# Characterization of Quantumness of non-Gaussian states under the influence
of Gaussian channel

Ramniwas Meenaa, Subhashish Banerjeeb
a,bDepartment of Physics, Indian Institute of Technology, Jodhpur,
India-342030
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(December 2022)

###### Abstract

The impact of a noisy Gaussian channel on a wide range of non-Gaussian input
states is studied in this work. The nonclassical nature of the states, both
input and output, is developed by studying the corresponding photon statistics
and quasi-probability distributions. It is found that photon addition has more
robust quantum mechanical properties as compared to the photon subtraction
case. The threshold value of the noise parameter corresponding to the
transition from partial negative ($W$ and $P$) and zero ($Q$) to completely
positive definite, at the center of phase space, depends not only on the
average number of thermal photons in the state, but also on the squeezing
parameter. In addition it is observed that the nonclassicality of the $k^{th}$
number filtrated thermal state could be further enhanced by adding photon(s).

Keywords— Non-Gaussian state, Gaussian Channel, Nonclassicality

##  1 Introduction

Quantum information is an application of quantum mechanics to information
theory, arising from its fundamental properties [1, 2]. It exploits concepts
such as state superposition, entanglement, and wave function collapse to
establish new paradigms in the field of information processing, for example,
for computing applications [3], cryptography [4], and simulation of quantum
phenomena [5]. To harness the quantum system behaviour it is necessary to
focus on quantum state engineering [6, 7]. Quantum engineering advances these
concepts by implementing them in a realistic scenario, as well as developing
algorithms, protocols, devices and systems.

The existence of quantum systems in complete isolation is a very rare
occurrence. Generally, the dynamics is that of an open system where the system
of interest evolves under the influence of interactions with its surroundings
[8, 9]. It is essential to understand how the environment affects quantum
systems, in order to investigate their properties. The idea of open quantum
system plays an important role in quantum state engineering. States
engineered, both Gaussian and non-Gaussian, are impacted by their environment
which could be modelled by (non-)Gaussian channels [10, 11]. These play an
important role in the physics of continuous-variable quantum systems. Here,
Gaussian channels have attracted considerable attention [12]. A study of
Gaussian channels has been conducted to study their impact on the evolution of
Gaussian states in the physics of continuous-variable quantum systems [12].
Over the past decade, a number of studies have been carried out on Gaussian
states, both from theoretical as well as experimental perspectives.
Nevertheless, many quantum technologies [13] beyond the realm of Gaussian
states, demand non-Gaussian elements to be introduced.

It should be noted that while Gaussian states have been a great success both
experimentally and theoretically [14, 15, 16], they have a significant
disadvantage when it comes to quantum technology: all Gaussian measurements
can be efficiently simulated [17]. A non-Gaussian operation was argued to be
necessary for the implementation of a universal quantum computer in pioneering
work on CV quantum computation [18]. It is particularly difficult to implement
common schemes based on the cubic phase gate in realistic setups [19, 20, 21,
22]. In addition, the information encoded by these protocols is highly non-
Gaussian in nature, such as the Gottesman-Kitaev-Preskill state [23]. However,
these states are quite difficult to create even though they could also be used
to implement non-Gaussian gates [24]. Although non-Gaussian states present
practical difficulties, in the CV regime one must venture into non-Gaussian
territory to achieve a quantum computational advantage [25]. In this sense, it
is important to have a general understanding of the properties of non-Gaussian
states and the way they behave.
In recent years, many proposals have been put forward to generate highly
nonclassical quantum states by appropriate operators acting on light fields
[26, 27, 28, 29, 30, 31, 32, 33, 34]. With the advent of quantum state
engineering [35, 36, 37, 38] and quantum information processing ([39] and
references therein), the study of non-classical properties of engineered
quantum states have become a prominent ﬁeld [40]. This is so because only the
presence of nonclassical features in a quantum state can provide quantum
supremacy. In the recent past, various techniques for quantum state
engineering have been developed [35, 36, 37, 38, 41, 42]. There are many
methods of quantum state construction. Prominent among these are the use of
the state superposition principle of quantum mechanics and operators acting on
the light field state. An important class of non-Gaussian states, the photon-
added thermal state (PATS), which does not exhibit squeezing, was introduced
in [43]. Multi-photon schemes can be realized in a laboratory, as the initial
thermal (even coherent) fields contain a very small number of photons. The
non-classicality of a single-photon-subtracted Gaussian state as well as a
photon-added-then-subtracted thermal state was investigated in [44]. Photon
subtraction and addition represented by bosonic annihilation and creation
operators $\hat{a}$ and $\hat{a}^{{\dagger}}$, respectively, have been
employed to transform a field state to a desired one. Focusing only on optics,
these methods rely mainly on the use of beam splitters and detectors, as well
as postselection measurements. Such techniques are useful in creating holes in
the photon number distribution [45] and in generating finite-dimensional
quantum states [37], both of which are non-classical. They are also useful in
realizing non-Gaussianity inducing operations, like photon addition and
subtraction [46, 47]. Recently, nonclassical properties of photon-added and
subtracted displaced Fock states were studied using various witnesses of
lower- and higher-order nonclassicality [48, 49]. Hole burning operations are
also very relevant, since the states investigated are extremely nonclassical
when quantified through a nonclassicality measure [50, 51]. The decoherence of
photon-added thermal, photon-added squeezed thermal states and photon-
subtraction squeezed thermal states have been studied in a photon-loss channel
in terms of negativity of Wigner function [52, 53, 54]. However, the
construction of photons filtered from the thermal state and subsequent photon
addition on it and the investigation of its quantum properties have not yet
been reported.

In this work, our aim is to study the impact of a Gaussian channel on various
non-Gaussian states viz. photon added thermal state (PATS), photon-subtracted
thermal state (PSTS), photon-added $k^{th}$-number filtered thermal state
(PAKFTS), photon-added squeezed thermal state (PASTS) and photon-subtracted
squeezed thermal state (PSSTS). By a comparison of input and output states for
various parameters of nonclassicality witness, for example, sub-Poissonian
photon statistics, Mandel $Q_{M}$ parameter, second ordered correlation
function $g^{2}$, zeros of $Q$ function, negative region of Wigner $W$ and
Sudarshan-Glauber $P$ functions, the effect of noise on non-classicality or
decoherence is brought out.

Making use of the technique of integration within an ordered product (IWOP) of
operators [55], quantum operators in optical fields are arranged into products
in a unified manner (normal ordering, antinormal ordering, Weyl (symmetric)
ordering). This technique was proposed to utilize the power of Dirac’s
symbolic method [56] and representation theory. An ordered product symbol is
constructed by arranging noncommutative operators in a way that enables them
to commute by the IWOP technique. However, the nature of the operators remains
unchanged, they are still $q$-numbers, instead of $c$-numbers. After the
integration over $c$-numbers within an ordered product is performed, one can
get rid of the normal ordering symbol.

The paper is organized as follows. The input non-Gaussian states are
introduced in Sec. II, followed by an introduction of bosonic Gaussian channel
in Sec. III. The resulting output states, after passing through the Gaussian
channel, are then discussed. In Sec. V, we describe the phase space
distribution for the input and output states. The photon statistics of the
input and output states are studied in Section VI. We then make our
conclusions.

##  2 Input State

Here we introduce the input Gaussian states to be subsequently passed through
the Gaussian quantum channel, discussed in Sec. III. As examples of Gaussian
states we consider the thermal and squeezed thermal states and apply
generalized $m$-photon addition/subtraction to the thermal and squeezed
thermal states, to generate the corresponding non-Gaussian states. We also
discuss hole burning in the thermal state.

###  2.1 Thermal State

The Hamiltonian of a single mode harmonic oscillator in thermal equilibrium is
$\hat{H}={\hat{a}^{\dagger}\hat{a}\hbar\omega}$ and the state is described by
the density operator $\hat{\rho}_{th}$ [57, 58],

$\displaystyle\hat{\rho}_{th}={\left(1-e^{-\beta\hbar\omega}\right)}e^{-\beta\hbar\omega\hat{n}},\>\beta=\frac{1}{k_{B}T},$
(2.1)

where, $\omega$ represents the frequency of a single mode in thermal
equilibrium at temperature $T$, $k_{B}$ is the Boltzmann constant, and
$\hat{n}=\hat{a}^{\dagger}\hat{a}$, the number operator.
The Eq. (2.1) can be written as follows in normal order:

$\displaystyle\hat{\rho}_{th}=A:e^{-A{\hat{a}^{{\dagger}}\hat{a}}}:,\>A=\left(1-e^{-\beta\hbar\omega}\right)=\frac{1}{1+n_{\text{th}}},$
(2.2)

where, $n_{th}=\left[\exp(\beta\hbar\omega)-1\right]^{-1}$ is the average
photon number in the thermal state and $:\hat{O}:$ is indicative of normal
ordering of operator $\hat{O}$.

