# Machine Learning Method for Functional Assessment of Retinal Models

Nikolas Papadopoulos∗, Nikos Melanitis†, Antonio Lozano‡, Cristina Soto-
Sanchez‡, Eduardo Fernandez‡
and Konstantina S. Nikita† $*$ Nikolas Papadopoulos is with the School of
Electrical and Computer Engineering, National Technical University of Athens,
Greece<EMAIL_ADDRESS>E.Fernandez, C. Soto-Sanchez and A.
Lozano are with Instituto de Bioingenieria, Universidad Miguel Hernandez,
Alicante, Spain${\dagger}$ K.S. Nikita and N. Melanitis are with the School of
Electrical and Computer Engineering, National Technical University of Athens,
Greece

###### Abstract

Challenges in the field of retinal prostheses motivate the development of
retinal models to accurately simulate Retinal Ganglion Cells (RGCs) responses.
The goal of retinal prostheses is to enable blind individuals to solve
complex, real-life visual tasks. In this paper, we introduce the functional
assessment (FA) of retinal models, which describes the concept of evaluating
the performance of retinal models on visual understanding tasks. We present a
machine learning method for FA: we feed traditional machine learning
classifiers with RGC responses generated by retinal models, to solve object
and digit recognition tasks (CIFAR-10, MNIST, Fashion MNIST, Imagenette). We
examined critical FA aspects, including how the performance of FA depends on
the task, how to optimally feed RGC responses to the classifiers and how the
number of output neurons correlates with the model’s accuracy. To increase the
number of output neurons, we manipulated input images - by splitting and then
feeding them to the retinal model - and we found that image splitting does not
significantly improve the model’s accuracy. We also show that differences in
the structure of datasets result in largely divergent performance of the
retinal model (MNIST and Fashion MNIST exceeded 80% accuracy, while CIFAR-10
and Imagenette achieved $\sim$40%). Furthermore, retinal models which perform
better in standard evaluation, i.e. more accurately predict RGC response,
perform better in FA as well. However, unlike standard evaluation, FA results
can be straightforwardly interpreted in the context of comparing the quality
of visual perception.

###### Index Terms:

retinal models, functional assessment, machine learning, retinal prosthesis,
visual recognition

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##  I INTRODUCTION

The increasing knowledge of visual systems along with technological advances
give novel results in prevention, limitation or even treatment of visual
impairment [1], [2]. However, retinal degenerative diseases, such as age-
related macular degeneration (AMD) and retinitis pigmentosa cannot be
effectively treated with surgery or medication [3]. Retinal prosthesis devices
aim to restore vision in such patients, by translating visual stimuli to
electrical stimulations that activate the retina. Then, the retina transmits
neural signals to the visual centers of the brain, which are responsible for
visual perception [4].

Although current retinal implants have managed to restore certain visual
functionalities, there are several technological and biological challenges to
be overcome. In particular, implants need to simulate natural retinal
processing, by incorporating models that faithfully predict the physiological
firing of retina cells to visual stimuli [3], [4]. Towards this direction, the
development of retinal models aims to simulate the biological neural
processing in the retina, by interpreting images to retinal responses.
Nowadays, advanced retinal models incorporate computer vision techniques [5],
[6] and, more recently, state-of-the-art models use Convolutional Neural
Networks (CNNs) [1], [2].

This paper proposes the functional assessment (FA) of retinal models, which
describes the concept of evaluating the performance of retinal models on
visual understanding tasks. This constitutes a divergence from the currently
common practice of evaluating a retinal model comparing the similarity of
model-generated and ground-truth RGC responses. Motivation for FA stems from
the observation that visual prostheses aim to restore the capacity of
individuals to comprehend their visual environment, thus we should directly
evaluate our models on such tasks. The need for FA of prosthetic (i.e.,
acquired through prostheses) vision has been raised in the literature [7]; FA
has been applied to evaluate vision in implantees [8] and in augmented-reality
interventions in people with severe vision impairment [9]. FA should be
focused on the visual functions that seem most important to the blind:
mobility, face recognition and reading [3]. In this context, we develop a
machine learning method for FA. Using images from well-established computer
vision datasets (Table I), we feed traditional classifiers with RGC responses
produced by the retinal model, in order to solve object and digit recognition
tasks. Our goal is to explore whether retinal models that faithfully reproduce
retina output show improved performance in visual understanding tasks and
also, draw conclusions on the quality of prosthetic vision that is attainable
by assessing the performance of retinal models directly on such tasks.

##  II METHODS

Figure 1: Functional assessment pipeline.

###  II-A Functional assessment pipeline

Functional assessment pipeline (Fig. 1) includes image preprocessing (Steps
II-A1, II-A2, II-A3, II-A4), feeding the retinal model with images (Step
II-A5), extracting and processing RGC responses (Steps II-A6, II-A7, II-A8)
and finally, feeding them to the classifiers. All steps, together with the
corresponding design decisions, are analyzed below.

####  II-A1 Input

Transform each image from rgb to grayscale, only for CIFAR-10 and Imagenette.
Then, normalize images in range (0, 1) and finally, reshape each image to (50,
50, 1) to match the input size of the retinal model.

