Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
00.0000/TQE.2024.DOI

This publication has emanated from research conducted with the financial
support of Science Foundation Ireland under Grant number 18/CRT/6222. For the
purpose of Open Access, the author has applied a CC BY public copyright
license to any Author Accepted Manuscript version arising from this
submission. This publication has emerged from research supported in part by a
grant from IEEE Antennas and Propagation Society Graduate Fellowship Program -
Quantum Technologies Initiative under Grant number ‘IEEE Funding –
1421.9040008’.

Corresponding author: Mouli Chakraborty (email: chakrabm@tcd.ie).

# A Machine Learning Approach for Optimizing Hybrid Quantum Noise Clusters for
Gaussian Quantum Channel Capacity

MOULI CHAKRABORTY1  ANSHU MUKHERJEE2  IOANNIS KRIKIDIS3  AVISHEK NAG4  and
SUBHASH CHANDRA 1  School of Natural Science, Trinity College Dublin, The
University of Dublin, College Green, Dublin 02, Ireland (e-mail:
<EMAIL_ADDRESS> School of Electrical and Electronic Engineering, University
College Dublin, Belfield, Dublin 04, Ireland (e-mail:
<EMAIL_ADDRESS> IRIDA Research Centre for Communication
Technologies, Department of Electrical and Computer Engineering, University of
Cyprus, 1678 Nicosia, Cyprus (e-mail: krikidis@ucy.ac.cy).  School of Computer
Science, University College Dublin, Belfield, Dublin 04, Ireland (e-mail:
<EMAIL_ADDRESS> School of Natural Science, Trinity College Dublin, The
University of Dublin, College Green, Dublin 02, Ireland (e-mail:
<EMAIL_ADDRESS>
###### Abstract

This research contributes to the advancement of quantum communication by
visualizing hybrid quantum noise in higher dimensions and optimizing the
capacity of the quantum channel by using machine learning (ML). Employing the
expectation maximization (EM) algorithm, the quantum channel parameters are
iteratively adjusted to estimate the channel capacity, facilitating the
categorization of quantum noise data in higher dimensions into a finite number
of clusters. In contrast to previous investigations that represented the model
in lower dimensions, our work describes the quantum noise as a Gaussian
Mixture Model (GMM) with mixing weights derived from a Poisson distribution.
The objective was to model the quantum noise using a finite mixture of
Gaussian components while preserving the mixing coefficients from the Poisson
distribution. Approximating the infinite Gaussian mixture with a finite number
of components makes it feasible to visualize clusters of quantum noise data
without modifying the original probability density function. By implementing
the EM algorithm, the research fine-tuned the channel parameters, identified
optimal clusters, improved channel capacity estimation, and offered insights
into the characteristics of quantum noise within an ML framework.

###### Index Terms:

Quantum Noise, Gaussian quantum channel, Gaussian Mixture Models, Expectation-
Maximization algorithm, Quantum channel capacity.

=-15pt

##  I Introduction

Quantum communication, an expanding area that merges quantum mechanics and
information theory, has the potential for unmatched security and effectiveness
in transmitting information. In addition, it can address issues related to
data and information security. Quantum communication entails transferring
quantum information through a quantum channel among various entities using
quantum-mechanical elements like photons. It facilitates functions such as
quantum cryptography [1], quantum teleportation [2], and quantum networks [3].
The inherent benefits of quantum communication are closely connected to
addressing issues related to quantum noise, which arises from the interaction
of quantum systems with their surroundings or during the measurement process
and can significantly hinder the accuracy and dependability of quantum
information transmission. Due to the high sensitivity of quantum systems to
external interferences, quantum noise can result in the degradation of
coherence and entanglement, which are vital aspects of quantum communication
[4]. Approaches such as quantum error correction and the utilization of
decoherence-free subspaces are aimed at maintaining the integrity of quantum
information in the presence of inevitable noise [5].

A thorough understanding of the noise that impacts these systems is essential
for accurately modeling and assessing their performance in quantum
technologies. Hybrid quantum-classical noise, which combines classical
Additive White Gaussian Noise (AWGN) and quantum Poisson noise, plays a
crucial role in this context. Quantum systems are inherently susceptible to
quantum noise, like Poisson noise, due to the discrete nature of light and
matter [6]. On the contrary, classical noise affects classical systems,
including AWGN, due to thermal fluctuations and electronic components. By
incorporating both forms of noise into a hybrid model, the complete range of
disturbances can be captured in systems that bridge the quantum and classical
domains [7]. Adopting a hybrid noise model enables a more accurate assessment
of system performance, covering error rates, information capacity, and
security analyses [8][9]. Particularly, it is critical for optimizing the
design and operation of quantum technology, significantly enhancing the
sensitivity and resolution of quantum sensing [10] and imaging. Consequently,
simulations that consider both quantum Poisson noise and classical AWGN
provide realistic testing environments essential for developing quantum error
correction techniques and creating robust quantum protocols resistant to noise
[11].

Quantum noise can disrupt quantum entanglement, leading to decoherence and
deterioration of entanglement, a process that poses significant challenges for
quantum communication and computing [12, 13]. Machine learning (ML) algorithms
can play a crucial role in the more efficient generation and detection of
entangled states, which is vital for various quantum communication protocols
[14], [15], thus improving the performance of quantum communication systems.
To illustrate how ML algorithms can enhance the generation of high-dimensional
entangled states with better fidelity and efficiency, a hybrid approach has
been suggested. This approach combines quantum control techniques with ML
algorithms to optimize the process of generating entanglement [16]. Generative
ML models can be utilized for quantum state tomography, including identifying
entangled states. This approach provides a more efficient and precise method
for characterizing quantum states, especially those related to entanglement
[17], [18], [19].

ML methods could have a crucial impact on improving the estimation and
optimization of quantum channel parameters, error correction, resource
allocation, and quantum information transmission. Progress has been notable in
quantum communication technology for free-space satellites [20]. However,
challenges related to quantum repeaters for long-distance quantum
communications still need to be addressed to establish a global quantum
network [21, 22]. Free-space quantum communication channels have encountered
difficulties such as loss in the quantum channel, security issues, and limited
data rates [22]. Conventional ML techniques offer a valuable approach to
characterizing the features of free-space quantum channels [20]. In a recent
study [23], supervised ML was employed to forecast the atmospheric strength of
a free-space quantum channel.

Estimating parameters in Gaussian quantum channels is a crucial task that
involves identifying and optimizing variables essential for transmitting
quantum states. A Gaussian Mixture Model (GMM) represents a probability
density function by a weighted sum of Gaussian component densities, providing
a flexible representation of non-Gaussian densities. With sufficient
components, GMMs can approximate any smooth function with arbitrary accuracy,
making them suitable for representing multivariate densities in practical
applications [24], [25]. The expectation maximization (EM) algorithm [26],
[27] being a robust iterative technique efficiently estimates the quantum
channel parameters maximizing the likelihood function. Maximum likelihood
significantly improves Gaussian quantum channels by reducing noise and errors
to enhance performance using updated parametric values. Focusing on Gaussian
states and operations, these channels can benefit from ML models trained to
forecast their evolution, thus refining quantum communication protocols [28].
In hybrid Quantum-Classical systems, ML aids in integrating Gaussian channels,
facilitating the efficient exchange of classical and quantum information. ML
techniques are essential for characterizing Gaussian quantum channels,
enabling the identification of crucial attributes such as noise properties and
entanglement effects, as well as detecting anomalies indicating external
interference or unexpected noise [29]. GMMs further elucidate the hybrid
quantum-classical noise model of these channels [7].

TABLE I:  Notation  $tr$ |  The trace of a matrix
---|---
$\mathbf{T(\cdot)}$ |  Trace-preserving map $\mathbf{T}$
$\mathbf{A}^{\dagger}$ |  The adjoint of matrix $\mathbf{A}$
${\mathbf{A}}^{T}$ |  The transpose of matrix $\mathbf{A}$
$\nu^{*}$ |  The complex conjugate of $\nu$
$\otimes$ |  The tensor product
$\mathcal{N}$ |  Gaussian density
$\ket{\cdot}$ |  The ket
$\bra{\cdot}$ |  The bra
$|\cdot|$ |  The norm

Figure 1: The flowchart of the research.

Furthermore, ML optimizes the preparation of the Gaussian state, enhancing
transmission fidelity through precise squeezing and displacement [10],[6]. As
parametric representations of the probability distribution of continuous
measurements, these models are adept at forming smooth approximations of
arbitrarily shaped densities, which is crucial for precise and reliable
estimation of channel characteristic. This enhances quantum state transmission
fidelity and performance. The iterative EM algorithm for calculating the
Maximum A Posteriori (MAP) [30], [31] estimation of GMM parameters from
training data is a widely accepted technique among researchers. This method
significantly aids in the estimation of parameters for Gaussian quantum
channels, optimizing the transmission of quantum states [32].

A mathematical model introduced in [7] tackles hybrid quantum-classical noise,
combining quantum Poisson noise with classical white Gaussian noise. Our study
advances quantum communication by integrating ML techniques to manage hybrid
quantum noise. Utilizing an iterative ML algorithm, we optimize quantum
channel parameters for precise capacity estimation and enhanced quantum noise
visualization in higher dimensions. Our method defines the quantum noise model
as a GMM, transforming it from an infinite to a finite mixture while
preserving the original probability density function. Through the EM
algorithm, we fine-tune channel parameters, yielding improved capacity
estimates. This innovative ML approach enhances our comprehension of quantum
noise’s impact on channel capacity, underscoring ML’s potential in quantum
communication advancement. In essence, the critical contributions of this
study can be summarized in three main aspects:

  * •

Our framework utilizes an iterative ML algorithm to dynamically adjust the
quantum channel parameters. This iterative process improves the accuracy of
estimating the quantum channel capacity. It facilitates the visualization of
the quantum noise model in higher dimensions. Initially, based on the hybrid
quantum noise model represented as a multivariate GMM that comprises an
infinite combination of multivariate Gaussian elements with mixing weights
obtained from a Poisson distribution, we have introduced a technique to
transform it into a finite mixture of multivariate Gaussians while retaining
the original Poisson mixing coefficients. This method aims to simplify the
complexity of the infinite mixture into a more manageable structure without
compromising the integrity of the probability density function (p.d.f.).