##### A. Photon Added Thermal State

The $m$-photon-added scheme, denoted by the mapping
$\hat{\rho}\rightarrow\hat{a}^{{\dagger}m}\hat{\rho}\hat{a}^{m}$, was first
proposed by Agarwal and Tara [43]. Theoretically, the photon added thermal
state (PATS) can be obtained by repeatedly operating the photon creation
operator $\hat{a}^{\dagger}$ on the thermal state. Thus, the density operator
for photon added thermal state $\hat{\rho}_{PATS}$ is given by,

$\displaystyle\hat{\rho}_{PATS}=\frac{\hat{a}^{{\dagger}m}\hat{\rho}_{th}\hat{a}^{m}}{Tr\left[\hat{a}^{{\dagger}m}\rho_{th}\left(\hat{a}\right)^{m}\right]}=N_{m}^{-1}:\hat{a}^{{\dagger}m}e^{-A\hat{a}^{\dagger}\hat{a}}\hat{a}^{m}:.$
(2.3)

Here, $m$ is positive integer number and $N_{m}$ is the normalization constant
which is determined in Appendix [A].

##### B. Photon Subtracted Thermal State

The photon subtracted thermal State (PSTS) can be obtained by repeatedly
operating the photon annihilator operator $\hat{a}$ on the thermal state.
Hence the density operator for the photon subtracted thermal state
$\hat{\rho}_{PSTS}$ is,

$\displaystyle\hat{\rho}_{PSTS}=\frac{\hat{a}^{m}\hat{\rho}_{th}\hat{a}^{{\dagger}m}}{Tr[\hat{a}^{m}\hat{\rho}_{th}\hat{a}^{{\dagger}m}]}.$

Any density operator $\hat{\rho}$ in terms of the coherent state, can be
written as,

$\hat{\rho}=\int{P(\alpha)|\alpha\rangle\langle\alpha|\frac{d^{2}\alpha}{\pi}},$

where $P(\alpha)$ is a weight function sometimes known as the Glauber-
Sudarshan $P$-function [59]. The $P$-function is a prominent quasi-probability
distribution [60]. The Eq. (2.1) can be written in terms of coherent state, as

$\displaystyle\hat{\rho}_{th}=\frac{1}{n_{\text{th}}}\int{\frac{d^{2}\alpha}{\pi}e^{-\frac{|\alpha|^{2}}{n_{\text{th}}}}|\alpha\rangle\langle\alpha|}.$
(2.4)

Using Eq. (2.4), we get following density operator $\hat{\rho}_{PSTS}$ for the
photon subtraction thermal state,

$\hat{\rho}_{PSTS}=N_{m^{-}}^{-1}\hat{a}^{m}\int{\frac{d^{2}\alpha}{\pi}e^{-\frac{|\alpha|^{2}}{n_{\text{th}}}}|\alpha\rangle\langle\alpha|}\hat{a}^{{\dagger}m}=N_{m^{-}}^{-1}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m}}e^{-\frac{|\alpha|^{2}}{n_{\text{th}}}}|\alpha\rangle\langle\alpha|},$
(2.5)

where, $m$ is positive integer number represent subtracted photons and
$N_{m^{-}}$ is the normalization constant which is determined in Appendix [A].

##### C. Photon Added Hole Burning Thermal State

In terms of Fock basis, the thermal state can be described as:

$\displaystyle\hat{\rho}_{th}=\left(1-e^{-\beta\hbar\omega}\right)\sum_{n=0}{e^{-\beta\hbar\omega\hat{n}}\left|n\right\rangle\left\langle
n\right|}=\left(1-e^{-\beta\hbar\omega}\right)\sum_{n=0}{e^{-\beta\hbar\omega
n}\left|n\right\rangle\left\langle n\right|}.$ (2.6)

There are a number of ways in which hole burning can be done for given states.
It is possible to determine the hole burning state at a particular position
(for example, the $k^{th}$ number state in the photon number distribution) by
filtering the $k^{th}$-number state from the thermal state. This hole burning
state, is also known as $k^{th}$-number filtered thermal state (KFTS).
In the KFTS, the density operator $\hat{\rho}_{KFTS}$ can be expressed as:

$\displaystyle\hat{\rho}_{\text{KFTS}}=\left(1-e^{-\beta\hbar\omega}\right)\sum_{n=0,n\neq\
k}{e^{-\beta\hbar\omega n}\left|n\right\rangle\left\langle
n\right|}=\left(1-e^{-\beta\hbar\omega}\right)\bigg{[}\sum_{n=0}{e^{-\beta\hbar\omega
n}\left|n\right\rangle\left\langle n\right|}-e^{-\beta\hbar\omega
k}\left|k\right\rangle\left\langle k\right|\bigg{]}.$

With the use of IWOP techniques, the above equation can be written as follows:

$\displaystyle\hat{\rho}_{KFTS}$
$\displaystyle=A\left[:e^{-A\hat{a}^{{\dagger}}\hat{a}}:-\>\frac{e^{-\beta\hbar\omega
k}}{k!}:{\hat{a}^{{\dagger}k}}e^{-\hat{a}^{{\dagger}}\hat{a}}{\hat{a}}^{k}:\right],$
(2.7)

where, $A=\left(1-e^{-\beta\hbar\omega}\right)=\frac{1}{1+n_{\text{th}}}.\\\ $

Theoretically, photon added KFTS can be obtained by operating the creation
operator on the $k^{th}$-number state filtration of thermal state. The density
operator for photon added $k^{th}$-number filtered thermal state (PAKFTS) can
be expressed as follows:

$\displaystyle\hat{\rho}_{\text{PAKFTS}}=N_{\text{km}}^{-1}\left[:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:-\>{\frac{e^{-\beta\hbar\omega
k}}{k!}:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:}\right].$
(2.8)

Here, $N_{km}$ is the normalization constant which is determined in Appendix
[A].

###  2.2 Squeezed Thermal State

The squeezed thermal state (STS) can be obtained from the thermal state by
applying the single mode squeezing operator,

$\displaystyle\hat{\rho}_{s}={S_{1}}^{{\dagger}}\hat{\rho}_{th}S_{1},$

where $S_{1}=\frac{1}{\sqrt{\mu}}\int
dq\left|\left.\frac{q}{\mu}\right\rangle\right.\left\langle\left.q\right|\right.$;
$\mu=e^{\lambda}$ is the single-mode squeezing operator with ${\lambda}$ being
the squeezing parameter [61, 57, 62].
The above method enables one to obtain the normally ordered form of STS
directly [63],

$\displaystyle\hat{\rho}_{s}=\frac{1}{\sqrt{A}}:\exp{\left[\frac{C}{2}(\hat{a}^{{\dagger}2}+\hat{a}^{2})+(B-1)\hat{a}^{{\dagger}}\hat{a}\right]}:,$
(2.9)

where we have set,

$A=n_{th}^{2}+(2n_{th}+1){\cosh}^{2}\lambda$, $B=\frac{n_{th}}{A}(n_{th}+1)$,
$C=\frac{2n_{th}+1}{2A}\sinh 2\lambda$.

##### A. Photon Added Squeezed Thermal State

The photon added squeezed thermal state (PASTS) can be obtained by repeatedly
applying creator operators to squeezed thermal states. From Eq. (2.9) a photon
added squeezed thermal state $\hat{\rho}_{PASTS}$ can be obtained as:

$\displaystyle\hat{\rho}_{PASTS}=\frac{N_{a,m}^{-1}}{\sqrt{A}}:\hat{a}^{{\dagger}m}\exp{\left[\frac{C}{2}(\hat{a}^{{\dagger}2}+{\hat{a}}^{2})+(B-1){\hat{a}}^{\dagger}\hat{a}\right]}\hat{a}^{m}:.$
(2.10)

Here, $m$ is a non-negative number and $N_{a,m}$ is the normalization that is
calculated in Appendix [A].