####  II-A2 Data augmentation

Implement data augmentation as a standard process to improve model training.
Two of the following transformations are applied to each image: rotate
$45^{o}$ or $-45^{o}$, use gaussian noise of scale = $0.1*255$ or scale =
$0.2*255$, crop images by 5 or 7 pixels from each side, translate images over
$x$ or $y$ axis by a percentage ranging from $-10\%$ to $+10\%$ of the image
size. Parameter values are selected, so as to produce different perspectives
of the same image and, at the same time, preserve the object for detection
inside the frame of the image. Data augmentation is implemented using imaug
library111https://github.com/aleju/imgaug.

####  II-A3 Split

Split image in $p^{2}$ parts, where $p=1,2,3,4$ and reshape the size of each
part to (50, 50). For $p=1$, we assume that the image is not split at all.
Motivation behind image splitting is to artificially increase the limited
number of output neurons ($60$ in our case) provided by the retinal model, so
as to increase the performance of the classifiers.

####  II-A4 Adjust

The goal of this step is to create a temporal dataset, as described in Section
II-C. It should be decided whether to repeat each image five times and get the
full response trend for t = 0 … 50 ms (Adjust=yes) or have one row per image
with ten image repetitions (see red frames in Fig. 2) and get a snapshot of
the response at t = 100 ms (Adjust=no)222We found that the retina response at
t=100 ms has high variance and thus, it could better distinguish different
objects (data not shown)..

####  II-A5 Feed

The retinal model, described in Section II-B, is fed with images and predicts
retinal responses for 60 neurons.

####  II-A6 Valid

In this step, the receptive fields of the neurons are identified using STA
Analysis [10]. If STA manages to compute the center of the receptive field for
a neuron, then we consider this neuron as valid. For the retinal model used,
only 12 valid neurons are found. Therefore, it should be decided whether to
keep all 60 neurons (Valid=no) or only the 12 valid ones (Valid=yes).

####  II-A7 Combine

If adjustment was previously implemented, there are five arrays of RGC
responses for every image, which are combined in one array, by applying
elementwise min or max transformation.

####  II-A8 Concatenate

If image splitting is implemented in Step II-A3, there is one array of 60 RGC
responses for every part of the split image. These arrays are then
concatenated to one larger array of size $a*(p^{2})$, where $a=60$ (or $a=12$
if valid neurons are selected) and $p=2,3,4$. The set of final arrays is the
training dataset for the classifiers and thus, the size of final arrays
represents the number of features for the classifiers.

###  II-B Retinal Model

We use a 3-layer CNN retinal model [2] that was trained to predict response
rates for sixty simultaneously recorded RGCs. This model was chosen, as it can
effectively predict retinal responses to natural images and, being trained
with natural images, it can model a wide range of retina’s biological
properties. To train the retinal model, we used an image dataset consisting of
$4890$ grayscale natural images of size $50$x$50$ pixels and the recorded
retinal responses (retinal responses were recorded at Prof. E. Fernandez lab)
[1]. Each frame, corresponding to $10$ ms of visual stimulus, was projected
onto the retina of a mouse for a total of $50$ ms. Thus, each frame was
repeated five times and the whole dataset consists of $24450$ natural images
of total duration $244.5$ s.

###  II-C Temporal dataset

The response of RGCs depends not only on the current stimulus, but also on
preceding stimulations. To model this temporal dependency, a temporal dataset
is created. For each image being projected onto the retina, we keep track of
the history of images projected before. The total number of frames used -the
actual image projected onto the retina (at $t=t_{n}$) plus the additional
image frames accounting for the stimulus history- is called temporal_interval.
Every row in the temporal dataset represents an input sample to the retinal
model. The number of image repetitions ($n$) represents the duration ($n*10$
ms, given that each frame corresponds to 10 ms of visual stimulus) that the
retina is exposed to a specific image (Fig. 2).

Figure 2: A clipping of the temporal dataset. Every row in the temporal
dataset represents an input sample to the retinal model and it has
temporal_interval frames (here, temporal_interval $=30$). In each row, the
number of image repetitions ($n=1,2,3,4,5$) represents the time that the image
has been projected onto the retina ($\Delta t=10,20,30,40,50$ ms
respectively). Black frames represent the intermediate stage between the
alternation of images, where no light is projected onto the retina.

###  II-D Design of simulations

We conducted simulations for all different combinations of design decisions
during data preprocessing. In particular, we trained four different
classifiers: MLP_500_100 (size of input layer = 500, size of hidden layer =
100), MLP_n_n/2 (size of input layer = n, size of hidden layer = n/2, where n
= number of features), SVM (kernel=‘rbf’) and Random Forest (max_features=12).
Hyperparameters were chosen based on the performance of classifiers in initial
simulations. We also chose four datasets (Table I): CIFAR-10, MNIST, Fashion
MNIST and Imagenette with ten different classes (ten object categories) each,
so as to compare classification tasks with equal number of classes. Then,
10000 samples from each dataset were augmented, creating in this way 30000
samples, which were randomly divided according to 70%/30% train/test split.
For each simulation, we trained each classifier ten times and we calculated
the mean and standard deviation of accuracy. We repeated all simulations for
two different parametrizations of the CNN retinal model (Section II-B):
RetModel1 with temporal_interval=30 and RetModel2 with temporal_interval=40.