  * •

We employ the EM algorithm to enhance the quantum channel parameters and
detect the best quantum noise groups. This improves the parameters of the
quantum channel and pinpoints optimal data clusters, resulting in a more
accurate determination of the quantum channel capacities using signal-to-noise
ratios. Comparative evaluation demonstrates that our ML-boosted parameters
significantly surpass the original approximations, emphasizing our model’s
effectiveness.

  * •

Our results reveal a comparative assessment that indicates that the ML-
enhanced channel parameters provide better estimations of the quantum channel
capacity compared to the initial parameter assumptions. This comparison
highlights the efficiency of integrating ML in optimizing quantum
communication channels. Moreover, our progress in visualizing quantum noise
data clusters in elevated dimensions provides a deep understanding of quantum
noise properties, showcasing the strong potential of ML to enhance quantum
communication technologies.

Overall, our work demonstrates the powerful use of ML in improving quantum
communication technologies, providing new avenues to comprehend and address
the issues caused by quantum noise in quantum channels. Please note, a
comprehensive list of the notations used throughout this paper is provided in
Table LABEL:NoationTable.

##  II System Model

Figure 2: A noisy Quantum Channel: A schematic diagram showing a single
quantum link between transmitter and receiver.

###  II-A Gaussian Quantum Channel Model

A qubit is represented as a linear combination of the basis states
$\boldsymbol{\ket{0}}$ and $\boldsymbol{\ket{1}}$, which can be written as:
$\boldsymbol{\ket{\psi}}=\alpha\boldsymbol{\ket{0}}+\beta\boldsymbol{\ket{1}}$,
where $\alpha,\beta$ are complex numbers such that
$|\alpha|^{2}+|\beta|^{2}=1$. This normalization condition ensures that the
total probability is 1. The density matrix formalism offers a more general way
to describe quantum states, including pure and mixed states. A single-state
vector describes a pure state, whereas a mixed state represents a statistical
mixture of several state vectors. For a qubit in pure state
$\boldsymbol{\ket{\psi}}$, the density matrix $\boldsymbol{\rho}$ is given by

$\boldsymbol{\rho}=\ket{\boldsymbol{\psi}}\bra{\boldsymbol{\psi}}=\begin{bmatrix}|\alpha|^{2}&\alpha^{*}\beta\\\
\alpha\beta^{*}&|\beta|^{2}\end{bmatrix},$

where $\alpha^{*}$ and $\beta^{*}$ are the complex conjugates of $\alpha$ and
$\beta$, respectively. The density matrix for a mixed state, which is a
probabilistic mixture of pure states $\boldsymbol{\ket{\psi_{i}}}$ with
probabilities $p_{i}$, is given by
$\boldsymbol{\rho}=\sum_{\begin{subarray}{c}i\end{subarray}}p_{i}\boldsymbol{\ket{\psi_{i}}}\boldsymbol{\bra{\psi_{i}}}$,
with the condition that $\sum_{\begin{subarray}{c}i\end{subarray}}p_{i}=1$.

In mathematical terms, a quantum channel is defined as a completely positive
trace-preserving map
${\boldsymbol{\rho}}\mapsto{\mathbf{T}}({\boldsymbol{\rho}})$ that takes
states, that is, density operators ${\boldsymbol{\rho}}$ acting on some
Hilbert space ${\mathbf{\mathcal{H}}}$, into states. Our work assumed that the
input and output Hilbert spaces are identical for simplicity. Every channel
can be conceived as a reduction of unitary evolution in a larger quantum
system. So for any channel ${\mathbf{T}}$ there exists a state
${\boldsymbol{\rho}}_{E}$ on a Hilbert space ${\mathbf{\mathcal{H}}}_{E}$, and
a unitary ${\mathbf{U}}$ such that

${\mathbf{T}}({\boldsymbol{\rho}})={tr_{E}[\mathbf{U}(\boldsymbol{\rho}\otimes\boldsymbol{\rho}_{E})\mathbf{U}^{\dagger}]}$
(1)

The system labeled $E$ serves as an environment, embodying degrees of freedom,
including observation, inducing a decoherence process. The channel is then a
local manifestation of the unitary evolution of the joint system. A Gaussian
channel [33, 34, 35, 36, 37] is now a channel of the form as in Eq.(1), where
${\mathbf{U}}$ is a Gaussian unitary, determined by a quadratic Bosonic
Hamiltonian, and ${\boldsymbol{\rho}}_{E}$ is a Gaussian state [34].

The simplest Gaussian channel is a lossless unitary evolution governed by a
quadratic Bosonic Hamiltonian

$\boldsymbol{\rho}\mapsto\mathbf{U}\boldsymbol{\rho}\mathbf{U^{\dagger}},\mathbf{U}=e^{i/2}\sum_{\begin{subarray}{c}k,l\end{subarray}}\mathbf{H}_{kl}\mathcal{R}_{k}\mathcal{R}_{l},$

with $\mathbf{H}$ being a real and symmetric $2n$ × $2n$ matrix, $\mathcal{R}$
represents the canonical coordinates for the quantum system with modes $n$,
that is, $n$ canonical degrees of freedom [38, 8]. These unitaries correspond
to a representation of the real symplectic group $Sp(2n,\mathbb{R})$, formed
by the real matrices for which $\mathbf{S}\boldsymbol{\sigma}\mathbf{S}^{T}$=
$\boldsymbol{\sigma}$, where matrix $\boldsymbol{\sigma}$ defines the
symplectic scalar product defined by
$\boldsymbol{\sigma}=\oplus^{n}_{i=1}\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$
[38, 8]. Such linear transformations preserve the commutation relations; the
relation between such a canonical transformation in phase space and the
corresponding unitary transformation in Hilbert space is given by
$\boldsymbol{S}=e^{{\boldsymbol{H\sigma}}}$. Gaussian unitaries are
omnipresent in physics, particularly in optics, and this is the reason why
Gaussian channels play such an important role. In particular, the action of
ideal beam splitters, phase shifters, and squeezers corresponds to symplectic
transformations. Depending on the context, it is appropriate or transparent to
formulate a Gaussian channel in the Schrodinger picture
$\boldsymbol{\rho}\mapsto\mathbf{T}_{\mathbf{A,Z}}(\boldsymbol{\rho})$ or to
define it as a transformation of covariance matrices

$\boldsymbol{\gamma}\mapsto{\mathbf{A}^{T}}\boldsymbol{\gamma}\mathbf{A}+\mathbf{Z}.$
(2)

Figure 3: The categorization of hybrid Quantum Noise : (a) Original data with
specific cluster; (b) Unlabeled data for finding the original clusters; (c)
GMM used for soft-clustering; (d) Responsibility of a cluster.

It is important to note that (2) is the most general form of a Gaussian
channel, where $\mathbf{A}$ serves the purpose of amplification, attenuation,
and rotation in phase space. In contrast, the contribution $\mathbf{Z}$ is a
noise term that can consist of quantum (required to make the mapped physical)
and classical noise. Interestingly, ${\mathbf{A}}$ can be any real matrix.
Therefore, any map
$\boldsymbol{\gamma}\mapsto{\mathbf{A}^{T}}\boldsymbol{\gamma}\mathbf{A}$ can
be made approximately, as long as ‘sufficient noise’ is added [8],[38].
According to (2), a single quantum link can be expressed as
$\boldsymbol{\gamma}\mapsto\boldsymbol{\gamma}+\mathbf{Z}$, where $\mathbf{Z}$
indicates that the noise term consists of the quantum and the classical part.
In (2), consider $\mathbf{A=I_{2x1}}$ as $\mathbf{A}$ can be any real matrix.
The only condition for the map above to be a quantum channel is to ensure that
sufficient noise is added [8],[38]. Therefore, ${\mathbf{A}^{T}}\mathbf{\gamma
A}$ is a scalar, and the corresponding noise $Z$ added to the system can be
viewed as a scalar. In terms of a mathematical equation, this unity quantum
communication channel can be expressed as $Y=X+Z$, where $X$, $Y$, and $Z$
correspond to the random variables of the transmitted signal, the received
signal, and the noise affecting the channel resulting from different unwanted
sources, respectively.

For a realistic quantum communication network carrying classical information,
it is assumed that the communication links suffer from classical and quantum
noise components with the assumption that the hybrid quantum-classical noise
is additive in nature, such that $Z=Z_{1}+Z_{2}$, where $Z_{1}$ is the Poisson
distributed quantum shot noise and $Z_{2}$ is the Gaussian distributed white
classical noise [7]. In this relation, Fig. 2 explains the theory in the
schematic diagram.

###  II-B Hybrid Noise model

For quantum communication, it is essential to model the realistic quantum
noise of a quantum channel. Previous studies show that quantum noise is widely
represented as Poissonian noise [39, 40, 41]. However, a practical quantum
link also suffers from noise sources that contribute non-quantum noises, such
as classical noise sources [42, 43, 44, 45]. Classical noise is the integrated
part of the quantum channel and must be addressed to model the realistic
quantum channel. In the semi-classical theory of photodetection, the shot
noise level corresponding to Poissonian photoelectron statistics is the
fundamental detection limit. Hence, the shot noise power level is often called
the quantum limit or the standard quantum limit of detection. For a similar
reason, shot noise is often called quantum noise [46]. By considering the
qubit noise as Poissonian and the classical noise as additive white Gaussian,
one can expect a hybrid quantum noise represented by the following p.d.f. [7],

$f_{Z}(z)=\sum_{i=0}^{\infty}\frac{e^{-\lambda}\lambda^{i}}{i!}\frac{1}{\sigma_{Z_{2}}\sqrt{2\pi}}e^{-\frac{1}{2}\Big{(}\frac{z-i-\mu_{Z_{2}}}{\sigma_{Z_{2}}}\Big{)}^{2}}.\\\
$ (3)

###  II-C Quantum Transmitted Signal Model

For the transmitted signal model, let us consider a Gaussian distributed
signal for the continuous-variable (CV) system. Mathematically, a state is
Gaussian if its distribution function in phase or its density operator in the
Fock space is Gaussian. Examples of Gaussian functions are well-known p.d.f.
of Normal distribution, Wigner function, etc. Important quantum information
processing experiments are done with quantum light, described as Gaussian
distributed. In our model, the input quantum signal $X$ in terms of qubits is
in a Gaussian state, and it can be characterized by the scalar random variable
$X$ having a Gaussian distribution with parameters $\mu_{X}$ and $\sigma_{X}$,
$X~{}\sim\mathcal{N}(x;\mu_{X},\,\sigma_{X}^{2})$ the p.d.f. $f_{X}$ of $X$
can be expressed as,

$f_{X}(x)=\frac{1}{\sigma_{X}\sqrt{2\pi}}e^{-\frac{1}{2}\big{(}\frac{x-\mu_{X}}{\sigma_{X}}\big{)}^{2}}$

where $\mu_{X}$ and $\sigma_{X}$ are the mean and standard deviation (s.d.) of
the distribution.