##### B. Photon Subtracted Squeezed Thermal State (PSSTS)

Additionally, we introduce the photon subtracted thermal state (PSSTS), which
can be generated by repeatedly applying the annihilator operator to squeezed
thermal states.
From Eq. (2.9) the photon added squeezed thermal state $\hat{\rho}_{PASTS}$
can be obtained as:

$\displaystyle\hat{\rho}_{PSSTS}=\frac{\hat{a}^{m}\hat{\rho}_{s}\hat{a}^{{\dagger}m}}{Tr[\hat{a}^{m}\hat{\rho}_{s}\hat{a}^{{\dagger}m}]}=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\hat{a}^{m}:\exp{\left[\frac{C}{2}(\hat{a}^{{\dagger}2}+{\hat{a}}^{2})+(B-1){\hat{a}}^{\dagger}\hat{a}\right]}:\hat{a}^{{\dagger}m}.$
(2.11)

Using the over-completeness relation for the coherent states, expressed as an
integral over the complex $\alpha$-plane, as

$\int{\frac{d^{2}\alpha}{\pi}|\alpha\rangle\langle\alpha|}=1$

in Eq. (2.11), we have:

$\displaystyle\hat{\rho}_{PSSTS}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\hat{a}^{m}|\alpha\rangle\langle\alpha|:\exp{\left[\frac{C}{2}(\hat{a}^{{\dagger}2}+{\hat{a}}^{2})+(B-1){\hat{a}}^{\dagger}\hat{a}\right]}:|\beta\rangle\langle\beta|\hat{a}^{{\dagger}m}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}|\alpha\rangle\langle\beta|.$
(2.12)

$N_{a,m^{-}}$ is the normalization calculated in Appendix [A].

##  3 Bosonic Gaussian Channels

Bosonic Gaussian channels are the results of zero-mean random Gaussian shifts
in the phase space of bosonic modes, which provide a completely positive map
(CPM) on the phase space [64]. By applying this CPM to an input state
$/hat[/rho]$, the output state $/mathrm{/Phi}_{s}/left(/hat{/rho}/right)$ may
be obtained:

$\displaystyle\mathrm{\Phi}_{s}\left(\hat{\rho}\right)=\int{\hat{D}_{z}\hat{\rho}\
{\hat{D}_{z}}^{\dagger}}G\left(z\right)d^{2}z.$ (3.1)

Here $\hat{D}_{z}=e^{z\hat{a}^{\dagger}-z^{\ast}\hat{a}}$ is the displacement
operator in the phase space of the bosonic mode, where the bosonic creation
and annihilation operators obey the commutation relation
$[\hat{a},\hat{a}^{\dagger}]=1$, and the classical-noise Gaussian
distribution, $G\left(z\right)=\frac{e^{-|z|^{2}/s}}{\pi s}$, has zero mean
and phase-space variance $\sigma^{2}=s$. The complex number
$\alpha=(p+iq)/\sqrt{2}$ corresponds to the phase-space points of a single-
mode harmonic oscillator described by position $q$ and momentum $p$.

Figure 1: Noisy Gaussian Channel

It is also possible to write the Eq. (3.1) in terms of Kraus operators as:

$\displaystyle\mathrm{\Phi}_{s}\left(\hat{\rho}\right)=\sum_{i}\hat{K}_{i}\rho{\hat{K}_{i}}^{\dagger},$
(3.2)

where, the Kraus operators $\hat{K}_{i}=\sqrt{G\left(z\right)\ }\hat{D}(z)$.
Here, the Gaussian channel $\phi_{s}(\rho)$ satisfies the semigroup property
with respect to the parameter $s$ (with $s\geq 0$). Hence $s$ can be used to
parameterize the evolution time [65].

##  4 Output State

As the input state passes through the noisy Gaussian channel, it gets
influenced by the channel resulting in the corresponding output state at the
other end, see Fig. (1). We now provide analytic expressions for various
output states corresponding to the input states. This is made possible by the
use of IWOP techniques.

###  4.1 Thermal State

##### A. Photon Added Thermal State

From Eq. (2.3), we have,

$\displaystyle\hat{D}_{z}\hat{\rho}_{PATS}{\hat{D}_{z}}^{\dagger}=N_{m}^{-1}:{{({\hat{a}}^{\dagger}-z^{\ast})}^{m}\exp{\bigg{[}-A({\hat{a}}^{\dagger}-z^{\ast}){(\hat{a}-z)}\bigg{]}}{(\hat{a}-z)}^{m}}:.$
(4.1)

Here, we use the following properties of displacement operator for normal
ordered function $F^{n}(\hat{a},\hat{a}^{{\dagger}})$-

$\displaystyle\hat{D_{z}}F^{n}(\hat{a},\hat{a}^{{\dagger}})\hat{{D_{z}}}^{\dagger}=F^{n}(\hat{a}-z,\hat{a}^{{\dagger}}-z^{\ast}).$
(4.2)

From Eqs. (3.1) and (4.1), shifting, $\left(\hat{a}-z\right)\rightarrow-z$,

$\displaystyle\mathrm{\Phi}_{s}\left(\hat{\rho}_{PATS}\right)=N_{m}^{-1}\int{\frac{d^{2}z}{\pi
s}:\left(z^{\ast}z\right)^{m}\exp\bigg{[}-\left(A+1/s\right)|z|^{2}+\frac{{\hat{a}}^{\dagger}z}{s}+\frac{\hat{a}z^{\ast}}{s}-\frac{{\hat{a}}^{\dagger}\hat{a}}{s}\bigg{]}}:.$

$\displaystyle\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
n}\exp[A|z|^{2}+Bz+Cz^{\ast}]=e^{-BC/A}\sum_{l=0}{\frac{n!m!}{l!(n-l)!(m-l)!(-A)^{n+m-l+1}}B^{m-l}C^{n-l}},$
(4.3) $Re(A)<0$.
$\displaystyle\int{\frac{d^{2}z}{\pi}\exp{\big{[}\zeta|z|^{2}+\xi z+\eta
z^{\ast}+fz^{2}+gz^{\ast
2}}\big{]}}=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp{\left[\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right]}$
(4.4) whose convergent condition is $Re(\zeta\mp f\mp g)<0$ and
$Re\left(\frac{\zeta^{2}-4fg}{\zeta\mp f\mp g}\right)<0$.

Using Eq. (4.3), along with the IWOP technique, we get the following output
state for PATS

$\displaystyle\mathrm{\Phi}_{s}\left(\hat{\rho}_{PATS}\right)=N_{m}^{-1}:\exp^{-\frac{A{\hat{a}}^{\dagger}\hat{a}}{As+1}}\sum_{l=0}^{m}\frac{{m!}^{2}\left({\hat{a}}^{\dagger}\right)^{m-l}\left(\hat{a}\right)^{m-l}s^{l}}{l!{\left(m-l\right)!)}^{2}\left(As+1\right)^{2m-l+1}}:.$
(4.5)

##### B. PSTS

For photon subtracted thermal state from Eq. (2.5),

$\displaystyle\hat{D}_{z}\hat{\rho}_{PSTS}{\hat{D}_{z}}^{\dagger}=N_{m^{-}}^{-1}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m}}e^{-\frac{|\alpha|^{2}}{n_{th}}}\hat{D}_{z}|\alpha\rangle\langle\alpha|{\hat{D}_{z}}^{\dagger}}.$

Using the following property for the displacement operator,

$\displaystyle\hat{D}_{z}|\alpha\rangle=\exp{\bigg{[}\frac{1}{2}(z\alpha^{\ast}-z^{\ast}\alpha)\bigg{]}}|{\alpha+z}\rangle,$
(4.6)

the output state for PSTS can be written as (using Eqs. (2.5), (4.6) and
(3.1)),

$\displaystyle{\Phi}_{s}\left(\hat{\rho}\right)=N_{m^{-}}^{-1}\int\int{\frac{d^{2}\alpha}{\pi}\frac{d^{2}z}{\pi
s}{(\alpha^{\ast}\alpha)^{m}}\exp{\bigg{[}-\frac{|\alpha|^{2}}{n_{th}}-\frac{|z|^{2}}{s}\bigg{]}}|{\alpha+z}\rangle\langle{\alpha+z}|}.$
(4.7)

##### C. PAKFTS

From Eq. (2.8), we have,

${\hat{D}}_{z}^{{\dagger}}\hat{\rho}_{\text{KFTS}}{\hat{D}}_{z}^{{\dagger}}=N_{\text{km}}^{-1}\left({\hat{D}}_{z}:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:{\hat{D}}_{z}^{{\dagger}}-\frac{e^{-\beta\hbar\omega
k}}{k!}{\hat{D}}_{z}:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:{\hat{D}}_{z}^{{\dagger}}\right).$
(4.8)