TABLE I: Description of the datasets used in simulations. Dataset | Image | Type of | # Classes
---|---|---|---
| size | classification |
CIFAR-10 [11] | 32x32 | objects | 10
MNIST [12] | 28x28 | digits | 10
Fashion MNIST [13] | 28x28 | clothing items | 10
Imagenette 333Available at https://github.com/fastai/imagenette/ | variable | objects | 10

##  III Results and Discussion

###  III-A Preprocessing decisions for functional assessment

Initially, we performed simulations with CIFAR-10 and MNIST, splitting images
in None, $4,9$ parts ($p=1,2,3$ respectively) for both Valid=yes/no (Step
II-A6). Then, we split images of the two more complex datasets, Fashion MNIST
and Imagenette, in None, $4,9,16$ parts for only Valid=no. In Fig. 3, we see
the percentage differences between 60 neurons (Split=None) and max splitting
(Split=9 or Split=16). We observe that image splitting (i.e. using more than
60 neurons) increases the performance mostly in MNIST (>10%) and less in
Fashion MNIST (<10%), while it does not further improve performance in
CIFAR-10 and Imagenette. This can be explained by structural differences
mentioned in Section III-B. We further assume that, by splitting images in too
many parts, the objects are difficult to be recognized due to oversegmentation
and so, performance does not significantly improve with splitting. In
addition, if we compare plots with Adjust=yes and those with Adjust=no (Fig.
3), we see that, keeping a snapshot of the retina response at a critical $t$
(at which retina response has high variance), produces similar results as
keeping the full trend of the retina response over time. By keeping only the
critical responses, we can also save significant computational resources and
time. Finally, we tested a set of different classifiers and we found that the
type of classifier is not an important factor -our conclusions remain
unchanged, irrespective of the classifier we use- even if the Random Forest
had the most efficient and consistent performance across the simulations (Fig.
3).

###  III-B Functional assessment performance on different datasets

Fig. 4 compares the maximum performance of classifiers between different
visual understanding tasks, i.e. different datasets (Table I). CIFAR-10 and
Imagenette achieve $\sim$40% accuracy, while MNIST and Fashion MNIST exceed
80%. We see that image resolution is not a crucial factor in the model’s
performance. Although Imagenette was compared to CIFAR-10 to test images with
higher resolution, it performs only slightly better -and in some cases- worse
than CIFAR-10. Furthermore, MNIST and Fashion MNIST have twice the accuracy of
CIFAR-10 and Imagenette, even if they consist of images with lower resolution.
Significant differences in performance between MNIST/Fashion MNIST and
CIFAR-10/Imagenette can be explained, if we take a closer look at the
structural differences between these datasets. Images in both MNIST and
Fashion MNIST have a dark background with the object for classification
situated in the middle. On the other hand, CIFAR-10 and Imagenette consist of
real-life images, with unclear background-foreground segregation and more
complex structures. This plays a key role, if we additionally consider the
retina only reacts in spatiotemporal changes among different image frames
[14]. This retina’s property favors both MNIST and Fashion MNIST, where only
the object under classification changes over different dataset images.

Figure 3: Sensitivity plots correlating the accuracy of classifiers (CIFAR-10,
MNIST, Fashion MNIST, Imagenette) with the number of neurons used as input
(n_features), for both Adjust=yes/no (Step II-A4).

Figure 4: Comparison of the maximum performance of classifiers between
datasets.

###  III-C Functional assessment of different retinal models

Finally, functional assessment is applied in order to compare the performance
of two different retinal models, RetModel1 and RetModel2 (Section II-D).
Results show that the retinal model with the lowest mean squared error (MSE)
(i.e. the one being closer to the biological retina) performs better across
the whole range of functional simulations. Therefore, it seems that functional
assessment is in accordance with standard evaluation techniques. Furthermore,
by functionally assessing a model, we get a direct and more easily
interpretable estimate of how well an implantee may perform in a visual task
of interest.

###  III-D Limitations and future directions of functional assessment

During the development of this work, several limitations were encountered. We
used a retinal model that has been trained to predict only a limited number of
RGC responses (Section II-B). However, a higher number of RGC responses could
better represent complex images and provide classifiers with a richer set of
features to solve more efficiently real-life visual tasks. Therefore, there is
a need to increase the number of RGC responses given by retinal models, either
by using improved experimental methods to collect the data or using a
different architecture for the retinal model.

Another limitation arises from choosing to process retinal responses, which
are one-dimensional firing-rate arrays, with traditional machine learning
classifiers. From a biological perspective, the visual pathway includes neural
processing both in the retina and mainly, in the brain’s visual centers, which
are responsible for higher visual functionalities, like object recognition.
Literature indicates that the brain’s visual centers can be effectively
modeled by deep neural architectures [15]. Deep learning networks have been
also used, in high-impact research on biological vision, to model the ventral
visual stream in order to elucidate retinal mechanisms [16]. Taking those
insights into consideration, we suggest that future efforts for FA should be
focused on combining end-to-end deep learning architectures to model both the
retina and the rest of the visual pathway. Moreover, we may train on tasks
with unsupervised methods, which produce biologically plausible ventral visual
system models and follow bio-plausible sensory learning procedures [17]. Given
also the interpretable nature of FA, future development involves explainable
models, providing insights into the relationship between image properties and
retina output [18].

##  IV Conclusion

In this work, we introduced the concept of functional assessment and designed
a machine learning framework for it. We investigated how retinal models,
trained to faithfully reproduce retina output, perform in visual understanding
tasks. We show that FA is comparable with the established evaluation method;
yet, FA provides a direct and easily interpretable way of assessment, based on
the performance on visual tasks. We also found that performance in FA is
closely dependent on the given task. Finally, image splitting, as a way to
increase the number of output neurons, does not significantly improve
accuracy; still, restoring functional vision requires that retinal models
interpret images to a larger number of RGC responses.