##  III Methodology

In this section, we start with a brief overview of the proposed analytical
framework. Beginning with the hybrid quantum noise model identified as an
infinite mixture of Gaussian with weightage coming from Poisson distribution,
this research aims to identify the quantum noise model with a finite number of
Gaussian keeping weightage from Poisson distribution. This will help determine
the clusters to visualize the quantum noise data in higher dimensions.

The ML algorithm has been used to update the parametric values of the quantum
noise model; the set of parameters is the Poisson parameter, Gaussian means
and variances, and their mixing coefficients. Primarily, the capacity for the
signal-to-noise ratio has been plotted based on the newly updated channel
parameters and compared with the existing capacity model of the quantum
channel [7]. Note that the transmitted signal has been considered Gaussian for
both models. Fig. 1 explains the flow of the research methodology.

Figure 4: The process of updating channel parameters and finding optimal
clusters of quantum noise using ML model

###  III-A Clustering with Gaussian Mixture Model

Let us start with the hybrid quantum noise model in (3),

$\begin{split}f_{Z}(z)&=\sum_{i=0}^{\infty}w_{i}.\mathcal{N}(z;\mu_{i}^{(z)},\,{\sigma_{i}^{(z)}}^{2}),\end{split}$
(4)

where $w_{i}=\frac{e^{-\lambda}\lambda^{i}}{i!}$ represent the weightage of
the mixtures of Gaussians, $\sum_{i=0}^{\infty}w_{i}=1$, $w_{i}\geq
0\quad\forall i$ and $\mathcal{N}(z;\mu_{i}^{(z)},\,{\sigma_{i}^{(z)}}^{2})$
is the Gaussian density in random variable $Z$, mean
$\mu_{i}^{(z)}=\mu_{Z_{2}}+i$ and
${\sigma_{i}^{(z)}}^{2}={\sigma_{Z_{2}}}^{2}$ with standard deviation
$\sigma_{i}^{(z)}$ and variance ${\sigma_{i}^{(z)}}^{2}$. These lead to
visualizing the noise model data in a scalar space or lower dimensions. In the
meantime, this infinite mixture can be approximated by a finite number of
components as

$\begin{split}f_{Z}(z)=\sum_{i=0}^{R}w_{i}.\mathcal{N}\Big{(}z;\mu_{i}^{(z)},\,{\sigma_{i}^{(z)}}^{2}\Big{)}.\end{split}$
(5)

This is the Gaussian mixture in the scalar variable $Z$, $w_{i}\geq
0\quad\forall i$ and $\sum_{i=0}^{R}w_{i}\approx 1$ for large $R$. However, a
higher-dimensional approach is needed to visualize the original hybrid quantum
noise data in the corresponding vector space as a scatter plot. This leads to
the noise p.d.f. in a higher dimension given by the corresponding Gaussian
mixture in a random vector $\mathbf{z}\in\mathbb{R}^{D}$ as

$f_{\boldsymbol{Z}}(\mathbf{z})=\sum_{i=0}^{R}w_{i}.\mathcal{N}\Big{(}\mathbf{z};\boldsymbol{\mu}_{i}^{(z)},\,{\boldsymbol{\Sigma}_{i}^{(z)}}\Big{)},$
(6)

where $w_{i}=\frac{e^{-\lambda}\lambda^{i}}{i!}$ represents the weightage of
the Gaussian mixtures, $\mathbf{z}$ is the random vector in $\mathbb{R}^{D}$,
$D$ denotes the dimension of the vector $\mathbf{z}$,
$\boldsymbol{\mu}_{i}^{(z)}$ is the mean vector ,
$\boldsymbol{{\Sigma}}_{i}^{(z)}$ is the covariance matrix of the
corresponding Gaussian density $\mathcal{N}$, and the multivariate Gaussian is
given by

$\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})\\\
=\frac{1}{(2\pi)^{D/2}|\boldsymbol{\Sigma}_{k}^{(z)}|^{1/2}}e^{-\frac{1}{2}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}\\{\boldsymbol{\Sigma}_{k}^{(z)}\\}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})}.$
(7)

Consider the hybrid quantum-classical noise data set
$\boldsymbol{Z}=\\{\mathbf{z}_{1},\mathbf{z}_{2},...,\mathbf{z}_{n}\\}$,
$\mathbf{z}_{i}\in\mathbb{R}^{D}$, where $D$ denotes the dimension of the data
points, respectively. The primary purpose of this research is to understand
the multidimensional hybrid quantum-classical noise data set by partitioning
it into $K$ number of clusters, maximizing the likelihood of the probabilistic
model. The hybrid quantum noise model in (6) can be expressed as

$f_{\boldsymbol{Z}}(\mathbf{z})=\sum_{t}f_{\boldsymbol{Z}}(\mathbf{z},t)=\sum_{t}{}f_{\boldsymbol{Z}}(\mathbf{z}|t)f_{\boldsymbol{Z}}(t),$
(8)

where $t$ is the latent variable, which approximates the distribution with a
mixture of Gaussians. To use the EM algorithm to identify the cluster of
components in the hybrid quantum-classical noise data set, a discrete random
variable must be introduced, which will pick the clusters (here $t$ being the
$t$th component of the mixture) that satisfy the condition that the points in
the cluster must be Gaussian distributed with the respective parametric value
of the cluster.

The EM algorithm performs better in classifying the classical data set and is
more flexible than the $K$-means algorithm for classical data. However, the
limitation of this research is that we should have compared the result with
the $K$-means algorithm for our hybrid quantum-classical noise data set. From
(6), this research guarantees the Gaussian distribution for each latent
variable $t$ conditional distribution $f_{\boldsymbol{Z}}(\mathbf{z}|t)$. In
(8), $f_{\boldsymbol{Z}}(\mathbf{z}|t)$ are Gaussian p.d.f.s, and
$f_{\boldsymbol{Z}}(t)$ are mixing coefficients. It is the weighting of the
distribution $f_{\boldsymbol{Z}}(\mathbf{z}|t)$ with a particular probability
$f_{\boldsymbol{Z}}(t)$. Together, these two quantities, the sum
$\sum_{t}{}f_{\boldsymbol{Z}}(\mathbf{z}|t)f_{\boldsymbol{Z}}(t)$ represents
the GMM, and $f_{\boldsymbol{Z}}(\mathbf{z}|t)$ is the posterior probabilities
of the latent variable $t$ given an observable $\mathbf{z}$. That is if one
observes a particular quantum data point $\mathbf{z}$, one can check the
probability that it belongs to a cluster set red, green, or blue depending on
the number of clusters considered case by case. For example, from the colour
code model shown in Fig. 3(a), if the point comes from the blue class, it is
coloured blue. If the point is in the green class, it shows the green dot. If
the point belongs to the red class, it will indicate the red-coloured dot.
However, the unclassified data are shown in Fig. 3(b). For the chosen data
point, the uncertainty of specification of the cluster lies when these colours
are mixed or overlapped, indicated with black circles in Fig. 3$(c)$, this is
called “soft cluster”. According to this probabilistic model, if a data point
is assigned, it is assigned the probability of belonging to one of the
classes. It is not specifically inferred that it belongs to the blue class,
but it is inferred that with this probability it belongs to the blue class. It
is the “responsibility” that the class will take for the point to be in the
class shown in Fig. 3 $(d)$. This is the setting for the GMM. It is a way to
classify assignments in a soft and probabilistic way.

Figure 5: The ground truth data of quantum noise sampling generated as a
convolution of Additive White Gaussian Noise and Poissonian quantum noise

Let us begin by considering the p.d.f. for the model as defined in (8). The
p.d.f. of the GMM, which models the distribution of data points in a
clustering scenario, is given by (8). The discrete latent variable
$t_{k}\in\\{0,1\\}$ for the $K$ clusters is represented by the prior

$f_{\boldsymbol{Z}}(t_{k}=1)=\frac{e^{-\lambda}\lambda^{k}}{k!},$

$k\in[0,K]$, $\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{i}}{i!}\approx 1$ for a
sufficient choice of $K$. The clusters of the underlying components are
Gaussians, with different parameters:

$f_{\boldsymbol{Z}}(\mathbf{z}|t_{k}=1)=\mathcal{N}\Big{(}\mathbf{z};\boldsymbol{\mu}_{k}^{(z)},\,{\boldsymbol{\Sigma}_{k}^{(z)}}\Big{)}.$

The quantity $t_{k}\in\\{0,1\\}$ indicates whether the data point $\mathbf{z}$
belongs to the $K$-th class or not. Specifically, $t_{k}=0$ means that
$\mathbf{z}$ does not belong to the $K$-th class ($t_{k}\notin K$-th class),
and $t_{k}=1$ means that $\mathbf{z}$ belongs to the $K$-th class ($t_{k}\in
K$-th class). The probability that the data point $\mathbf{z}$ belongs to the
$K$-th class is denoted by $f_{\boldsymbol{Z}}(t_{k}=1)$. Consequently, the
joint probability distribution of the data point $\mathbf{z}$ and the class
membership $t_{k}$ can be expressed as

$f_{\boldsymbol{Z}}(\mathbf{z},t_{k}=1)=f_{\boldsymbol{Z}}(\mathbf{z}|t_{k}=1)f_{\boldsymbol{Z}}(t_{k}=1)$
$=\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\Big{(}\mathbf{z};\boldsymbol{\mu}_{k}^{(z)},\,{\boldsymbol{\Sigma}_{k}^{(z)}}\Big{)}$

and the full generic model is given by

$f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu},\boldsymbol{\Sigma})=\sum_{t}f_{\boldsymbol{Z}}(\mathbf{z},t)\\\
=\sum_{k}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\Big{(}\mathbf{z};\boldsymbol{\mu}_{k}^{(z)},\,{\boldsymbol{\Sigma}_{k}^{(z)}}\Big{)}.$
(9)

