-\acute{P}_{\nu_{\mu}\rightarrow\nu_{e}}(\acute{L}_{p}).$
(41)

Here, in our calculations, $E^{loc}_{0}(r_{B})$ i.e., the energy of the
massless neutrino measured by the observer at the detector situated at the
radial distance $r_{B}$ from the center of the gravitational source is used.
In order to plot the variations of ${\cal C}_{\mu}$ in Eq. (37) in terms of
the proper distance $L_{p}$ we take $E^{loc}_{0}(r_{B})=3\times 10^{2}$ TeV.
For neutrinos propagating outwards, we suppose $r_{A}=10^{8}$ Km and $2\times
10^{8}$ Km $\leq L_{p}\leq 4\times 10^{8}$Km and for neutrinos propagating
inwards, $\acute{r}_{A}=4\times 10^{8}$ Km and $1.5\times 10^{8}$ Km
$\leq\acute{L}_{p}\leq 3\times 10^{8}$Km. The corresponding diagrams are found
in Fig. 2(a) and Fig. 2(b).

Figure 2: Variations of ${\cal C}_{\mu}$ as a function of $L_{p}$, for
neutrinos radially propagating (a) outwards and (b) inwards the gravitational
source. Blue (solid) and orange (dashed) curves represent the ${\cal C}_{\mu}$
in the flat and curved space time, respectively.

It is notable from the two diagrams in Fig. 2 that although as we expect, the
wave length shows local dependence in both outwards and inwards propagation,
we have no decrease in the maximum value of the quantum coherence.

##  5 Conclusion

One of the important questions in physics is about the quantum effects of the
gravitation, either we have a quantum gravity or the gravitation can play any
role at the quantum scales. The most important quantum aspect that plays role
in neutrino oscillation is the quantum coherence. In this paper, we have
studied the gravitational effects on quantum coherence in neutrino oscillation
via both qualitative (through investigating the violation of LGI) and
quantitative (by calculating $l_{1}$-norm) manners.

In case of the former, we have rewritten the parameter $K_{3}$ that is known
as the simplest form of the LGI, in terms of the proper distance $L_{p}$. Next
we constructed the parameter $K_{3}$ for two flavor neutrinos propagating
radially outwards and inwards between a neutrino source and the two detectors
placed in a Schwarzschild metric. Then we have given a numerical example for
the variations of the parameter $K_{3}$ with respect to the energy $E_{0}$ in
the weak field approximation. The results are demonstrated by Fig. 1. We
concluded that the gravitational effects decrease the maximum value of the
parameter $K_{3}$ for some energy intervals. More explicitly, there exists
some energy intervals for which we have no LGI violation for neutrinos
propagating in curved space-time but we do have the LGI violation in flat
space-time. In case of the latter, we have calculated the $l_{1}$-norm, ${\cal
C}(\rho)$, for the above physical situation. We reckon that although a local
dependence in wave lengths of the oscillation is observable in ${\cal
C}(\rho)$ when the neutrinos propagate in curved space-time, the maximum
amount of the ${\cal C}(\rho)$ remains unchanged (please see Fig. 2.).

In this study, the considered setup seems to be a gedanken experiment and the
distances chosen for the LGI experiment are somewhat exaggerated. However, the
issues presented in this study may be considerable for future experiments
which may be designed to examine the decoherence effects for long baseline
experiments by the messaging particles other than neutrinos between satellites
at the vicinity of the Earth. Recently, some efforts have been made to create
a quantum network in space [36, 37, 38]. Therefore, it can be helpful to study
the various quantum resources such as quantum coherence in the curved space-
time.

Furthermore, we have treated neutrinos by plane wave and the decoherence
effects due to the separation of the mass eigenstate wave packets (for
instance see Ref. [39]) have been ignored. The wave packet decoherence effects
in the presence of the strong gravitational fields have been studied in Ref.
[40]. However, the plane wave assumption does not make our discussion
inaccurate because modifications due to neutrino localization (wave packet
approach) only play an important role when the propagation length is of the
order of the coherence one. Therefore, we do not see any damping in the
maximum value of $l_{1}$-norm. But under the same condition, we have a damping
in the LGI which is due to the constructive and destructive effects in some
terms constructing $K_{3}$. Thus, this statement shows that the LGI violation
and $l_{1}$-norm are independent measure criteria of coherence. This reminds
us of some mixed states such as the Werner state [41], in which there are
regions where we have no CHSH violation and, therefore, no nonlocality, but
still get a nonzero value for the measure quantities of the entanglement.

In general, there exists another scenario for decoherence of neutrino states
in vacuum; the effects caused by the real and microscopic virtual black holes,
speculated from the quantum gravity considerations, can lead to the
decoherence of neutrino states (for instance see [21]). These theories could
have an impressive role on the neutrino oscillation but there is no empirical
indication for them and they are bounded via various studies performed in this
field [42, 43, 44, 45]. Meanwhile, considering such a hypothesis is a new
issue that goes beyond the scope of the current article.

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