From Eqs. (4.2), (2.8) and (4.8), shifting,
$\left(\hat{a}-z\right)\rightarrow-z$,

$\displaystyle\phi_{s}{\left(\hat{\rho}_{\text{PAKFTS}}\right)}=N_{\text{km}}^{-1}\int{\left(:(z^{\ast}z)^{m}e^{-A|z|^{2}+{\frac{(\hat{a}^{\dagger}-z\ast)(\hat{a}-z)}{s}}}:-\>\frac{e^{-\beta\hbar\omega
k}}{k!}:(z^{\ast}z)^{m+k}e^{-|z|^{2}+{\frac{(\hat{a}^{\dagger}-z\ast)(\hat{a}-z)}{s}}}:\right)\frac{d^{2}z}{\pi
s}}.$

Using Eq. (4.4), the output state for PAKFTS can be written as

$\phi_{s}{\left(\hat{\rho}_{\text{PAKFTS}}\right)}=N_{\text{km}}^{-1}\bigg{[}:e^{-\frac{A\hat{a}^{{\dagger}}\hat{a}}{As+1}}\sum_{l=0}^{m}\frac{m!^{2}\hat{a}^{{\dagger}m-l}\hat{a}^{m-l}s^{l}}{l!(m-l)!^{2}(As+1)^{2m-l+1}}:\\\
-\>\frac{e^{-\beta\hbar\omega
k}}{k!}:e^{-\frac{\hat{a}^{{\dagger}}\hat{a}}{s+1}}\sum_{l=0}^{m+k}\frac{(m+k)!^{2}\hat{a}^{{\dagger}m+k-l}\hat{a}^{m+k-l}s^{l}}{l!(m+k-l)!^{2}(s+1)^{2(m+k)-l+1}}:\bigg{]}.$
(4.9)

###  4.2 For Squeezed Thermal State

##### A. Noisy PASTS

For PASTS, we have (using Eqs. (4.2) and (2.10)),

$\displaystyle\hat{D_{z}}\hat{\rho}_{PASTS}\hat{{D_{z}}}^{\dagger}=\frac{N_{a,m}^{-1}}{\sqrt{A}}:\left({\hat{a}}^{\dagger}-z^{\ast}\right)^{m}\exp\left[\frac{C}{2}(({{\hat{a}}^{\dagger}-z^{\ast})}^{2}+\left(\hat{a}-z\right)^{2})+(B-1)({\hat{a}}^{\dagger}-z^{\ast})(\hat{a}-z)\right]{(\hat{a}-z)}^{m}:.$
(4.10)

From Eqs. (3.1) and (4.10),

$\displaystyle{\Phi}_{s}\left(\hat{\rho}_{PASTS}\right)$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int{\frac{d^{2}z}{\pi\
s}:{({\hat{a}}^{\dagger}-z^{\ast})}^{m}\exp{\left[\frac{C}{2}(({{\hat{a}}^{\dagger}-z^{\ast})}^{2}+{(\hat{a}-z)}^{2})+(B-1)({\hat{a}}^{\dagger}-z^{\ast})(\hat{a}-z)-\frac{\left|z\right|^{2}}{s}\right]}{(\hat{a}-z)}^{m}:}.$

The advantage of using the IWOP technique is that the operators $\hat{a}$ and
$\hat{a}^{{\dagger}}$ can be regraded as parameters in the integrations and
can be permuted, allowing for the shifting $(z-\hat{a}\longrightarrow z)$,

$\displaystyle{\Phi}_{s}\left(\hat{\rho}_{PASTS}\right)$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int{\frac{d^{2}z}{\pi
s}:{(z^{\ast}z)}^{m}\exp\left[\frac{C}{2}({z^{\ast}}^{2}+z^{2})+\left(B-1-1/s\right)|z|^{2}+\frac{{\hat{a}}^{\dagger}z}{s}+\frac{\hat{a}z^{\ast}}{s}-\frac{{\hat{a}}^{\dagger}\hat{a}}{s}\right]}:$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{Y}^{m}\int{\frac{d^{2}z}{\pi
s}:{\exp\left[\frac{C}{2}({z^{\ast}}^{2}+z^{2})-Y|z|^{2}+\frac{{\hat{a}}^{\dagger}z}{s}+\frac{\hat{a}z^{\ast}}{s}-\frac{{\hat{a}}^{\dagger}\hat{a}}{s}\right]}:},$

where $Y=(-B+1+1/s)$.

Using the Eq. (4.4), we get the following output state for PASTS

$\displaystyle\mathrm{\Phi}_{s}\left(\hat{\rho}_{PASTS}\right)=\frac{N_{a,m}^{-1}}{s\sqrt{A}}:\exp\left(-\frac{{\hat{a}}^{\dagger}\hat{a}}{s}\right)\partial_{Y}^{m}\left[{(Y^{2}-C^{2})}^{-1/2}\exp\left[\frac{\frac{C}{2}({{\hat{a}}^{{\dagger}2}}+{\hat{a}}^{2})-Y{\hat{a}}^{\dagger}\hat{a}}{s^{2}(Y^{2}-C^{2})}\right]\right]:.$
(4.11)

##### B. PSSTS

From Eq. (2.12) we have,

$\displaystyle\hat{D}_{z}\hat{\rho}_{\text{PSSTS}}0{\hat{D}_{z}}^{{\dagger}}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int{\frac{d^{2}\alpha}{\pi}{\frac{d^{2}\beta}{\pi}}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}{\hat{D}_{z}}{|{\alpha}\rangle\langle{\beta}|}{{\hat{D}_{z}}^{\dagger}}}.$

From the above and Eq. (3.1), the output state for PSSTS is seen to be

$\displaystyle\phi_{s}\left(\hat{\rho}_{PSSTS}\right)=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}-\frac{|z|^{2}}{s}\right]}{\hat{D}_{z}}|{\alpha}\rangle\langle{\beta}|{\hat{D_{z}}^{{\dagger}}}.$

Using the Eq. (4.6), the above equation can be expressed as

$\displaystyle\phi_{s}(\hat{\rho}_{PSSTS})$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\frac{1}{2}\\{z(\alpha^{\ast}-\beta^{\ast})-z^{\ast}(\alpha-\beta)\\}-\frac{|\alpha|^{2}+|\beta|^{2}}{2}-\frac{|z|^{2}}{s}\right]}|{\alpha+z}\rangle\langle{\beta+z}|.$
(4.12)

##  5 Quasi-Probability Distribution

An important aspect in the quest for the understanding of the quantumness of
states, discussed above, would be the study of the corresponding quasi-
probability distributions, for example, the $W$, $P$ and $Q$ distributions.
These can be studied on a common platform by the use of Cahill and Glauber’s
$\kappa$-parameterized function [66],

$\displaystyle\chi\left(\gamma,\kappa\right)={Tr\left[\hat{\rho}\exp{\left(\gamma\hat{a}^{\dagger}-\gamma^{*}\hat{a}+\left(\kappa/2\right){|\gamma|}^{2}\right)}\right]},$
(5.1)

where $\kappa=-1,\>0,\>1$ corresponds to $Q$, $W$ and $P$ quasi-probability
distributions, respectively. $P_{\kappa}\left(\alpha\right)$ are connected to
the family of characteristics functions in CV systems through the complex
Fourier transform,

$\displaystyle
P_{\kappa}\left(\alpha\right)=\frac{1}{\pi}\int\exp\left(\gamma^{\ast}\alpha-\gamma\alpha^{\ast}\right)\chi\left(\gamma,\kappa\right)\frac{d^{2}\gamma}{\pi}.$
(5.2)

These distributions are called quasi-probability distributions, because even
though they sum up to unity, their behavior is not entirely consistent with
that expected of probability distributions. In particular, there are
(infinitely many) quantum states $\rho$ for which the function
$P_{\kappa}\left(\alpha\right)$ is not a regular probability distribution for
some values of $\kappa$, as it can assume negative values or even be singular
in certain points of the phase space [67, 68, 60, 69, 70]. An exception is the
case $\kappa=-1$, which corresponds to the Husimi ‘$Q$-function’ [71] and
represents a non-negative and regular distribution for any quantum state
$\rho$. Zeros of $Q$ function are a witness of non-classicality [72]. The case
$\kappa=0$ corresponds to the so-called Wigner ‘$W$-function’ [73]. Wigner
function’s negativity is another witness of non-classicality. The
symmetrically ordered characteristic function would thus be
$\chi\left(\gamma,0\right)$. Finally, $\kappa=1$ yields the so-called
‘$P$-representation’, which was introduced independently by Glauber and
Sudarshan [74]. The $P$ distribution can become negative or even singular
(namely, more singular than a Dirac $\delta$) when the state $\rho$ deviates
from a mixture of coherent states. For this reason, the regularity and
positivity of the $P$-representation is often adopted as an indicator of
‘classicality’ of a CV state $\rho$ [75].