The posterior or conditional probability of $t$ (the latent cluster) given a
point $\mathbf{z}$ is

$f_{\boldsymbol{Z}}(t_{k}=1|\mathbf{z})=\frac{f_{\boldsymbol{Z}}(t_{k}=1)f_{\boldsymbol{Z}}(\mathbf{z}|t_{k}=1)}{f_{\boldsymbol{Z}}(\mathbf{z})}\\\
=\frac{f_{\boldsymbol{Z}}(t_{k}=1)f_{\boldsymbol{Z}}(\mathbf{z}|t_{k}=1)}{\sum_{j}{}f_{\boldsymbol{Z}}(t_{j}=1)f_{\boldsymbol{Z}}(\mathbf{z}|t_{j}=1)}\\\
=\frac{\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\Big{(}\mathbf{z};\boldsymbol{\mu}_{k}^{(z)},\,{\boldsymbol{\Sigma}_{k}^{(z)}}\Big{)}}{\sum_{j}{}\frac{e^{-\lambda}\lambda^{j}}{j!}\mathcal{N}\Big{(}\mathbf{z};\boldsymbol{\mu}_{k}^{(z)},\,{\boldsymbol{\Sigma}_{k}^{(z)}}\Big{)}}=\gamma(z).$

Inferring a latent cluster for an observed point $\mathbf{z}$ probabilities
can be assigned to this point that belongs to one of these unobserved
clusters. $f_{\boldsymbol{Z}}(t_{k}=1|\mathbf{z})=\gamma(z)$ is the formula
for obtaining the conditional posterior for the observed data point
$\mathbf{z}$ that belongs to one of the latent clusters. We will call $\gamma$
the responsibility that class $K$ takes to explain the data point
$\mathbf{z}$.

Figure 6: The initial guess of Gaussian Mixture clusters of bivariate hybrid
quantum noise sample data

EM produced a set of these parametric values
$w,\boldsymbol{\mu}^{(z)},\boldsymbol{\Sigma}^{(z)}$ to infer what parameters
are the most optimal and which parameters lead to the distribution that most
likely explains the hybrid quantum-classical noise data set. Therefore, we
need to define the log-likelihood function and want to maximize it to find
optimal parameters that describe the hybrid quantum-classical noise
dataset.The log-likelihood of the hybrid quantum noise can be expressed as

$\begin{split}&\ln
f_{\boldsymbol{Z}}\Big{(}\mathbf{z};w,\boldsymbol{\mu}^{(z)},\,\boldsymbol{\Sigma}^{(z)}\Big{)}\\\
&=\sum_{n=1}^{N}\ln\Big{(}\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})\Big{)}\end{split}$
(10)

where $N$ is referred to as the total number of hybrid quantum noise data
points named sample data. To maximize the log-likelihood function, in (10),
the expression cannot be further simplified due to the logarithmic summation
over the total number of clusters. However, one of the solutions can be to use
the ML-based EM algorithm. One needs to improve iteratively or log-likelihood
using this algorithm.

###  III-B Maximization of log-likelihood to find optimal parametric values
for classifying the clusters:

We need to maximize $\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu}^{(z)},\boldsymbol{\Sigma}^{(z)})$
w.r.t.
$\frac{e^{-\lambda}\lambda^{k}}{k!},\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}$
$\forall k=1(1)K$, where $K$ is the total number of clusters. The problem is
convex. Maximizing by looking for stationary points of logarithmic likelihood
implies optimizing with respect to parameters
$\frac{e^{-\lambda}\lambda^{k}}{k!},\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}$.
Therefore, we set the partial derivative of the log-likelihood function with
respect to the mean vector $\boldsymbol{\mu}_{k}^{(z)}$ of the $K$-th class to
zero

$\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu}^{(z)},\boldsymbol{\Sigma}^{(z)})=0,$
(11)

and solve it for $\boldsymbol{\mu}_{k}^{(z)}$. However, this equation does not
have a closed-form solution. The stationary points depend on the posterior
$\gamma(t_{n,k})$. It can find local minima using the so-called alternative
update algorithm of the (expected) posterior iterative algorithm
$\gamma(t_{n,k})$ and maximization for parameters
$\frac{e^{-\lambda}\lambda^{k}}{k!},\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}$.
In (11), if we take the derivative, it depends on the parameters
$\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}$ and
$\frac{e^{-\lambda}\lambda^{k}}{k!}$. Instead of this, we could write the
solution for $\boldsymbol{\mu}_{k}^{(z)}$ as a function of responsibility
$\gamma(t_{n,k})$, that is,
$\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(z;\frac{e^{-\lambda}\lambda^{k}}{k!},\boldsymbol{\mu}_{k}^{(z)},\mathbf{\Sigma_{k}^{(z)}})=0$
$\implies\boldsymbol{\mu}_{k}^{(z)}$ is a function of $\gamma(t_{n,k})$ and
$\gamma(t_{n,k})$ depends on
$\frac{e^{-\lambda}\lambda^{k}}{k!},\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}$.
The proposed solution is to group all remaining parameters in this term of
responsibility. Therefore, the term of responsibility still depends on other
parameters $\frac{e^{-\lambda}\lambda^{k}}{k!},\boldsymbol{\Sigma}_{k}^{(z)},$
even on $\mathbf{\mu_{k}}$. Using the EM algorithm, to fix the posterior
probability or the responsibility $\gamma(t_{n,k})$ and solve for the case
$\boldsymbol{\mu}_{k}^{(z)}$ and all the other parameters, one can update the
responsibilities and again iterate this solution for maximization and solve
for the parameters.

Figure 7: 1st iteration result under the updating of component of Gaussian
Mixture clusters of the bivariate hybrid quantum noise sample data

######  Lemma 1.

The maximizing log-likelihood in (10) w.r.t. $\boldsymbol{\mu}_{k}^{(z)}$ by
setting derivative of the log-likelihood w.r.t. $\boldsymbol{\mu}_{k}^{(z)}$
to $0$

$\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu}^{(z)},\mathbf{\Sigma^{(z)}})=0,$

gives

$\boldsymbol{\mu}_{k}^{(z)}=\frac{\sum_{n=1}^{N}\gamma(t_{n,k})\mathbf{z}_{n}}{\sum_{n=1}^{N}\gamma(t_{n,k})}$

###### Proof:

Please refer to Appendix A. ∎

Figure 8: The result after iteration size 5 for the updating of component of
Gaussian Mixture clusters of the bivariate hybrid quantum noise sample data

Now, to maximize the log-likelihood with respect to the mixing coefficient, $\
\frac{e^{-\lambda}\lambda^{k}}{k!}$ by considering a constrained optimization
problem using the Lagrange multiplier method. To maximize log-likelihood
w.r.t. $\frac{e^{-\lambda}\lambda^{k}}{k!}$ we have to set the derivative of
the log-likelihood w.r.t. $\frac{e^{-\lambda}\lambda^{k}}{k!}$ to $0$, under
the constraint that the sum of the probabilities
$\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}$, add up to one, that is,
$\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}=1$.

Let the Lagrange multipliers be denoted by

$L(\frac{e^{-\lambda}\lambda^{k}}{k!},\Lambda):=f(\mathbf{z}_{n})+\Lambda(g(\mathbf{z}_{n})-c)$

where $f(\mathbf{z}_{n})=\sum_{n=1}^{N}\ln
f_{\boldsymbol{Z}}(\mathbf{z}_{n};\frac{e^{-\lambda}\lambda^{k}}{k!},\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})$
, $g(\mathbf{z}_{n})=\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}$ and
$c=1$. Let us set the partial derivative of the Lagrange multipliers w.r.t.
$\frac{e^{-\lambda}\lambda^{k}}{k!}$ that is $\frac{\partial}{\partial
w_{k}}L(w,\Lambda)=0$ gives

$\frac{\partial}{\partial\frac{e^{-\lambda}\lambda^{k}}{k!}}\Bigg{[}\sum_{n=1}^{N}\ln
f_{\boldsymbol{Z}}(\mathbf{z}_{n};w,\boldsymbol{\mu},\boldsymbol{\Sigma})\\\
+\Lambda\bigg{\\{}\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}-1\bigg{\\}}\Bigg{]}=0\\\
\implies\sum_{n=1}^{N}\frac{1}{f_{\boldsymbol{Z}}(\mathbf{z}_{n};w,\boldsymbol{\mu},\boldsymbol{\Sigma})}\frac{\partial}{\partial\frac{e^{-\lambda}\lambda^{k}}{k!}}f_{\boldsymbol{Z}}(\mathbf{z}_{n};w,\boldsymbol{\mu},\boldsymbol{\Sigma})\\\
+\Lambda\frac{\partial}{\partial\frac{e^{-\lambda}\lambda^{k}}{k!}}\bigg{\\{}\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}-1\bigg{\\}}=0$
(12)
$\implies\sum_{n=1}^{N}\frac{\frac{\partial}{\partial\frac{e^{-\lambda}\lambda^{k}}{k!}}\Big{[}\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})\Big{]}}{\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})}$
$+\Lambda\frac{\partial}{\partial\frac{e^{-\lambda}\lambda^{k}}{k!}}\bigg{\\{}\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}-1\bigg{\\}}=0$
$\implies\frac{e^{-\lambda}\lambda^{k}}{k!}=-\frac{1}{\Lambda}\sum_{n=1}^{N}\gamma(t_{n,k}).$

Now solve it for $\Lambda$. For that, we consider the constraint part
$\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}=1.$. Now, setting derivative
of the Lagrange multipliers w.r.t. $\Lambda$ to zero ,that is
$\frac{\partial}{\partial\Lambda}L(\frac{e^{-\lambda}\lambda^{k}}{k!},\Lambda)=0\implies\frac{\partial}{\partial\Lambda}\big{\\{}f(\mathbf{z}_{n})+\Lambda(g(\mathbf{z}_{n})-c)\big{\\}}=0\implies\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}-1=0\\\
\implies-\frac{1}{\Lambda}\sum_{n=1}^{N}\sum_{k=1}^{K}\gamma(t_{n,k})=1\implies\Lambda=-N$,
because
$\sum_{k=1}^{K}\gamma(t_{n,k})=\sum_{k=1}^{K}f_{\boldsymbol{Z}}(t_{n,k}|\mathbf{z})=1$,
this is due to the sum of all classes. Finally, this gives us

$w_{k}=\frac{1}{N}\sum_{n=1}^{N}\gamma(t_{n,k}).$

Now we need to calculate the expression for $\boldsymbol{\Sigma}_{k}^{(z)}$.