## References

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# Quantum Correlations and Coherence in a Moving Unruh-deWitt Detector

S. Bhuvaneswari Centre for Nonlinear Science (CeNSc), PG & Research Department
of Physics, Government College for Women (Autonomous), Kumbakonam, Tamil Nadu,
India. R. Muthuganesan Department of Physics, Faculty of Nuclear Sciences and
Physical Engineering, Czech Technical University in Prague, Br̆ehová 7, 115 19
Praha 1-Staré Mĕsto, Czech Republic, Email<EMAIL_ADDRESS>R. Radha
111Corresponding author

###### Abstract

In this paper, we investigate the quantum correlations and coherence of two
accelerating Unruh-deWitt detectors coupled to a scalar field in $3+1$
Minkowski space-time. We show that the entanglement is completely destroyed in
the limit of infinite acceleration while the local quantum uncertainty and
$l_{1}$-norm of coherence remain nonzero. In addition, we also highlight the
role of Unruh temperature and energy spacing of detectors on quantum
correlations for different choices of initial states.

Keywords: Quantum Correlation, Coherence, Unruh-deWitt detector, Uncertainty.

##  1 Introduction

Quantum resource theory is an essential tool to understand the nature of
physical systems from the perspective of nonlocality and provides an
operational interpretation of various quantum effects [1]. Quantum coherence,
superposition, entanglement and quantum correlations beyond entanglement are
few notable resources which can prove to be advantageous over the classical
algorithms. Understanding these resources in physical systems continue to be a
formidable and interesting task even today. The quantum information processing
is usually studied within the framework of entanglement-versus-separability .
The entanglement is considered to be the only early version of nonlocal
aspects of quantum systems [2, 3, 4] and is demonstrated through the violation
of Bell inequality [5]. The seminal work of Werner and Almedia et al. reveals
that the entanglement is an incomplete manifestation of nonlocality implying
that quantum correlations cannot only be limited to entanglement [6, 7] and
the separable states can also come in handy in the implementation of some
specific quantum tasks. To study this quantum correlation beyond entanglement,
Ollivier and W. H. Zurek introduced a measure called quantum discord which
captures the nonlocality of separable (unentangled) states[8].

Other than entanglement, many other types of quantum correlations have been
discovered in recent years [8, 9, 10]. As a result, a number of quantum
correlation concepts have emerged with each being motivated by specific
applications in quantum information science competing for recognition as the
most accurate quantumness metric. For example, the skew-information and
quantum Fisher information play an important role in parameter estimation and
bring out the limitations on the variance of the observable. As stated, each
quantum correlation captures different quantumness in quantum systems due to
their distinct type of measurement techniques. Among them, skew information
based quantum correlation measures such as local quantum uncertainty (LQU) and
uncertainty-induced nonlocality (UIN) are widely used as a quantumness measure
to characterize quantum/physical systems. Girolami et al. [11] have introduced
a realizable discord-like nonclassical correlation measure for bipartite
systems known as local quantum uncertainty. Further, this measure quantifies
the uncertainty in a quantum state arising due to its noncommutativity with
the measured local observables. The LQU is defined in terms of the skew
information. The UIN provides an alternative to capture the nonlocal effects
via local measurements. Both LQU and UIN possess the closed formula for
qubit–qudit ($2\otimes n$ dimensional) systems. Apart from the correlation
quantifiers, the quantum coherence is a faithful nonclassicality indicator
even for a single qubit system and is a consequence of superposition of
quantum states. Like quantum correlations, coherence is also considered to be
critical in the development of quantum technology [12, 13, 15, 16, 17, 18, 19,
20]. The quantification of quantum coherence has been recently developed by
Baumgratz et al [21]. Recently, the coherence measure based on the
$l_{1}$-norm as a valid measure has been introduced and characterized in
different contexts [23, 24, 25]. Further, the interplay between the quantum
coherence measure and correlations is also studied [27, 26].

In relativistic field theory, exploiting the formulation of the detector
models, two well-known effects such as Unruh and Hawking effects have been
studied. In this formulation, idealized particles are considered as detectors
similar to a two-level system following the classical worldline and whose
internal states are coupled to the field. In recent times, Unruh-deWitt (UdW)
detector model has been devised as a computational element. The intervention
of the environment on the system may cause decoherence and degrades the unique
features of quantum system. Assuming that the quantum fluctuations of the
background field plays the role of an environment, the Unruh temperature
$T_{U}=a/2\pi$($a$ is acceleration) plays the role of decoherence and destroys
the properties of coupled UdW detectors. Here, the pair of detectors is
considered as an open system and its dynamics is governed by the master
equation [28]. Recently, different approaches have been employed to study the
quantum effects like entanglement, quantum correlations and coherence [31, 29,
30, 32, 33, 34]. Moreover, the quantum correlation and entropic uncertainty
relation are also investigated in a multipartite scenario[35, 36].

In this article, we focus on the generation and retention of quantumness in
two-UdW detectors interacting with the scalar field. To quantify the
quantumness of the detectors, we use the entanglement, local quantum
uncertainty, uncertainty-induced nonlocality and $l_{1}$-norm of coherence. We
show that the entanglement decreases monotonically and vanishes at a finite
temperature. The coherence and LQU exhibit a remarkable difference namely,
revival of quantum correlation after vanishing at a finite time. Further, we
study the dynamics of a product and an incoherent state and the investigation
reveals that the dynamics generates the correlation and coherence from the
product and incoherent state respectively.