We now take up the states introduced above and obtain their characteristic
function and thence the probability distribution function. This is done for
both the input as well as the output states, obtained by sending the input
state through the bosonic Gaussian channel. This enables a characterization of
quantumness in the states as well as the contrast obtained due to the effect
of the channel, which brings out the impact of noise.

###  5.1 Thermal States

##### A. Photon Added Thermal State

####  5.1.1 Input State

We begin with the characteristic function (CF),

$\displaystyle\chi\left(\gamma,\kappa\right)={Tr\left[e^{\gamma\hat{a}^{\dagger}}\hat{\rho}_{PATS}e^{-\gamma^{\ast}\hat{a}}\right]}e^{\frac{\kappa+1}{2}{|\gamma|^{2}}}.$
(5.3)

Since,

$\displaystyle{Tr\left[e^{\gamma\hat{a}^{\dagger}}\hat{\rho}_{PATS}e^{-\gamma^{\ast}\hat{a}}\right]}=\int{(\alpha^{\ast}\alpha)^{m}\exp{\left[-A|\alpha|^{2}+\gamma\alpha^{\ast}-\gamma^{\ast}\alpha\right]}\frac{d^{2}\alpha}{\pi}},$

making use of Eq. (4.3),

Figure 2: Quasi-Probability Distribution functions, viz., Husimi $Q$-function
(1st row), Wigner $W$-function (2nd row) and Sudarshan-Glauber $P$-function
(3rd row) for thermal state (1st column), one photon added-thermal state (2nd
column) and two photon-added thermal state (3rd column). Here input PATS are
represented by green colour and output PATS are represented by yellow (noise
parameter $s=0.3$) and red (noise parameter $s=1$).

$\displaystyle{Tr\left[e^{\gamma\hat{a}^{\dagger}}\hat{\rho}_{PATS}e^{-\gamma^{\ast}\hat{a}}\right]}=N_{m}^{-1}\sum_{l=0}^{m}{\frac{{m!}^{2}{({-\gamma}^{\ast}\gamma)}^{m-l}}{l!(({m-l)!)}^{2}A^{2m-l+1}}e^{-\frac{\left|\gamma\right|^{2}}{A}}}.$

Using this in Eq. (5.3), the CF for the input state is,

$\displaystyle\chi_{in}\left(\gamma,\kappa\right)$
$\displaystyle=N_{m}^{-1}\sum_{l=0}^{m}{\frac{{m!}^{2}{({-\gamma}^{\ast}\gamma)}^{m-l}}{l!(({m-l)!)}^{2}A^{2m-l+1}}e^{-\left(\frac{1}{A}-\frac{\kappa+1}{2}\right)\left|\gamma\right|^{2}}}.$
(5.4)

From Eqs. (5.4) and (5.2),

$\displaystyle
P_{\kappa}\left(\alpha\right)=\frac{N_{m}^{-1}}{\pi}\sum_{l=0}^{m}{\frac{{m!}^{2}{(-1)}^{m-l}}{l!(({m-l)!)}^{2}A^{2m-l+1}}}\int{\frac{d^{2}\gamma}{\pi}(\gamma^{\ast}\gamma)^{m-l}\exp{\left[-\left(\frac{1}{A}-\frac{\kappa+1}{2}\right)\left|\gamma\right|^{2}+\gamma^{\ast}\alpha-\gamma\alpha^{\ast}\right]}}.$

Again using Eq. (4.3), we get the following quasi-probability function for the
input state,

$\displaystyle
P_{in}\left(\kappa,\alpha\right)=\frac{N_{m}^{-1}}{\pi}\sum_{l=0}^{m}{\left[\frac{{m!}^{2}\left(-1\right)^{m-l}}{l!\>A^{2m-l+1}}\sum_{h=0}^{m-l}{\left[\frac{(-1)^{m-l-h}|\alpha|^{2(m-l-h)}}{h!(({m-l-h)!)}^{2}\left(\frac{1}{A}-\frac{\kappa+1}{2}\right)^{2m-2l-h+1}}\exp\left(\frac{-|\alpha|^{2}}{\left(\frac{1}{A}-\frac{\kappa+1}{2}\right)}\right)\right]}\right]}.$
(5.5)

####  5.1.2 Output State

Similarly, for noisy PATS at the output, we have,

$\displaystyle{Tr\left[e^{\gamma\hat{a}^{\dagger}}\phi_{s}(\hat{\rho}_{PATS})e^{-\gamma^{\ast}\hat{a}}\right]}=N_{m}^{-1}\sum_{l=0}^{m}\frac{{m!}^{2}s^{l}}{l!\left(As+1\right)^{2m-l+1}}\sum_{b=0}^{m}{\frac{{({-\gamma}^{\ast}\gamma)}^{m-l-b}}{b!\
(({m-l-b)!)}^{2}{\left(\frac{A}{As+1}\right)}^{2m-l+1}}\exp{\left[-{\frac{As+1}{A}}\left|\gamma\right|^{2}\right]}}.$

Using the above expression for the trace in Eq. (5.3), we get the following CF
for the output state,

$\displaystyle\chi_{out}\left(\gamma,\kappa\right)=N_{m}^{-1}\sum_{l=0}^{m}\frac{{m!}^{2}s^{l}}{l!\left(As+1\right)^{2m-l+1}}\sum_{b=0}^{m}{\frac{{({-\gamma}^{\ast}\gamma)}^{m-l-b}}{b!(({m-l-b)!)}^{2}{\left(\frac{A}{As+1}\right)}^{2m-l+1}}\exp{\left[-\left(\frac{As+1}{A}-\frac{\kappa+1}{2}\right)\left|\gamma\right|^{2}\right]}}.$
(5.6)

From Eqs. (5.6) and (5.2),

$P_{out}\left(\gamma,\kappa\right)=N_{m}^{-1}\sum_{l=0}^{m}\frac{{m!}^{2}s^{l}}{l!\left(As+1\right)^{2m-l+1}}\sum_{b=0}^{m}{\frac{(-1)^{m-l-h}}{b!(({m-l-b)!)}^{2}{\left(\frac{A}{As+1}\right)}^{2m-l+1}}}\\\
\times\int{{({-\gamma}^{\ast}\gamma)}^{m-l-b}\exp{\left[-\left(\frac{As+1}{A}-\frac{\kappa+1}{2}\right)\left|\gamma\right|^{2}+\gamma\alpha^{\ast}-\gamma^{\ast}\alpha\right]}\frac{d^{2}\gamma}{\pi^{2}}}.$

Using Eq. (4.3), the quasi-probability distribution for the output state is
obtained as,

$\displaystyle P_{out}\left(\alpha,\kappa\right)$
$\displaystyle=\frac{N_{m}^{-1}}{\pi}\sum_{l=0}^{m}\frac{{m!}^{2}s^{l}}{l!\left(As+1\right)^{2m-l+1}}\sum_{b=0}^{m-l}\frac{\left(-1\right)^{m-l-b}}{b!\>\left(\frac{A}{As+1}\right)^{2m-2l-b+1}}$
$\displaystyle\times\sum_{h=0}^{m-l-b}\left[\frac{(-1)^{m-l-
b-h}|\alpha|^{2(m-l-b-h)}}{h!(({m-l-
b-h)!)}^{2}\left({\frac{As+1}{A}}-\frac{\kappa+1}{2}\right)^{2m-2l-2b-h+1}}\right]\exp\left[\frac{-|\alpha|^{2}}{\left({\frac{As+1}{A}}-\frac{\kappa+1}{2}\right)}\right].$
(5.7)

In Figure (2) we have plotted quasi-probability distribution functions for
several values of $n_{th}$ (characterizing temperature $T$ of the photon), $m$
(number of photon added), and noise parameter $s$. For $\kappa=-1,\>0$ or $1$
we get the Husimi $Q$, Wigner $W$ and Sudarshan-Glauber $P$ functions,
respectively. The contrast between the input and output, after passing through
the Gaussian channel, is brought out in each sub-figure. At $m=0$ all quasi-
probability distributions are seen to be Gaussian. As photons are added to the
thermal state, the quasi-probability distribution dips at the center,
indicative of non-Gaussian behavior. For higher value of $m$ the $Q$-function
tends to zero at the center and width also increases. The $W$\- function and
$P$-function have negative regions for ($m>0$), indicative of quantumness. The
distributions become wider as we increase average photon number and noise. For
output PATS it can be seen that peak value decreases and the negative region
shifts to the positive (for the $W$ and $P$-function) region and zeros of $Q$
also tend to shift peak values. These are signatures of decoherence, heralding
the onset of classicality.