######  Lemma 2.

The maximization of the log-likelihood in (10) w.r.t.
$\boldsymbol{\Sigma}_{k}^{(z)}$ by setting

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu},\boldsymbol{\Sigma})=0,$

gives

$\boldsymbol{\Sigma}_{k}^{(z)}=\frac{\sum_{n=1}^{N}\gamma(t_{n,k})(\mathbf{z}_{n}-\mu_{k}^{(z)})(\mathbf{z}_{n}-\mu_{k}^{(z)})^{T}}{\sum_{i=1}^{N}\gamma(t_{n,k})}.$

###### Proof:

Please refer to Appendix B. ∎

Now define the effective number of points in cluster K by

$N_{k}=\sum_{n=1}^{N}\gamma(t_{n,k}).$

Hence, the solution for
$\frac{e^{-\lambda}\lambda^{k}}{k!},\mu_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}$
(dependent on the posterior) can be written as

$\begin{split}&\boldsymbol{\mu}_{k}^{(z)}=\frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(t_{n,k})\mathbf{z}_{n},\\\
&\frac{e^{-\lambda}\lambda^{k}}{k!}=\frac{N_{k}}{N},\\\
&\boldsymbol{\Sigma}_{k}^{(z)}=\frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(t_{n,k})(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}.\\\
\end{split}$ (13)

###  III-C Expectation-Maximization Algorithm (EM):

The EM algorithm is a robust technique for estimating parameters in GMMs. This
iterative algorithm comprises two primary steps: the expectation step (E step)
and the maximization step (M step). Initially, the GMM parameters are
established, which involve initialization of the means
$\boldsymbol{\mu}_{k}^{(z)}$, covariances $\boldsymbol{\Sigma}_{k}^{(z)}$, and
the probabilities of the mixture component expressed as
$\frac{e^{-\lambda}\lambda^{k}}{k!}$ for each component $k$. In step E, the
algorithm updates the expected posterior probabilities, or responsibilities
$\gamma(z_{n,k})$, for each data point that concerns each component of the
mixture. This step estimates the degree to which each component generates each
data point. Subsequently, the M step aims to maximize the log-likelihood of
the model given the current parameter estimates, adjusting
$\boldsymbol{\mu}_{k}^{(z)}$, $\boldsymbol{\Sigma}_{k}^{(z)}$, and the
probabilities based on the fixed posteriors established in the E step. The
algorithm alternates between these E and M steps until it meets a convergence
criterion, such as a negligible change in the log-likelihood or after a
specified number of iterations. This iterative process allows the EM algorithm
to efficiently find local minima in the optimization landscape of the GMM
parameter estimation problem.

1

2

3

Input: $\lambda$, $\boldsymbol{\mu}_{k,0}^{(z)}$,
$\boldsymbol{\Sigma}_{k,0}^{(z)}$, $k=1(1)K$

4

Output: $\boldsymbol{\mu}_{k,m}^{(z)}$, $\boldsymbol{\Sigma}_{k,m}^{(z)}$

5

6$m\leftarrow 1$

7repeat

8

9  Calculate the responsibilities $\gamma(z_{n,k,m})$ for each data point
$\mathbf{z}_{n}$ and each component $k$ according to
$\gamma(z_{n,k})=\frac{\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})}{\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})}$;

10

11
$w_{k,m}=\frac{e^{-\lambda}\lambda^{k}}{k!}=\frac{1}{N}\sum_{n=1}^{N}\gamma(\mathbf{z}_{n,k,m})$;

12

13
$\boldsymbol{\mu}_{k,m}^{(z)}=\frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(\mathbf{z}_{n,k,m})\mathbf{z}_{n}$
;

14

15
$\boldsymbol{\Sigma}_{k,m}^{(z)}=\frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(\mathbf{z}_{n,k,m})(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})(\mathbf{z}_{n}-\boldsymbol{\mu}_{k,m}^{(z)})^{T}$;

16

17 $m\leftarrow m+1$;

18

19until  _convergence_ ;

Algorithm 1 Expectation Maximization Algorithm for finding the optimal
$\frac{e^{-\lambda}\lambda^{k}}{k!}$, $\boldsymbol{\mu}_{k}^{(z)}$,
$\boldsymbol{\Sigma}_{k}^{(z)}$

In our probabilistic framework, which models responsibility-classical noise as
a combination of several Gaussian distribution functions, as detailed in
Equation (6), we outline the EM algorithm within Algorithm 1.

###  III-D The convergent of EM algorithm in the context of log-likelihood
function

Convergence of the EM algorithm is typically understood in terms of the log-
likelihood function of the observed data. In terms of monotonicity, the EM
algorithm ensures that the log-likelihood function of the observed data does
not decrease with each iteration [47]. Under general conditions, the EM
algorithm converges to a local maximum (or saddle point in some cases) of the
log-likelihood function. The convergence rate is typically linear but can slow
near the maximum. The specific convergence rate can depend on the proximity of
the initial parameter estimates to the actual values and the shape of the log-
likelihood surface [48]. The convergence of the EM algorithm is guaranteed
under certain regularity conditions, including certain derivatives of the log-
likelihood function and the boundedness of the parameter space. However, these
conditions can sometimes only be strictly met in practice, leading to
convergence to local optima or, in rare cases, divergence if assumptions are
violated severely [49].

For practical considerations, the EM algorithm’s convergence is usually
monitored by checking the change in the log-likelihood value or the change in
parameter estimates between successive iterations. Convergence is assumed when
these changes fall below a predetermined threshold. It is also common to set a
maximum number of iterations to prevent infinite looping in cases where
convergence is slow or the algorithm oscillates.

Figure 9: The result after iteration size 10 for the updating of component of
Gaussian Mixture clusters of the bivariate hybrid quantum noise sample data.

###  III-E Discusssion on selecting GMM in comparison to existing ML models

GMM stands out among other ML models due to its flexibility, probabilistic
foundation, and ability to model complex distributions. First, its flexibility
in modeling distributions allows it to approximate almost any continuous
density function, making it ideal for datasets with multiple subpopulations or
clusters. Unlike single-Gaussian models, GMMs can effectively capture diverse
data structures [26]. Second, GMM offers soft clustering, assigning to each
data point a probability of belonging to each cluster, enabling nuanced
grouping and analysis [50]. This contrasts with hard clustering methods like
K-means. Third, GMM’s probabilistic framework inherently accommodates
uncertainty in data, ensuring robust modeling results in noisy or incomplete
datasets [51]. Lastly, GMM facilitates parameter estimation via the
Expectation-Maximization (EM) algorithm, iteratively refining data likelihood
under the model. Additionally, model selection criteria such as Bayesian or
Akaike Information Criterion are directly applicable to GMMs, helping to
determine the optimal number of components [52][53].

##  IV Application in Maximization of Quantum Channel Capacity

In quantum channel modeling, the interplay between the Poisson parameter and
the GMM is fundamental to accurately simulating quantum noise. The Poisson
parameter, $\lambda$, quantifies the average photon count detected over a
specific interval, highlighting the quantum nature of light and the stochastic
process of photon emission and detection. This parameter is essential for
modeling photon statistics in quantum optics [40, 39], where photon arrivals
at a detector are typically Poisson distributed, reflecting the quantized
nature of light. On the contrary, the GMM provides a probabilistic framework
for modeling quantum noise, encompassing both quantum and classical noise
influences on quantum states during transmission [53]. This allows for a more
comprehensive representation of various noise sources and their statistical
behaviors in a quantum channel. The connection between these two models lies
in their joint capability to more accurately represent the complexity of
quantum noise [54]. Although the Poisson parameter $\lambda$ captures the
quantum mechanical aspects of the photon distribution, the GMM accounts for a
broader range of noise scenarios, including classical noise effects such as
thermal noise and detector inefficiencies [38]. This blend facilitates nuanced
understanding and optimization of quantum channels, enabling the development
of sophisticated error correction codes and communication protocols tailored
to mitigate the specific noise characteristics encountered in quantum
information transfer.

Figure 10: The result after iteration size 15 under the updating of component
of Gaussian Mixture clusters of the bivariate hybrid quantum noise sample
data.

Taking into account hybrid quantum noise and the Gaussian transmitted signal
model as discussed above [7], the capacity of the quantum channel can be
expressed as follows:
$C=\sum_{i=0}^{R}\frac{e^{-\lambda}\lambda^{i}}{i!}\Bigg{(}-\log_{2}\Big{(}\frac{e^{-\lambda}\lambda^{i}}{i!}\Big{)}+\frac{1}{2}\log_{2}{\Big{(}(2\pi
e)^{M}\big{|}{\boldsymbol{\Sigma}}_{i}^{(y)}\big{|}\Big{)}}\\\
+\log_{2}\Big{(}\sum_{j=0}^{R}\frac{e^{-\lambda}\lambda^{j}}{j!}\mathcal{N}\Big{(}\boldsymbol{\mu}_{i}^{(z)};\boldsymbol{\mu}_{j}^{(z)},\,{\boldsymbol{\Sigma}}_{i}^{(z)}+{\boldsymbol{\Sigma}}_{j}^{(z)}\Big{)}\Big{)}\Bigg{)}$.

Figure 11: The Final results corresponding to iteration size 20 under the
updating of component of Gaussian Mixture clusters of the bivariate hybrid
quantum noise sample data.