The paper is structured as follows. In Sec. 2, we review the quantifier of
quantum correlation and coherence measures . In Sec. 3, we introduce the
physical model under our consideration and thermal state of two-UdW detectors.
The investigations on quantum correlation and coherence are presented in Sec.
4. Finally, the conclusions are drawn in Sec. 5.

##  2 Quantum Correlation Measures

In this section, we review some of the popular quantum correlation measures to
be investigated in this article. For this purpose, we consider an arbitrary
bipartite state $\rho$ shared by the subsystems $a$ and $b$ in the separable
Hilbert space $\mathcal{H}^{a}\otimes\mathcal{H}^{b}$.

###  2.1 Entanglement

Entanglement is a nonnegative real function of a state $\rho$ which cannot
increase under the local operations and classical communication (LOCC) and is
zero for unentangled states. To quantify the amount of entanglement associated
with the two-qubit physical system $\rho$ under consideration, various
measures have been introduced. The concurrence is the most popular measure of
entanglement for mixed bipartite systems and is defined as [37]

$\displaystyle C(\rho)=\text{max}\\{0,\leavevmode\nobreak\
\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$ (1)

where $\lambda_{i}$ are eigenvalues of the matrix
$R=\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$
$(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4})$. Here,
$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$
is a spin flipped matrix with * denoting the complex conjugate in the
computational basis. The function $C(\rho)$ ranges from 0 to 1 and its minimal
and maximal values correspond to separable and entangled states respectively.

###  2.2 Local quantum uncertainty

The local quantum uncertainty (LQU) is a more reliable measure of the
quantumness of bipartite states which goes beyond entanglement. In recent
times, researchers have paid wide attention to this discord-like measure. This
is essentially due to its easy computation and the fact that it enjoys all
necessary properties of being a faithful measure of quantum correlation. It is
shown that LQU is non-zero for the separable state even in the absence of
entanglement. For a bipartite state $\rho$, the LQU is defined as the minimal
skew information attainable with a single local measurement. Mathematically,
it is defined as [11],

$\displaystyle\mathcal{Q}_{\mathcal{I}}(\rho)=\leavevmode\nobreak\
^{\text{min}}_{H^{a}}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{I}(\rho,H^{a}\otimes\mathds{1}^{b}),$ (2)

where $H^{a}$ is anylocal observable on the subsystem $a$, $\mathds{1}^{b}$ is
the $2\times 2$ identity operator acting on the system $b$ and

$\displaystyle\mathcal{I}(\rho)=-\frac{1}{2}\text{Tr}\left([\sqrt{\rho},H^{a}\otimes\mathds{1}^{b}]^{2}\right)$
(3)

is the skew information which provides an analytical tool to quantify the
information content in the state $\rho$ with respect to the observable
$H^{a}$, $[\cdot,\cdot]$ is the commutator operator. Here, the information
content of $\rho$ about $H^{a}$ is quantified by how much the measurement of
$H^{a}$ on the state is uncertain. The measurement outcome is certain if only
if the state is an eigenvector of $H^{a}$. On the other hand, if it is a
mixture of eigenvectors of $H^{a}$, the uncertainty is only due to the
imperfect knowledge of the state. For pure bipartite states, the local quantum
uncertainty reduces to the linear entropy of entanglement and vanishes for
classically correlated states. For any $2\times n$ dimensional bipartite
system, the closed formula of LQU is computed as

$\displaystyle\mathcal{U}(\rho)=\leavevmode\nobreak\
1-\text{max}\\{\omega_{1},\omega_{2},\omega_{3}\\}.$ (4)

Here, $\omega_{i}$ are the eigenvalues of matrix $W$ and the matrix elements
are defined as

$\displaystyle\omega_{ij}=\text{Tr}[\sqrt{\rho}(\sigma_{i}^{a}\otimes\mathds{1}^{b})\sqrt{\rho}(\sigma_{j}^{a}\otimes\mathds{1}^{b})],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ i,j=1,2,3,$ (5)

where $\sigma_{i}$ represents the Pauli spin matrices.

###  2.3 Uncertainty-induced nonlocality

Next, we employ another important skew information based measure ,namely
uncertainty-induced nonlocality (UIN) and it can be considered as an updated
version of measurement-induced nonlocality (MIN) [9]. It is defined as [22]

$\displaystyle\mathcal{Q}(\rho)=\leavevmode\nobreak\
^{\text{max}}_{H^{a}}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{I}(\rho,H^{a}\otimes\mathds{1}^{b}).$ (6)

This measure also satisfies all the necessary axioms of a valid measure of
bipartite quantum correlation and is reduced to entanglement monotone for
$2\times n$ dimensional pure states. It also possesses a closed formula [22].