##### B. Photon Subtracted Thermal State

####  5.1.3 Input State

We begin, as before, with the characteristic function,

$\displaystyle\chi\left(\gamma,\kappa\right)={Tr\left[e^{-\gamma^{\ast}\hat{a}}\hat{\rho}_{PSTS}e^{\gamma\hat{a}^{\dagger}}\right]}e^{\frac{\kappa-1}{2}}{|\gamma|^{2}},$
(5.8)

where,

$\displaystyle{Tr\left[e^{-\gamma^{\ast}\hat{a}}\hat{\rho}_{PSTS}e^{\gamma\hat{a}^{\dagger}}\right]}=N_{m^{-}}^{-1}\sum_{l=0}^{m}{\frac{{m!}^{2}{({-\gamma}^{\ast}\gamma)}^{m-l}}{l!(({m-l)!)}^{2}(\frac{1}{n_{th}})^{2m-l+1}}e^{-n_{th}\left|\gamma\right|^{2}}}.$

Using this in Eq. (5.8), the characteristic function comes out to be,

$\displaystyle\chi_{in}\left(\gamma,\kappa\right)=N_{m^{-}}^{-1}\sum_{l=0}^{m}{\frac{{m!}^{2}{({-\gamma}^{\ast}\gamma)}^{m-l}}{l!(({m-l)!)}^{2}(\frac{1}{n_{th}})^{2m-l+1}}e^{-\left(n_{th}-\frac{(\kappa-1)}{2}\right)\left|\gamma\right|^{2}}}.$
(5.9)

From Eqs. (5.9) and (5.2),

$\displaystyle
P_{in}\left(\kappa,\alpha\right)=\frac{N_{m^{-}}^{-1}}{\pi}\sum_{l=0}^{m}{\left[\frac{{m!}^{2}\left(-1\right)^{m-l}}{l!\
(\frac{1}{n_{th}})^{2m-l+1}}\sum_{h=0}^{m-l}{\left[\frac{(-1)^{m-l-h}\left|\alpha\right|^{2(m-l-h)}}{h!\
(({m-l-h)!)}^{2}\
\left(n_{th}-\frac{\kappa-1}{2}\right)^{2m-2l-h+1}}\exp\bigg{(}\frac{-|\alpha|^{2}}{\left(n_{th}-\frac{\kappa-1}{2}\right)}\bigg{)}\right]\
}\right]}.$ (5.10)

Figure 3: Quasi-Probability Distribution functions e.g Husimi $Q$-function
(1st row), Wigner $W$-function (2nd row) and Sudarshan-Glauber $P$-function
(3rd row) for thermal state (1st column), various photon subtracted-thermal
state (2nd and 3rd columns). Input PSTS are represented by green colour and
output PSTS are represented by red (noise parameter $s=0.2$) and blue (noise
parameter $s=0.8$).

####  5.1.4 Output State

For the noisy PSTS at the output, we have

$\displaystyle{Tr\left[e^{-\gamma^{\ast}\hat{a}}{\Phi}_{s}\left(\hat{\rho}_{PSTS}\right)e^{\gamma\hat{a}^{\dagger}}\right]}$
$\displaystyle=N_{m^{-}}^{-1}\int\int{\frac{d^{2}\alpha}{\pi}\frac{d^{2}z}{\pi}{(\alpha^{\ast}\alpha)^{m}}\exp{\bigg{[}-\frac{|\alpha|^{2}}{n_{th}}-\frac{|z|^{2}}{s}+\gamma(\alpha^{\ast}+z^{\ast})-\gamma^{\ast}(\alpha+z)\bigg{]}}}$
$\displaystyle=N_{m^{-}}^{-1}\sum_{l=0}^{m}{\frac{{m!}^{2}{({-\gamma}^{\ast}\gamma)}^{m-l}}{l!(({m-l)!)}^{2}(\frac{1}{n_{th}})^{2m-l+1}}e^{-\left(n_{th}+s\right)\left|\gamma\right|^{2}}}$
(5.11)

Using the trace in Eq. (5.8), we get the following expression for the
characteristic function,

$\displaystyle\chi_{out}\left(\gamma,\kappa\right)=N_{m^{-}}^{-1}\sum_{l=0}^{m}{\frac{{m!}^{2}{({-\gamma}^{\ast}\gamma)}^{m-l}}{l!(({m-l)!)}^{2}(\frac{1}{n_{th}})^{2m-l+1}}e^{-\left(n_{th}+s-\frac{(\kappa-1)}{2}\right)\left|\gamma\right|^{2}}}.$
(5.12)

From Eqs. (5.12) and (5.2), we can get the quasi-probability distribution
function for the output state as:

$\displaystyle
P_{out}\left(\kappa,\alpha\right)=\frac{N_{m^{-}}^{-1}}{\pi}\sum_{n=0}^{m}{\left[\frac{{m!}^{2}\left(-1\right)^{m-l}}{l!(\frac{1}{n_{th}})^{2m-l+1}}\sum_{n=0}^{m-l}{\left[\frac{(-1)^{m-l-h}|\alpha|^{2(m-l-h)}}{h!(({m-l-h)!)}^{2}\left(n_{th}+s-\frac{\kappa+1}{2}\right)^{2m-2l-h+1}}\exp\left[\frac{-|\alpha|^{2}}{\left(n_{th}+s-\frac{\kappa+1}{2}\right)}\right]\right]}\right]}.$
(5.13)

In figure (3) we have plotted quasi-probability distribution functions for
several values of $n_{th}$ (characterizing temperature $T$ of the photon), $m$
(number of photon subtracted), and noise parameter $s$. For $\kappa=-1$,
$\kappa=0$ and $\kappa=1$ we get Husimi $Q$, Wigner $W$ and Sudarshan-Glauber
$P$-functions, respectively. At $m=0$ all quasi-probability distribution
functions are Gaussian but as we subtract photons from the thermal state we
get very small dip at the center. For higher values ($\leavevmode\nobreak\
m>40$, not shown here) of $m$ the $Q$-function can be observed to tend to zero
at the center, with a corresponding increase in width. The $W$ and $P$
functions also dip at the center but do not attain negative values. The dip
becomes wider as we increase the average photon number and for higher values
of $m$, one can have regions which exhibit negative values. For output PSTS it
can be seen that peak values decrease and the dip (negative for higher $m$)
region shifts upward, quicker than that for the PATS, exhibiting signs of
decoherence. With increase in noise, the states again attain their Gaussian
behavior. Hence, increase in noise facilitates the transition from non-
Gaussian to Gaussian behavior.