This formula delineates the capacity of the quantum channel for scenarios
where both the noise $\boldsymbol{Z}$ and the received signal $\boldsymbol{Y}$
are represented as random vectors of dimension size $M$. In scalar analogy,
that is, when the noise $Z$ with scalar p.d.f., the expression of capacity
reduces to
$C=\sum_{i=0}^{R}\frac{e^{-\lambda}\lambda^{i}}{i!}\Bigg{[}-\log_{2}\Big{(}\frac{e^{-\lambda}\lambda^{i}}{i!}\Big{)}+\frac{1}{2}\log_{2}{\Big{(}2\pi
e{\sigma}_{i}^{(y)}\Big{)}}+\log_{2}\Bigg{(}\sum_{j=0}^{R}\frac{e^{-\lambda}\lambda^{j}}{j!}\mathcal{N}\Big{(}{\mu_{i}}^{(z)};{\mu_{j}}^{(z)},\,{{\sigma}_{i}^{(z)}}^{2}+{{\sigma}_{j}^{(z)}}^{2}\Big{)}\Bigg{)}\Bigg{]}$,
by putting $M=1$, and replacing
$\big{|}{\boldsymbol{\Sigma}}_{i}^{(y)}\big{|}$ by ${\sigma}_{i}^{(y)}$,
$\boldsymbol{\mu}_{i}^{(z)}$ by $\mu_{i}^{(z)}$, $\boldsymbol{\mu}_{j}^{(z)}$
by $\mu_{j}^{(z)}$, ${\boldsymbol{\Sigma}}_{i}^{(z)}$ by
${{\sigma}_{i}^{(z)}}^{2}$ and $\boldsymbol{\Sigma}_{j}^{(z)}$ by
${{\sigma}_{j}^{(z)}}^{2}$, where each vector is replaced by its scalar
analogue [7]. Again $\mu_{i}^{(z)}=\mu_{Z_{2}}+i$ $\forall i$,
${{\sigma}_{i}^{(z)}}^{2}={\sigma_{Z_{2}}}^{2}$ $\forall i$, and
${{\sigma}_{i}^{(y)}}^{2}={\sigma_{X}}^{2}+{\sigma_{Z_{2}}}^{2}$ $\forall i$.
Therefore, the capacity of the quantum channel is given by

$\begin{split}C&=\sum_{i=0}^{R}\frac{e^{-\lambda}\lambda^{i}}{i!}\Bigg{[}-\log_{2}\Big{(}\frac{e^{-\lambda}\lambda^{i}}{i!}\Big{)}\\\
&+\frac{1}{2}\log_{2}{\Big{(}2\pi
e\big{(}{\sigma_{X}}^{2}+{\sigma_{Z_{2}}}^{2}\big{)}\Big{)}}\\\
&+\log_{2}\Bigg{(}\sum_{j=0}^{R}\frac{e^{-\lambda}\lambda^{j}}{j!}\frac{1}{\sqrt{2}\sigma_{Z_{2}}\sqrt{2\pi}}e^{-\frac{1}{2}\Big{(}\frac{i-j}{\sqrt{2}\sigma_{Z_{2}}}\Big{)}^{2}}\Bigg{)}\Bigg{]}.\end{split}$
(14)

The updated channel parameters will be used to plot the new capacity graph
with the signal-to-noise ratio using the EM algorithm. Based on the initial
guess of the channel parameters, they will be compared with the capacity graph
with the signal-to-noise ratio.

##  V Numerical Analysis

###  V-A Hybrid quantum noise data visualization

In this subsection, we visualize the original underlying hybrid quantum-
classical noise data set in the vector field and investigate its properties.
Consider our hybrid quantum-classical noise model given by (6), is a GMM
consisting of a finite number of mixtures in Gaussians with a set of mixing
coefficients. The weightage (coefficient) will decide the contributions of the
Gaussian component, which has probability properties, explicitly represented
by the prior $f_{\boldsymbol{Z}}(t_{k}=1)=\frac{e^{-\lambda}\lambda^{k}}{k!},$
$k\in[0,K],\frac{e^{-\lambda}\lambda^{k}}{k!}\in[0,1],\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\approx
1$ where $t_{k}\in\\{0,1\\}$ is the discrete latent variable for the $K$
clusters. In our model, the weightage comes from the Poisson distribution, so
it is necessary to carefully choose the values of these parameters for the
initial guess so that the initial condition
$\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\approx 1$ will satisfy up to
a specifically defined accuracy, and not only that these values will depend on
the chosen value of $\lambda$. The higher value of $\lambda$ will be
responsible for the more significant number of components (value of $K$).
Since we intend to reduce the number of the mixture for simplicity of
visualization, choose the value of $\lambda$ as low as possible. The physical
interpretation of the lower value of $\lambda$ is the controlled photon
emission per second.

Figure 12: The log-likelihood of the bivariate hybrid quantum noise
probability density function. The saturation of the curve implies the
convergent of the EM algorithm for the cluster finding.

For example, set first the accuracy of
$\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\approx 1$ to be at least
$85\%$, that is, $\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}>0.85$. Now,
if one sets $\lambda=2$, the initial guess of the mixing coefficients can be
$0.271,0.271,0.135,0.180,0.009$ (as these are the most probable values in the
Poisson distribution with $\lambda=2$), the sum of these values $5$ gives
$0.947$ ($>0.85$) which means that the number of clusters (that is, the value
of $K$) can be considered as $5$. In another example, let $\lambda=5$, the
initial guess of the mixing coefficients can be
$0.175,0.175,0.146,0.140,0.104,0.084,0.065$ (since these are the most probable
values in the Poisson distribution with $\lambda=5$), the sum of these values
of $7$ gives $0.889$ ($>0.85$) which means that the number of clusters ( that
is, the value of $K$) can be considered as $7$. However, suppose that we
consider the five highest probable values $0.175,0.175,0.146,0.140,0.104$ from
the Poisson distribution with $\lambda=5$ to keep the number of clusters as
$K=5$. In that case, we will get the sum of $0.74<0.85$, which does not
satisfy the initial condition.

Higher values of $\lambda$ will give a flatter Poisson distribution and a
lower value of the mixing coefficient or weightage $w_{i}$, leading to more
components in the Gaussian mixture. However, it aims to reduce the number of
components in the mixture to visualize the classified clusters in the modeling
quantum data. This will simplify our understanding of the underlying hybrid
quantum-classical noise model. As the constraint $\sum_{i=0}^{R}w_{i}\approx
1$ for large $R$ to be satisfied up to a specific error tolerance, we set an
accuracy level of $85\%$ by setting $|1-\sum_{i=0}^{R}w_{i}|\leq 0.15$, that
is, we tolerate the error $15\%$. We should have a precision of at least
$85\%$ while keeping the optimal number of clusters to visualize the hybrid
quantum-classical noise data set.

Figure 13: The capacity of the quantum channel with respect to SNR for the
number of components $5$ and variance of $\sigma_{Z_{2}}^{1}=1$.

In quantum cryptography, Alice’s light source is crucial for generating the
photons dispatched to Bob. It has been assumed throughout that Alice transmits
photons singly to Bob. Transmitting multiple photons per pulse could
inadvertently benefit Eve, the eavesdropper, by providing additional
information. The issue is exacerbated with each additional photon per pulse.
For instance, Eve’s chances of accurately determining the basis and bit value
for many pulses with three photons per pulse increase significantly. Thus,
ensuring that each light pulse contains only a single photon is paramount,
though challenging in practice. The conventional approach involves using a
pulsed laser and substantially attenuating its output to achieve a low mean
number of photons per pulse, $\lambda$. A laser light of a single frequency
exhibits Poissonian photon statistics. With a small $\lambda$, most pulses
will be photon-free, a minority will have one photon, and an even smaller
fraction will contain multiple photons. For example, selecting $\lambda=0.1$
is common in current quantum cryptography but results in a concerning scenario
in which $5\%$ of the pulses contain more than one photon for every pulse with
exactly one photon [46].

Let us visualize the model with $\lambda=2$. Our first objective is to find
the optimal number of clusters $K$ with at least $85\%$ accuracy, and this is
the most probable value for $\frac{e^{-\lambda}\lambda^{k}}{k!}$ from the
Poisson distribution with parameter $\lambda=2$. For the initial guess of the
mixing coefficient $\frac{e^{-\lambda}\lambda^{i}}{i!}\forall i=0,1,2,3,4$,
choose $w_{1}=0.135$, $w_{2}=0.271$, $w_{3}=0.271$, $w_{4}=0.18$, and
$w_{1}=0.09$. Now, if the condition $|1-\sum_{i=0}^{R}w_{i}|\leq 0.15$ is
satisfied, it can proceed with the values and obtain the acceptable number of
clusters. If the condition does not satisfy, try different values of the
Poisson distribution with parameter $\lambda=2$. So, let us calculate
$|1-\sum_{i=0}^{R}w_{i}|=1-(0.135+0.271+0.271+0.18+0.09)=1-0.94735=0.05265<0.15$,
that is, our accuracy is $\approx 95\%$, and we tolerate the error $\approx
5\%$ allowed by our limitation limit. This will give us the optimal number of
clusters $K$, $5$ within the error tolerance. Therefore, we rewrite our noise
probability density function in five Gaussian mixtures,

$f_{\boldsymbol{Z}}(\mathbf{z})=\sum_{k=0}^{K}w_{k}.\mathcal{N}\Big{(}\mathbf{z};\boldsymbol{\mu}_{k}^{(z)},\,\boldsymbol{\Sigma}_{k}^{(z)}\Big{)}$

Figure 14: The capacity of the quantum channel with respect to SNR for the
number of components $5$ and variance $\sigma_{Z_{2}}^{2}=0.687$.

We aim to find the optimal parametric values for $\boldsymbol{\mu}_{k}^{(z)}$
and $\boldsymbol{\Sigma}_{k}^{(z)}$ for $k=0,1,2,3,4$. Note that a general
Gaussian mixture of five components needs an initial guess of the parameter
value $15$, and at the end of the result optimal values should be obtained for
these parameters $15$. However, we have fewer degrees of freedom in choosing
the parametric values discussed below.

This program simulation was run in an Anaconda3 Jupyter notebook with Python
to find the final values of the parameters, and this led to the classification
of the clusters and visualization of the components of the hybrid-quantum
classical data set. Since our number of mixing components, a.k.a. the number
of clusters, is $5$, we start with the initial guess or the
$\boldsymbol{\mu}_{0},\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2},\boldsymbol{\mu}_{3},\boldsymbol{\mu}_{4}$,
and
$\boldsymbol{\Sigma}_{0},\boldsymbol{\Sigma}_{1},\boldsymbol{\Sigma}_{2},\boldsymbol{\Sigma}_{3},\boldsymbol{\Sigma}_{4}$.
In the hybrid quantum-classical noise model (6), we calculated the mean vector
$\boldsymbol{\mu}_{k}^{(z)}=(\mu_{Z_{2}}+k)\mathbf{I}_{2x1}$, and covariance
matrix as $\boldsymbol{\Sigma}_{k}^{(z)}=\sigma_{Z_{2}}^{2}\mathbf{I}_{2x2}$
for $k=0,1,2,3,4$. Here, $\mathbf{I}$ represents the identity matrix of the
corresponding dimensions. So, start with the initial choice for mean
$\mu_{Z_{2}}=0$, and variance $\sigma_{Z_{2}}^{2}=1$. Let us visualize the
hybrid quantum-classical noise data set by plotting
$\mathcal{N}(\mathbf{z};\boldsymbol{\mu}_{k}^{(z)},\,{\boldsymbol{\Sigma}}_{k}^{(z)})$
for $k=0,1,2,3,4$ given by Fig. 5. This unlabeled quantum noise data set will
be the ground truth for the EM algorithm to find the clusters.