###  2.4 $C_{l_{1}}$-norm coherence

Quantum coherence is an important resource for information processing and a
manifestation of the quantum superposition principle. Recently, its
quantification has been formulated and a set of conditions to be satisfied by
any proper measure of coherence has been identified [23, 24, 25]. The distance
based quantum coherence measure

$\displaystyle C(\rho)=\leavevmode\nobreak\
^{\text{min}}_{\delta\in\mathcal{I}}\leavevmode\nobreak\ d(\rho,\delta)$ (7)

is the minimal distance between $\rho$ and a set of incoherent quantum states
$\delta\in\mathcal{I}$. Recently, a faithful measure of coherence using
$l_{1}$-norm has been identified and is defined as the sum of absolute values
of all off-diagonal elements of $\rho$ [21]

$\displaystyle C_{{l_{1}}}(\rho)=\sum_{i\neq j}|\rho_{ij}|.$ (8)

The above definition is a basis dependent measure and is crucial in the
identification of phase transition in physical models [38, 39, 40].

##  3 Physical Model

To understand the behaviors of quantum correlations in two accelerating UdW
detectors in $3+1$-dimensional Minkowski space-time [31], we first introduce
the Hamiltonian of the physical system under consideration. Considering the
detector as a two-level atomic system and a valid qubit system, we consider
the two accelerating UdW detectors in $3+1$-dimensional Minkowski space time
as an open quantum system for our investigation. The total Hamiltonian of the
combined system becomes

$\displaystyle H=\frac{\omega}{2}\Sigma_{3}+H_{\Phi}+\mu H_{I}$ (9)

where $\omega$ is the energy level spacing of the atom and $\Sigma_{3}$ is one
of the symmetrized bipartite operators.
$\Sigma_{i}\equiv\sigma_{i}^{a}\otimes\mathbf{1}^{b}+\mathbf{1}^{a}\otimes\sigma_{i}^{b}$
is defined by Pauli matrices $\\{\sigma^{(\alpha)}_{i}|i=1,2,3\\}$ with
superscripts $\\{\alpha=a,b\\}$ labelling distinct atoms. $H_{\Phi}$ is the
Hamiltonian of free massless scalar fields $\Phi(t,\mathbf{x})$ satisfying
standard Klein-Gordon relativistic equation. The interaction Hamiltonian
between the atoms and the fluctuating field bath in a dipole form can be
written as
$H_{I}=(\sigma_{2}^{(a)}\otimes\mathbf{1}^{(b)})\Phi(t,\mathbf{x}_{1})+(\mathbf{1}^{(a)}\otimes\sigma_{2}^{(b)})\Phi(t,\mathbf{x}_{2})$
[41].

Here, we consider the coupling between the detectors and the environment
($\mu\leq 1$) and the initial state of the composite system is
$\rho_{tot}(0)=\rho_{ab}(0)\otimes|0\rangle\langle 0|$, with $\rho_{ab}(0)$
being the initial state of the detectors and $|0\rangle$ is the field vacuum
(environment). The time evolution of $\rho_{tot}(0)$ is unitary governed by
von Neumann equation $\dot{\rho}_{tot}(\tau)=-i[H,\rho_{tot}(\tau)]$, where
$\tau$ is the proper time of the atom. Due to the environment decoherence or
dissipation on the system $\rho_{ab}$, the density matrix is governed by a
Lindblad master equation and evolves non-unitarily in the following form [42,
43]

$\displaystyle\frac{\partial\rho_{ab}(t)}{\partial
t}=-i[H_{\tiny\mbox{eff}},\rho_{ab}(t)]+\mathcal{L}\left[\rho_{ab}(t)\right]$
(10)

where

$\displaystyle\mathcal{L}\left[\rho\right]=\sum_{\begin{subarray}{c}i,j=1,2,3\\\
\alpha,\;\beta=a,b\end{subarray}}\frac{C_{ij}}{2}\big{[}2\sigma_{j}^{(\beta)}\rho_{ab}\sigma_{i}^{(\alpha)}-\\{\sigma_{i}^{(\alpha)}\sigma_{j}^{(\beta)},\rho_{ab}\\}\big{]}$
(11)

describes the evolution due to the interaction between the detectors and
external field.

After introducing the Wightman function of scalar field
$G^{+}\left(x,x^{\prime}\right)=\left\langle
0\left|\Phi(x)\Phi\left({x}^{\prime}\right)\right|0\right\rangle$, its Fourier
transform

$\displaystyle\mathcal{G}(\lambda)=\int_{-\infty}^{\infty}d\tau\leavevmode\nobreak\
e^{i\lambda\tau}G^{+}(\tau)=\int_{-\infty}^{\infty}d\tau\leavevmode\nobreak\
e^{i\lambda\tau}\left\langle\Phi(\tau)\Phi(0)\right\rangle$ (12)

determines the coefficients $C_{ij}$ by a decomposition

$\displaystyle
C_{ij}=\frac{\gamma_{+}}{2}\delta_{ij}-i\frac{\gamma_{-}}{2}\epsilon_{ijk}\delta_{3,k}+\gamma_{0}\delta_{3,i}\delta_{3,j}$
(13)

where

$\displaystyle\gamma_{\pm}=\mathcal{G}(\omega)\pm\mathcal{G}(-\omega),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\gamma_{0}=\mathcal{G}(0)-\gamma_{+}/2.$ (14)

Moreover, the interaction with external scalar field would also induce a Lamb
shift contribution for the effective Hamiltonian of the detector
$H_{\mbox{\tiny eff}}=\frac{1}{2}\tilde{\omega}\sigma_{3}$ in terms of a
renormalized frequency
$\tilde{\omega}=\omega+i[\mathcal{K}(-\omega)-\mathcal{K}(\omega)]$ where
$\mathcal{K}(\lambda)=\frac{1}{i\pi}\mbox{P}\int_{-\infty}^{\infty}d\omega\frac{\mathcal{G}(\omega)}{\omega-\lambda}$
is a Hilbert transform of Wightman function.