##### C. PAKFTS

####  5.1.5 Input State

Now, from Eq. (2.8), we have,

$\displaystyle\text{Tr}\left[e^{\gamma{\hat{a}}^{{\dagger}}}\rho
e^{-\gamma^{*}\hat{\text{\ a}}}\right]=N_{\text{km}}^{-1}\text{\
Tr}\left[{e^{\gamma{\hat{a}}^{{\dagger}}}:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:e^{-\gamma^{*}\hat{\text{\
a}}}}^{\ }\right]-\ N_{\text{km}}^{-1}\ \frac{e^{-\beta\hbar\omega
k}}{k!}\text{\
Tr}\left[{e^{\gamma{\hat{a}}^{{\dagger}}}:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:\
e^{-\gamma^{*}\hat{\text{\ a}}}}^{\ }\right].$

Using Eq. (5.3) and the above, we get the following CF,

$\chi_{in}(\gamma,\kappa)=N_{\text{km}}^{-1}\bigg{[}\exp\left\\{-\left(\frac{1}{A}-\frac{\kappa+1}{2}\right)|\gamma|^{2}\right\\}\sum_{l=0}^{m}\frac{{m!}^{2}{\left(-\gamma^{*}\gamma\right)^{\
}}^{m-l}s^{l}}{l!\>((m-l)!)^{2}A^{2m-l+1}}\\\ -\frac{e^{-\beta\hbar\omega
k}}{k!}\exp\left\\{-\left(1-\frac{\kappa+1}{2}\right)|\gamma|^{2}\right\\}\sum_{l=0}^{m+k}\frac{{(m+k)!}^{2}{\left(-\gamma^{*}\gamma\right)^{\
}}^{m+k-l}s^{l}}{l!\>((m+k-l)!)^{2}}\bigg{]}.$ (5.14)

From Eqs. (5.14), (5.2) and (4.4), the corresponding probability distribution
is obtained as,

$\displaystyle P_{in}\left(\kappa,\alpha\right)$
$\displaystyle=\frac{N_{\text{km}}^{-1}}{\mathbf{\pi}}\bigg{[}\sum_{l=0}^{m}{\frac{{m!}^{2}\
(-1)^{m-l}}{l!\>A^{2m-l+1}}\sum_{h=0}^{m-l}\frac{(-1)^{m-l-h}|\alpha|^{2(m-l-h)}\exp\left(-\frac{\left|\mathbf{\alpha}\right|^{2}}{\left(\frac{1}{A}-\frac{\kappa+1}{2}\right)}\right)}{h!\>((m-l-h)!)^{2}\left(\frac{1}{A}-\frac{\kappa+1}{2}\right)^{2m-2l-h+1}}}$
$\displaystyle-\frac{e^{-\beta\hbar\omega
k}}{k!}\sum_{l=0}^{m+k}{\frac{{(m+k)!}^{2}(-1)^{m+k-l}}{l!(s+1)^{2(m+k)-l+1}}\sum_{h=0}^{m+k-l}\frac{(-1)^{m+k-l-h}|\alpha|^{2(m+k-l-h)}\exp\left(-\frac{\left|\mathbf{\alpha}\right|^{2}}{\left(1-\frac{\kappa+1}{2}\right)}\right)}{h!\>((m+k-l-h)!)^{2}\left(1-\frac{\kappa+1}{2}\right)^{2m+2k-2l-h+1}}}\bigg{]}.$
(5.15)

Figure 4: Quasi-Probability Distribution functions, Husimi $Q$-function (1st
row), Wigner $W$-function (2nd row) and Sudarshan-Glauber $P$-function (3rd
row). The input PAKFTS are represented by green colour, while the output
PAKFTS are depicted by yellow (noise parameter $s=0.2$) and red (noise
parameter $s=0.5$). Here 1st column ($m=0$, $k=1$), 2nd column ($m=1$, $k=0$)
and 3rd column ($m=1$, $k=l$) are PAKFTS (corresponding to input and output
states).

####  5.1.6 Output State

Similarly, for noisy PAKFTS at the output, we have,

$\displaystyle\text{Tr}\left[e^{\gamma{\hat{a}}^{{\dagger}}}{\phi_{s}(\rho_{PAKFTS})e}^{-\gamma^{*}\hat{\text{\
a}}}\right]$
$\displaystyle=N_{\text{km}}^{-1}\bigg{[}\sum_{l=0}^{m}\frac{{m!}^{2}\>\text{Tr}\left[{e^{\gamma{\hat{a}}^{{\dagger}}}:{{\hat{a}}^{{\dagger}m-l}\exp\left(-\frac{A{\hat{a}}^{{\dagger}}\hat{a}}{\text{As}+1}\right)\hat{a}^{m-l}}:e^{-\gamma^{*}\hat{\text{\
a}}}}^{\ }\right]\ s^{l}}{l!\>((m-l)!)^{2}(As+1)^{2m-l+1}}$
$\displaystyle-\frac{e^{-\beta
k}}{k!}\sum_{l=0}^{m+k}\frac{{(m+k)!}^{2}\>\text{Tr}\left[{e^{\gamma{\hat{a}}^{{\dagger}}}:{{\hat{a}}^{{\dagger}m+k-l}\
{\exp\left(-\frac{{\hat{a}}^{{\dagger}}\hat{a}}{s+1}\right)}\hat{a}^{m+k-l}}:e^{-\gamma^{\ast}\hat{\text{\
a}}}}^{\ }\right]s^{l}}{l!\>((m+k-l)!)^{2}(s+1)^{2(m+k)-l+1}}\bigg{]}.$

Using the above equation with Eq. (5.3), we get the following CF,

$\displaystyle\chi_{out}(\gamma,\kappa)$
$\displaystyle=N_{\text{km}}^{-1}\bigg{[}\sum_{l=0}^{m}{\frac{{m!}^{2}\
s^{l}}{l!\>(As+1)^{2m-l+1}}\sum_{h=0}^{m-l}\frac{{\left(-\gamma^{*}\gamma\right)^{\
}}^{m-l-h}\exp\left\\{-\left(\frac{As+1}{A}-\frac{\kappa+1}{2}\right)|\gamma|^{2}\right\\}}{h!\>((m-l-h)!)^{2}\left(\frac{As+1}{A}\right)^{2m-2l-h+1}}}$
$\displaystyle-\frac{e^{-\beta
k}}{k!}\sum_{l=0}^{m+k}{\frac{{(m+k)!}^{2}s^{l}}{l!\>(s+1)^{2(m+k)-l+1}}\sum_{h=0}^{m+k-l}\frac{{\left(-\gamma^{*}\gamma\right)^{\
}}^{m+k-l-h}\exp\left\\{-\left((s+1)-\frac{\kappa+1}{2}\right)|\gamma|^{2}\right\\}}{h!\>((m+k-l-h)!)^{2}\>(s+1)^{2m+2k-2l-h+1}}}\bigg{]}.$
(5.16)

From Eqs. (5.16), (5.2) and (4.4),

${P}_{out}=\frac{N_{\text{km}}^{-1}}{\mathbf{\pi}}\bigg{[}\sum_{l=0}^{m}{\frac{{m!}^{2}\
s^{l}}{l!(As+1)^{2m-l+1}}\sum_{h=0}^{m-l}\frac{I}{h!\>((m-l-h)!)^{2}\left(\frac{As+1}{A}\right)^{2m-2l-h+1}}}\\\
-\frac{e^{-\beta
k}}{k!}\sum_{l=0}^{m+k}{\frac{{(m+k)!}^{2}s^{l}}{l!\>(s+1)^{2(m+k)-l+1}}\sum_{h=0}^{m+k-l}{\text{\
}\frac{I^{`}}{h!\>((m+k-l-h)!)^{2}\>(s+1)^{2m+2k-2l-h+1}}}}\bigg{]},$

where,

$I=\sum_{b=0}^{m-l-h}{\frac{{(m-l-h)!}^{2}{(-1)^{m-l-h-b}|\alpha|^{2(m-l-
h-b)}}}{b!{(m-l-
h-b)!}^{2}\left(\frac{As+1}{A}-\frac{\kappa+1}{2}\right)^{2m-2l-h+1}}\exp\left[-\frac{\left|\mathbf{\alpha}\right|^{2}}{\left(\frac{As+1}{A}-\frac{\kappa+1}{2}\right)}\right]},$
(5.17)

and

$I^{`}=\sum_{b=0}^{m+k-l-h}{\frac{{(m+k-l-h)!}^{2}{{(-1)^{m+k-l-
h-b}|\alpha|^{2(m+k-l-h-b)}}}}{b!{(m+k-l-
h-b)!}^{2}\left(s+1-\frac{\kappa+1}{2}\right)^{2m+2k-2l-h+1}}\exp\left[-\frac{\left|\mathbf{\alpha}\right|^{2}}{\left(s+1-\frac{\kappa+1}{2}\right)}\right]}.$
(5.18)

In the figure (4) quasi-probability distribution functions are depicted for
several values of $n_{th}$, $m$, and noise parameter $s$. For $\kappa=-1$,
$\kappa=0$ and $\kappa=1$ we get Husimi $Q$, Wigner $W$ and Sudarshan-Glauber
$P$-functions, respectively. At $m=0$, $k=1$ all quasi-probability
distributions are Gaussian but as we add photons to the KFTS we get a dip at
the center for the $Q$-function and negative regions for $W$ and $P$
functions. It was observed that for small values of $k$, the nonclassicality
achieved, indicated by the negative values of the $W$ and $P$ functions, was
higher as compared to the larger $k$ scenario. This suggests that hole burning
around the $n=0$ state impacts the nature of the thermal state, Gaussian in
nature, more than for larger $n$, where the tail regions of the Gaussian are
targetted. For output states, as we increase noise parameter $s$, there is a
decrease in the centered dip and the states tend to their original Gaussian
form. Further increase in the noise $s$ would eventually revert the states to
their Gaussian form.