In the process of setting the initial estimate for the Gaussian parameters,
let us start with
$\boldsymbol{\mu}_{0}=\mu_{Z_{2}}\mathbf{I}_{2x1}=\mathbf{0}_{2x1},\boldsymbol{\Sigma}_{0}=\sigma_{Z_{2}}^{2}\mathbf{I}_{2x2}=\mathbf{I}_{2x2}$,
based on the values chosen above of mean $\mu_{Z_{2}}$ and variance
$\sigma_{Z_{2}}^{2}$. Consider the precalculated initial estimate for the
mixing coefficients, which is Poisson distributed: $w_{0}=0.135$,
$w_{1}=0.271$, $w_{2}=0.271$, $w_{3}=0.18$, and $w_{4}=0.09$. Fig. 6 shows the
estimated clusters to choose the parametric values above for the initial
guess. The following figures show how the iteration of intermediate steps
iteratively updates the parametric values; the corresponding plots for the
estimated clusters are shown accordingly in Fig. 7, Fig. 8, Fig. 9, Fig. 10,
and Fig. 11. In particular, Fig. 11 shows the optimal clusters resulting from
the EM algorithm. Fig. 12 shows the logarithmic likelihood of the density
function of the hybrid quantum noise; the saturation of the function confirms
the convergence of the EM algorithm, which implies that we obtained the final
optimal set of the cluster for the quantum noise data set. Also, from Fig. 10
and Fig. 11, we can infer that the system model is convergent since updating
does not change the clusters.

We started with the initial estimate for the parametric values $w_{0}=0.135$,
$w_{1}=0.271$, $w_{2}=0.271$, $w_{3}=0.18$, $w_{4}=0.09$, and
$\boldsymbol{\mu}_{0}=\mathbf{0}_{2X1}$, and
$\boldsymbol{\Sigma}_{0}=\mathbf{I}_{2X2}$. We plotted the figures for the
cases where the number of iterations =$1,5,10,15,20$ is shown in Fig. 7, Fig.
8, Fig. 9, Fig. 10, and Fig. 11 respectively. Finally, we get optimal values
for the parameters as follows:

$w_{0}^{\prime}=0.1871,w_{1}^{\prime}=0.3321,w_{2}^{\prime}=0.2916,$
$w_{3}^{\prime}=0.1487,w_{4}^{\prime}=0.0403,$

and

$\boldsymbol{\mu}_{0}^{\prime}=\begin{bmatrix}0.4180\\\ 0.0895\end{bmatrix},$
$\boldsymbol{\Sigma}_{0}^{\prime}=\begin{bmatrix}0.8910&-0.1011\\\
-0.1011&0.9452\end{bmatrix}$.

From the algorithm, the optimal value is obtained as $w_{0}^{\prime}=0.1871$,
this $\implies\frac{e^{-\lambda}\lambda^{0}}{0!}=0.1871$,
$\implies\lambda=1.6757\approx 2$ in integer values. Therefore, as a
verification, this application of the EM algorithm for finding the optimal
cluster in the classification of bivariate hybrid-quantum data can update the
optimal parametric values of Gaussian components and helps to identify the
optimal clusters but does not affect the Poissonian parameter $\lambda$, which
is the number of photon counts in the system.

However, the iterative EM algorithm has successfully updated the other two
channel parameters. We will use the updated parameters to compare the quantum
channel capacity for the Gaussian transmitted signal.

###  V-B Capacity comparison based on updated channel parameters and initial
guess of parameters

Consider the expression of quantum channel capacity in (14) involving the
channel parameter $\sigma_{Z_{2}}$ only to reflect any change in capacity, as
we have shown that there is no change in the value of $\lambda$, and
$\mu_{Z_{2}}$ does not involve the expression.

So, start with the initial value of
$\boldsymbol{\Sigma}_{0}=\sigma_{Z_{2}}^{2}\mathbf{I}_{2x2}$ where
$\sigma_{Z_{2}}=1$ $\longleftrightarrow$ $|\boldsymbol{\Sigma}_{0}|=1$, and
updated value of $\boldsymbol{\Sigma}_{0}^{\prime}$
$\implies|\boldsymbol{\Sigma}_{0}^{\prime}|=0.829\longleftrightarrow\sigma_{Z_{2}}=0.829\implies\sigma_{Z_{2}}^{2}=0.687$.

This implies that the EM algorithm will lower the Gaussian noise variance by
optimal clustering. Now, let us compare the capacity with respect to the
signal-to-noise ratio (SNR) when $\sigma_{Z_{2}}^{2}=1$ and $0.687$ based on
the cluster size $5$. In Fig. 13, we have plotted the benchmark result as the
capacity of the quantum channel with respect to SNR for the number of
components $5$ and the variance of Gaussian noise, $\sigma_{Z_{2}}^{2}=1$.
Fig. 14 we plot the capacity of the quantum channel with respect to the
signal-to-noise ratio for the number of components $5$ and the variance
$\sigma_{Z_{2}}^{2}=0.687$. Fig. 14 gives a better result in capacity
estimation compared to Fig. 13.

##  VI Conclusion

This work enhances quantum communication through ML, focusing on a hybrid
quantum noise model that combines quantum Poissonian noise with classical
white Gaussian noise. Traditionally visualized in lower dimensions, this noise
has been reinterpreted as a finite GMM, with a Poisson distribution
determining mixing weights. The goal was to simplify the model for easier
visualization without compromising the accuracy of the original noise
distribution. Using the EM algorithm, we update the channel parameters and
identify optimal clusters, allowing a more precise representation of quantum
noise in higher dimensions. When the capacities of the quantum channels were
compared under initial and updated parameters in varying signal-to-noise
ratios, it was demonstrated that ML significantly improves the accuracy of the
parameters. This approach offers a more intuitive understanding of quantum
noise and aids in the precise identification of noise clusters, marking a
substantial advancement in the field.

## Appendix

###  VI-A Proof of Lemma 1

The aim is to compute an expression for $\boldsymbol{\mu}_{k}^{(z)}$ as a
function of responsibility $\gamma$, this can be done by setting a partial
derivative of log-likelihood w.r.t. $\boldsymbol{\mu}_{k}^{(z)}$ as zero ,
given by $\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu}^{(z)},\mathbf{\Sigma^{(z)}})=0$

$\implies\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\sum_{n=1}^{N}\ln
f_{\boldsymbol{Z}}(\mathbf{z}_{n};w,\boldsymbol{\mu}^{(z)},\boldsymbol{\Sigma}^{(z)})=0\\\
\implies\sum_{n=1}^{N}\frac{\frac{e^{-\lambda}\lambda^{k}}{k!}\frac{\partial}{\partial\mu_{k}^{(z)}}\\{\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}\\}}{\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})}=0$
(15)

To evaluate the density derivative of
$\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}$
w.r.t. $\boldsymbol{\mu}_{k}^{(z)}$ given by,
$\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}$

$=-\frac{1}{2}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}\\\
\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\Big{\\{}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})\Big{\\}}$

Moreover, the density derivative w.r.t. $\boldsymbol{\mu}_{k}^{(z)}$is given
by

$\frac{\partial}{\partial\boldsymbol{\mu}_{k}^{(z)}}\Big{\\{}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})\Big{\\}}$

$=-2(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}$

This can simply done by using the matrix rule, $\frac{\partial}{\partial
x}\\{x^{T}Ax\\}=x^{T}A^{T}+x^{T}A$, and setting
$x=\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)},A:={\mathbf{\Sigma}_{k}^{(z)}}^{-1}$
and also note $\boldsymbol{\Sigma}_{k}^{(z)}$ is symmetric.

Therefore, the ultimate equation for calculating the expression of
$\boldsymbol{\mu}_{k}^{(z)}$ as a function of the responsibility $\gamma$ is
derived as follows:

$\displaystyle\sum_{n=1}^{N}\Bigg{[}\frac{\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}}{\sum_{j=1}^{K}\frac{e^{-\lambda}\lambda^{j}}{j!}\mathcal{N}\big{(}\mathbf{z}_{n};\mathbf{\mu_{j}},\mathbf{\Sigma_{j}}\big{)}}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}\\{\boldsymbol{\Sigma}_{k}^{(z)}\\}^{-1}\Bigg{]}=0$
(16)

since each $\gamma(t_{n,k})$ being a probability is a scalar, therefore
$\gamma(t_{n,k})^{T}=\gamma(t_{n,k})$ gives

$\boldsymbol{\mu}_{k}^{(z)}=\frac{\sum_{n=1}^{N}\gamma(t_{n,k})\mathbf{z}_{n}}{\sum_{n=1}^{N}\gamma(t_{n,k})}.$
(17)

###  VI-B Proof of Lemma 2

To compute an expression for $\boldsymbol{\Sigma}_{k}^{(z)}$ as a function of
responsibility $\gamma$, let us set the partial derivative of log-likelihood
w.r.t. $\boldsymbol{\Sigma}_{k}^{(z)}$ by setting zero , given by
$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\log
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu},\boldsymbol{\Sigma})=0$.
Before compute an expression for $\boldsymbol{\Sigma}_{k}^{(z)}$, it is
necessary to simplify the partial derivative log-likelihood w.r.t.
$\boldsymbol{\Sigma}_{k}^{(z)}$ given by,

$\begin{split}&\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu},\boldsymbol{\Sigma})\\\
&=\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\sum_{n=1}^{N}\ln\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})\\\
&=\sum_{n=1}^{N}\frac{\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\left(\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})\right)}{\sum_{k=1}^{K}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})},\\\
\end{split}$ (18)

as
$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\frac{e^{-\lambda}\lambda^{j}}{j!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{j},\boldsymbol{\Sigma}_{j})=0\,\forall
j\neq k\in{1,\dots,N}.$ To compute the derivative
$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})$
in numerator of (LABEL:eq_expression_for_sigma1), we use the logarithmic
derivative, expressed as
$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln\left(\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)})\right)$.
This method simplifies the differentiation process by taking advantage of the
logarithm’s properties to break down the product and exponential components
within the function. Note that