Following a trajectory of the accelerating detectors, one can find that the
field Wightman function fulfills the Kubo-Martin-Schwinger (KMS) condition,
i.e., $G^{+}(\tau)=G^{+}(\tau+i\beta)$, where $\beta\equiv 1/T_{U}=2\pi/a$.
Translating it into frequency space, one finds that

$\displaystyle\mathcal{G}(\lambda)=e^{\beta\omega}\mathcal{G}(-\lambda).$ (15)

Using translation invariance $\langle
0|\Phi(x(0))\Phi(x(\tau))|0\rangle=\langle
0|\Phi(x(-\tau))\Phi(x(0))|0\rangle$ and after some algebraic manipulations,
we find that eq.(14) can be resolved as

$\displaystyle\gamma_{+}=$
$\displaystyle\int_{-\infty}^{\infty}d\tau\leavevmode\nobreak\
e^{i\lambda\tau}\langle
0|\left\\{\Phi(\tau),\Phi(0)\right\\}|0\rangle=\left(1+e^{-\beta\omega}\right)\mathcal{G}(\omega),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (16) $\displaystyle\gamma_{-}=$
$\displaystyle\int_{-\infty}^{\infty}d\tau\leavevmode\nobreak\
e^{i\lambda\tau}\langle
0|\left[\Phi(\tau),\Phi(0)\right]|0\rangle=\left(1-e^{-\beta\omega}\right)\mathcal{G}(\omega).$
(17)

It should be be noted that eq. (17) holds true for generic interacting fields.
Considering the two-atom state in Bloch representation, we compute t he final
equilibrium state of two-UdW detectors asymptotically as

$\displaystyle\rho_{ab}=\left(\begin{array}[]{cccc}\varrho_{11}&0&0&0\\\
0&\varrho_{22}&\varrho_{23}&0\\\ 0&\varrho_{23}&\varrho_{22}&0\\\
0&0&0&\varrho_{44}\end{array}\right)$ (22)

where the matrix elements are

$\displaystyle\displaystyle\varrho_{11}$ $\displaystyle=$
$\displaystyle\frac{(3+\Delta_{0})(\gamma-1)^{2}}{4\left(3+\gamma^{2}\right)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\displaystyle\varrho_{44}=\frac{(3+\Delta_{0})(\gamma+1)^{2}}{4\left(3+\gamma^{2}\right)},$
$\displaystyle\displaystyle\varrho_{22}$ $\displaystyle=$
$\displaystyle\frac{3-\Delta_{0}-(\Delta_{0}+1)\gamma^{2}}{4\left(3+\gamma^{2}\right)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\displaystyle\varrho_{23}=\frac{\Delta_{0}-\gamma^{2}}{2\left(3+\gamma^{2}\right)},$
(23)

with the parameter

$\displaystyle\gamma\equiv\gamma_{-}/\gamma_{+}=\frac{1-e^{-\beta\omega}}{1+e^{-\beta\omega}}=\tanh(\beta\omega/2).$
(24)

It should be mentioned that the parameter depends solely on the Unruh
temperature $T_{U}$ and characterizes the thermal nature of Unruh effect.
Further, the dimensionless parameter
$\Delta_{0}=\sum_{i}\text{Tr}[\rho_{ab}(0)\sigma^{a}_{i}\otimes\sigma^{b}_{i}]$
provides the choice of initial state and ranges as
$-3\leqslant\Delta_{0}\leqslant 1$.

Figure 1: Thermal quantum correlation quantified by (a) Entanglement, (b)
$l_{1}$ norm of coherence, (c) LQU and (d) UIN of UdW detector as a function
of Unruh temperature $T_{U}$ for different initial states for $\omega=1$.

##  4 Results and Discussions

Figure 2: Behaviors of quantum correlation measures (a) Entanglement (b)
$l_{1}$ norm of coherence, (c) LQU and (d) UIN of UdW detector as a function
of Unruh temperature $T_{U}$ for $\Delta_{0}=0$.

Figure 3: Behaviors of quantum correlation measures (a) Entanglement (b)
$l_{1}$ norm of coherence, (c) LQU and (d) UIN of UDW detector as a function
of Unruh temperature $T_{U}$ for different initial states for
$\Delta_{0}=0.5$.

We now study the dynamics of quantum correlations measures quantified in terms
of entanglement, $C_{l_{1}}$-norm of coherence , local quantum uncertainty and
uncertainity induced nonlocality.

In general, Unruh effect is recognized as an environment decoherence. Hence,
we study the quantum correlations as a function of Unruh temperature $T_{U}$.
In Fig. (1), we plot the quantum correlation measures as a function of Unruh
temperature for different choices of initial conditions $\Delta_{0}$ with a
fixed value of $\omega$. For a given initial state, the entanglement is
maximum at $T=0$. As temperature increases, the concurrence is a monotonically
decreasing function of Unruh temperature $T_{U}$ and abruptly vanishes . The
entanglement completely vanishes for $\Delta_{0}=1$. Further, one can observe
that the increase of $\Delta_{0}$ also reduces the quantum of entanglement
between the detectors. In order to compare the entanglement with other
correlation measures such as coherence, LQU and UIN, we plot these measures in
Fig (1b & c) as a function of Unruh temperature. As Unruh temperature $T_{U}$
increases, the quantum correlation measures decrease initially and reach a
dark point (zero). While increasing the temperature $T_{U}$ further, both LQU
and coherence start to grow from the dark point and then correlations increase
with the Unruh temperature $T_{U}$ unconventionally. It is quite obvious that
both the measures are non-monotonic functions of $T_{U}$. In general, if we
immerse any static two-qubit in a thermal bath, the quantumness of the system
degrades with the temperature of the thermal bath. But here, on the contrary,
we observe the revival of quantum correlations and coherence due to the
acceleration of the qubits. In addition, the dark point is shifted towards
higher temperatures with the increase of $\Delta_{0}$. Similar to
entanglement, UIN decreases from the maximum values with the increase of
temperature and saturates at a nonzero constant value. A similar observation
is noticed using trace distance correlation measure [32].