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}=\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}\\\
\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln\Big{(}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}\Big{)},$

plugging all together gives the partial derivative of log-likelihood w.r.t.
$\boldsymbol{\Sigma}_{k}^{(z)}$ as,

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu},\boldsymbol{\Sigma})\\\
=\sum_{n=1}^{N}\gamma(t_{n,k})\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln\left(\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}\right).$
(19)

Note here that

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln\Big{(}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}\big{(}\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\boldsymbol{\Sigma}_{k}^{(z)}\big{)}\Big{)}\\\
=\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\Big{\\{}\ln\bigg{(}\frac{e^{-\lambda}\lambda^{k}}{k!}\bigg{)}\Big{\\}}+\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\Big{\\{}\ln\Big{(}\frac{1}{(2\pi)^{D/2}|\boldsymbol{\Sigma}_{k}^{(z)}|^{1/2}}\Big{)}\Big{\\}}\\\
+\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\Big{\\{}-\frac{1}{2}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})\Big{\\}}$

For the first part, $\ln\big{(}\frac{e^{-\lambda}\lambda^{k}}{k!}\big{)}$ is
constant w.r.t. $\boldsymbol{\Sigma}_{k}^{(z)}$, hence
$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\Big{\\{}\ln\frac{e^{-\lambda}\lambda^{k}}{k!}\Big{\\}}=0$.
For the second part,

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\Big{\\{}\ln\Big{(}\frac{1}{(2\pi)^{D/2}|\boldsymbol{\Sigma}_{k}^{(z)}|^{1/2}}\Big{)}\Big{\\}}=-\frac{1}{2}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}$

This can be computed using the matrix rule : $\frac{\partial}{\partial
A}|A|=|A|(A^{-1})^{T};$ with $A=\boldsymbol{\Sigma}_{k}^{(z)}$, and note
$\boldsymbol{\Sigma}_{k}^{(z)}$ is symmetric . For the third part, the partial
derivative of
$\frac{1}{2}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})$
w.r.t. $\boldsymbol{\Sigma}_{k}^{(z)}$ can be computed using the matrix rule:
$\frac{\partial}{\partial\mathbf{A}}(\mathbf{x^{T}A^{-1}x})=\mathbf{-A^{-1}xx^{T}A^{-1}},$
considering,
$\mathbf{x}=\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)};\mathbf{A}=\boldsymbol{\Sigma}_{k}^{(z)}$,
which is given by

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\Big{\\{}-\frac{1}{2}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})\Big{\\}}$
$=\frac{1}{2}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}.$

Therefore, we have

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln\Big{(}\frac{e^{-\lambda}\lambda^{k}}{k!}\mathcal{N}(\mathbf{z}_{n};\boldsymbol{\mu}_{k}^{(z)},\mathbf{\boldsymbol{\Sigma}_{k}^{(z)}})\Big{)}\\\
=-\frac{1}{2}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}+\frac{1}{2}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}.$

Hence, the final expression of the partial derivative of log-likelihood w.r.t.
$\boldsymbol{\Sigma}_{k}^{(z)}$ by

$\frac{\partial}{\partial\boldsymbol{\Sigma}_{k}^{(z)}}\ln
f_{\boldsymbol{Z}}(\mathbf{z};w,\boldsymbol{\mu},\boldsymbol{\Sigma})=\sum_{n=1}^{N}\gamma(t_{n,k})\Bigg{\\{}-\frac{1}{2}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}\\\
+\frac{1}{2}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})(\mathbf{z}_{n}-\boldsymbol{\mu}_{k}^{(z)})^{T}{\boldsymbol{\Sigma}_{k}^{(z)}}^{-1}\Bigg{\\}}.$
(20)

Finally, maximizing log-likelihood w.r.t. $\boldsymbol{\Sigma}_{k}^{(z)}$ by
setting the derivative zero gives

$\sum_{n=1}^{N}\gamma(t_{n,k})\\{\boldsymbol{\Sigma}_{k}^{(z)}\\}^{-1}\\\
\left[(\mathbf{z}_{n}-\mu_{k}^{(z)})(\mathbf{z}_{n}-\mu_{k}^{(z)})^{T}\\{\boldsymbol{\Sigma}_{k}^{(z)}\\}^{-1}-1\right]=0.$
(21)

Therefore we have the expression for $\boldsymbol{\Sigma}_{k}^{(z)}$ depending
on the responsibility $\gamma$ as,

$\boldsymbol{\Sigma}_{k}^{(z)}=\frac{\sum_{n=1}^{N}\gamma(t_{n,k})(\mathbf{z}_{n}-\mu_{k}^{(z)})(\mathbf{z}_{n}-\mu_{k}^{(z)})^{T}}{\sum_{i=1}^{N}\gamma(t_{n,k})}$

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|  Mouli Chakraborty is a doctoral researcher at Trinity College Dublin,
affiliated with the Science Foundation Ireland Centre for Research Training in
Advanced Networks for Sustainable Societies (SFI CRT ADVANCE). She began her
academic journey with a B.Sc. (Honours) in Mathematics from St. Xavier’s
College, Calcutta, in 2016. This led her to earn an M.Sc. in Mathematics with
distinction from the Ramakrishna Mission Vivekananda Educational and Research
Institute, India, in 2019. During her academic pursuits, Mouli contributed as
a junior researcher at St. Xavier’s College, Calcutta (2015-2016), delved into
postgraduate research at Jadavpur University, Kolkata (2016-2017), and served
as a research assistant at RKMVERI (2017-2019). Before her current role at
Trinity College Dublin, she enriched her experience as a Visiting Researcher
at the International Centre for Theoretical Sciences (ICTS) of the Tata
Institute of Fundamental Research in Bengaluru, India. M. Chakraborty’s
research portfolio spans Mathematical aspects of Quantum Communication,
Quantum Information Processing, Quantum Computing, Machine Learning, Data
Analytics, and the emerging field of Quantum Machine Learning. In recognition
of her work, she was honored with the IEEE Antennas and Propagation Society
Graduate Fellowship Program- Quantum Technologies Initiative 2021.
Furthermore, she proudly represents the academic community as a student member
of IEEE.
---|---
|  Anshu Mukherjee is currently an Assistant Professor in the School of
Electrical and Electronic Engineering at the University College Dublin (UCD),
Ireland. He received a Bachelor of Technology (B.Tech.) degree in Information
and Telecommunication Engineering from the SRM Institute of Science and
Technology, Chennai, India 2018. Venturing further into academia, Dr.
Mukherjee pursued research at the esteemed Indian Institute of Technology
(IIT) Patna. Continuing his pursuit of knowledge, he earned his Ph.D. from UCD
in 2023. His research interests include wireless security, physical layer
security, energy efficiency, and the application of optimization and machine
learning in wireless communication, with a keen interest in quantum
communication. Dr. Mukherjee is a member of the Institute of Electronics and
Electrical Engineers (IEEE).
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|  Ioannis Krikidis (Fellow, IEEE) received the Diploma degree in computer
engineering from the Computer Engineering and Informatics Department,
University of Patras, Greece, in 2000, and the M.Sc. and Ph.D. degrees in
electrical engineering from the Ecole Nationale Superieure des
Telecommunications (ENST), Paris, France, in 2001 and 2005, respectively. From
2006 to 2007, he worked as a Postdoctoral Researcher with ENST, Paris, France.
From 2007 to 2010, he was a Research Fellow with the School of Engineering and
Electronics, University of Edinburgh, Edinburgh, U.K. He is currently an
Associate Professor with the Department of Electrical and Computer
Engineering, University of Cyprus, Nicosia, Cyprus. His current research
interests include wireless communications, cooperative networks, 6G
communication systems, wireless powered communications, and intelligent
reflecting surfaces. He was the recipient of the Young Researcher Award from
the Research Promotion Foundation, Cyprus, in 2013, and the recipient of the
IEEE ComSoc Best Young Professional Award in Academia, 2016, and the IEEE
SIGNAL PROCESSING LETTERS Best Paper Award 2019. He has received the
prestigious ERC Consolidator Grant. He serves as an Associate Editor for IEEE
TRANSACTIONS ON WIRELESS COMMUNICATIONS and a Senior Editor for IEEE WIRELESS
COMMUNICATIONS LETTERS. He has been recognized by the Web of Science as a
Highly Cited Researcher from 2017 to 2021.
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|  Avishek Nag is currently an Assistant Professor in the School of Computer
Science at University College Dublin in Ireland. Dr. Nag received the BE
(Honours) degree from Jadavpur University, Kolkata, India, in 2005, the MTech
degree from the Indian Institute of Technology, Kharagpur, India, in 2007, and
the PhD degree from the University of California, Davis in 2012. He worked as
a research associate at the CONNECT Centre for Future Networks and
Communication at Trinity College Dublin before joining University College
Dublin. His research interests include but are not limited to Cross-layer
optimization in Wired and Wireless Networks, Network Reliability, Mathematics
of Networks (Optimisation, Graph Theory), Network Virtualisation, Software-
Defined Networks, Machine Learning, Data Analytics, Quantum Key Distribution,
and Internet of Things. Dr Nag is a senior member of the Institute of
Electronics and Electrical Engineers (IEEE) and also the outreach lead for
Ireland for the IEEE UK and Ireland Blockchain Group.
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|  Subhash Chandra is an Assistant Professor in the School of Natural Sciences
at Trinity College Dublin, Ireland. Dr. Chandra received an MTech degree in
Physics in 2008 from the Indian Institute of Technology, Kharagpur, India, and
a PhD in physics from the Dublin Institute of Technology in 2013. He worked as
a postdoctoral researcher at the Centre for Industrial and Engineering Optics
(IEO), Dublin Institute of Technology. Then, worked as a research fellow at
the School of Engineering before becoming an assistant professor at the School
of Natural Sciences at Trinity College Dublin. His research interests span
across quantum physics, optics, and computing. Research focuses on optics,
quantum dots, quantum physics, plasmonic, Monte Carlo Ray Tracing simulation,
data analytics, and more, particularly designing light management technology
for photovoltaic devices and daylighting. He is developing light management
software for optimising Luminescent Solar Devices to maximise solar energy
harvesting
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