We observe the nonlocal features between the detectors even in the absence of
entanglement which are captured through the $C_{l_{1}}$\- norm of coherence
and skew information based measures. The entanglement fails to completely
quantify the nonlocality of the UdW detector for $\Delta_{0}=1$. On the other
hand, the correlation measures beyond entanglement are encapsulated in UdW
detector even in the absence of entanglement.

Next, we analyze the role of energy spacing $\omega$ on the quantum
correlation measures. For this purpose, we plot them as a function of Unruh
temperature $T_{U}$ with specific energy spacing of detectors $\omega=1,3,5$.
It is obvious that entanglement always degrades monotonously with increasing
Unruh temperature, i.e., no revival of entanglement can happen for any initial
state preparation. This clearly demonstrates the distinction between the
entanglement and coherence. In Fig. (1)a, we observe that the entanglement
vanishes at low temperatures $T_{U}=0.5$ for $\Delta_{0}=0$. For the same
initial state $(\Delta_{0}=0)$, comparing Fig. (1)a with Fig. (2)a, we observe
that the entanglement is induced in the higher temperature regime while
increasing the values of energy spacing of detectors from $\omega=1$ to
$\omega=3$. If we increase the value of $\omega$ further, one observes similar
monotonic decreasing behavior of the correlation measures while the
entanglement survives for sufficiently higher temperatures compared to lower
values of $\omega$. On the other hand, the other correlation measures also
increase with the increase of energy spacing. In Fig. (3), we have plotted the
correlation measures as a function of $T_{U}$ for another initial state such
as $\Delta_{0}=0.5$. Here again, the entanglement decreases monotonously while
one sees the revival of LQU and coherence measures.

Figure 4: The Density of quantum correlation measures (a) Coherence, (b) LQU,
(c) Entanglement and (d) UIN of UDW detector as a function of $T_{U}$ and
$\Delta_{0}$ for $\omega=3$

Now, we illustrate our results with an example. If we choose the initial state
of two-detector in a product form

$\displaystyle\rho_{\text{in}}=\rho_{a}(0)\otimes\rho_{b}(0),$ (25)

correlations of the initial state become zero. The state of each detector can
be written in Bloch form as

$\displaystyle\rho_{a}(0)=\frac{1}{2}\left(I+n\cdot\sigma\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\rho_{b}(0)=\frac{1}{2}\left(I+m\cdot\sigma\right)$ (26)

where $\mathbf{n}$ and $\mathbf{m}$ are two unit Bloch vectors. Without loss
of generality, taking $\mathbf{n}=(0,0,1)$ and
$\mathbf{m}=(0,\sin\theta,\cos\theta)$, we have
$\Delta_{0}=\mathbf{n}\cdot\mathbf{m}=\cos\theta$ where $\theta\in[0,\pi]$ is
an angle between two vectors giving $\Delta_{0}\in[-1,1]$.

In Fig. (4), we have plotted the density of quantum correlation measures as a
function of $T_{U}$ and $\Delta_{0}$ with $\omega=3$. Again, it is obvious
that the entanglement is generated during time evolution of the detectors for
$\Delta_{0}=(-1,0)$ and there is no entanglement induced in the range
$\Delta_{0}=(0,1)$. Here again, we notice that the entanglement vanishes at
low temperatures and is unable to capture the complete manifestation of
nonlocal attributes of UD detectors. On the other hand, the quantum
correlation measures quantify more nonlocal aspects of UD detectors compared
to entanglement. It can be noticed that the generation of entanglement and
quantum correlations in the initial product state arises due to the
acceleration of the system.

##  5 Conclusions

In this paper, we have studied the quantum coherence and correlations of two
accelerating Unruh- DdeWitt detectors coupled to a scalar background in 3 + 1
Minkowski spacetime. We have employed the entanglement, local quantum
uncertainty, uncertainty-induced nonlocality and $l_{1}$-norm of coherence as
the quantumness quantifiers. Results of the paper indicate the revival of LQU
and coherence while entanglement does not. We have also shown that the energy
spacing of detectors induces the nonlocality between the detectors. The
distinction between the static and accelerating qubit systems is observed in
terms of generation of quantumness in a product state and revival mechanism of
quantum correlations.

Further, the results of our manuscript can have wider ramifications in
understanding the relativistic quantum information processing from the
perspective of quantum correlation measures.

## Acknowledgment

Authors are indebted to the referees for their critical comments to improve
the contents of the manuscript. SB and RR wish to acknowledge the financial
support received from the Council of Scientific and Industrial Research
(CSIR), Government of India under Grant No. 03(1456)/19/EMR-II.

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