###  5.2 Squeezed Thermal States

##### A. PASTS

####  5.2.1 Input State

From Eq. (2.10), we have,

$\displaystyle{Tr\left[e^{\gamma\hat{a}^{{\dagger}}}\hat{\rho}_{PASTS}e^{-\gamma^{\ast}\hat{a}}\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int{\langle\alpha|}:\exp{\left(\gamma\hat{a}^{\dagger}\right)}\hat{a}^{{\dagger}m}\exp{\left[\frac{C}{2}(\hat{a}^{{\dagger}2}+{\hat{a}}^{2})+(B-1){\hat{a}}^{\dagger}\hat{a}\right]}\hat{a}^{m}\exp\left(-\gamma^{\ast}\hat{a}\right):{|\alpha\rangle}\frac{d^{2}\alpha}{\pi}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{X}^{m}\int{\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\alpha^{2})+X\alpha^{\ast}\alpha+\gamma\alpha^{\ast}-\gamma^{\ast}\alpha\right]}\frac{d^{2}\alpha}{\pi}}.$

Using the Eq. (4.4),

$\displaystyle{Tr\left[e^{\gamma\hat{a}^{{\dagger}}}\hat{\rho}_{PASTS}e^{-\gamma^{\ast}\hat{a}}\right]}=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{X}^{m}{(X^{2}-C^{2})}^{-1/2}\exp\left[\frac{\frac{C}{2}({{\gamma}^{\ast}}^{2}+\gamma^{2})-X{\left|\gamma\right|^{2}}}{(X^{2}-C^{2})}\right],$

where, $X=(1-B)$. Using this in Eq. (5.3), the following expression for the
characteristic function is obtained,

$\displaystyle\chi_{in}\left(\gamma,\kappa\right)=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{X}^{m}{(X^{2}-C^{2})}^{-1/2}\exp\left[\frac{\frac{C}{2}({{\gamma}^{\ast}}^{2}+\gamma^{2})-X{\left|\gamma\right|^{2}}}{(X^{2}-C^{2})}+\frac{\kappa+1}{2}{|\gamma|}^{2}\right].$
(5.19)

From Eqs. (5.19) and (5.2), the quasi-probability distribution is seen to be

$\displaystyle
P_{in}\left(\alpha,\kappa\right)=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{X}^{m}\left[{(X^{2}-C^{2})}^{-1/2}{(X_{*}^{2}-C_{*}^{2})}^{-1/2}\exp\left(\frac{(C_{*}/2)({{\alpha}^{\ast}}^{2}+\alpha^{2})-X_{*}{|\alpha|}^{2}}{(X_{*}^{2}-C_{*}^{2})}\right)\right],$
(5.20)

where, $C_{*}=\frac{C}{(X^{2}-C^{2})}$ and
$X_{*}=\frac{X}{(X^{2}-C^{2})}-\frac{\kappa+1}{2}$.

Figure 5: Quasi-Probability Distribution functions, Husimi $Q$-function (1st
row), Wigner $W$-function (2nd row) and Sudarshan-Glauber $P$-function (3rd
row) for input PASTS are represented by green and output PASTS are represented
by yellow and red colors, respectively. Here the 1st column represents STS,
the 2nd column one photon added-STS and the 3rd column two photon added STS,
depicting both input and output PASTS for various values of the noise
parameter $s$, $n_{th}$ and the squeezing parameter $\lambda$.

####  5.2.2 Output State

Similarly, for noisy PASTS at the output, we get the following expression for
the characteristic function,

$\displaystyle\chi_{out}\left(\gamma,\kappa\right)=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\left(Y^{2}-C^{2}\right)^{-1/2}\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)^{-1/2}\exp\left[\frac{C_{1}}{2}\left({\gamma^{\ast}}^{2}+\gamma^{2}\right)-Y_{1}\gamma^{\ast}\gamma\right].$
(5.21)

The calculations are detailed in Appendix [B].

From Eqs. (5.21) and (5.2),

$P_{out}\left(\alpha,\kappa\right)=\frac{1}{\pi}\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\left(Y^{2}-C^{2}\right)^{-1/2}\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)^{-1/2}\\\
\times\int{\exp\left[\frac{C_{1}}{2}\left({\gamma^{\ast}}^{2}+\gamma^{2}\right)-Y_{1}\gamma^{\ast}\gamma+\gamma^{\ast}\alpha-\gamma\alpha^{\ast}\right]}\frac{d^{2}\alpha}{\pi}.\\\
$

Using Eq. (4.4), we get following expression for quasi-probability
distribution for output PASTS,

$P_{out}\left(\alpha,\kappa\right)=\frac{1}{\pi}\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\left(Y^{2}-C^{2}\right)^{-1/2}\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)^{-1/2}\left(Y_{1}^{2}-C_{1}^{2}\right)^{-1/2}\\\
\times\exp\left[\frac{\frac{C_{1}}{2}\left({\alpha^{\ast}}^{2}+\alpha^{2}\right)-Y_{1}\alpha^{\ast}\alpha}{\left({Y_{1}}^{2}-{C_{1}}^{2}\right)}\right].$
(5.22)

Figure (5) depicts quasi-probability distribution functions as a function of
$n_{th}$, number of photons added $m$, squeezing parameter $\lambda$, and
noise parameterized by $s$. Even though the PASTS are seen to have
qualitatively similar phase space distributions as PATS, they exhibit
comparatively more negative regions for the $W$ and $P$ functions for $m>0$
and for smaller values of {$n_{th},\lambda$}. This brings out the role of
squeezing in highlighting the quantum features of the states. Furthermore, the
quantum signatures in the PASTS quasi-probability distributions are seen up to
a (small) threshold value of the squeezing parameter $\lambda$. For the output
states, the cutoff value of squeezing parameter (for exhibiting quantumness)
decreases as we increase the noise parameter $s$.

##### B. PSSTS

####  5.2.3 Input State

Figure 6: Quasi-Probability Distribution functions, Husimi $Q$-function (1st
row), Wigner $W$-function (2nd row) and Sudarshan-Glauber $P$-function (3rd
row) for input PSSTS are represented by green color while output PSSTS are in
red and blue. The 1st column represents the STS, while the 2nd and 3rd columns
depict, one photon subtracted-STS and two photon subtracted-STS, respectively,
for both input and output states, for various values of parameters, such as,
average number of thermal photons, noise parameter $s$ and squeezing parameter
$\lambda$.

From Eqs. (2.12) and (5.1),

$\displaystyle\chi_{in}\left(\gamma,\kappa\right)=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\exp{\left(A_{1}\gamma^{\ast
2}+A_{2}\gamma^{2}-A_{3}|\gamma|^{2}\right)}\bigg{]}\bigg{|}_{u=1},$ (5.23)

which is calculated in appendix [B].
From Eqs. (5.23) and (5.2), we get following expression for quasi-probability
distribution for input PSSTS,

$\displaystyle P_{in}\left(\alpha,\kappa\right)$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\pi\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\int\frac{d^{2}\gamma}{\pi}\exp{\left(A_{1}\gamma^{\ast
2}+A_{2}\gamma^{2}-A_{3}|\gamma|^{2}+\gamma^{\ast}\alpha-\gamma\alpha^{\ast}\right)}\bigg{]}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\pi\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}(A_{3}^{2}-4A_{1}A_{2})^{-1/2}\exp{\left(\frac{A_{1}\alpha^{2}+A_{2}\alpha^{\ast
2}-A_{3}|\alpha|^{2}}{(A_{3}^{2}-4A_{1}A_{2})}\right)}\bigg{]}\bigg{|}_{u=1}.$
(5.24)

####  5.2.4 Output State

Similarly, For noisy PSSTS at the output, we have following characteristic
function,

$\displaystyle\chi_{out}\left(\gamma,\kappa\right)=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\exp{\left(-\left(N_{1}-\frac{(\kappa-1)}{2}\right)|\gamma|^{2}+N_{2}\gamma^{2}+N_{3}\gamma^{\ast
2}\right)}\bigg{]}\bigg{|}_{u=1},$ (5.25)

which is calculated in Appendix [B].
From Eqs. (5.25) and (5.2), we get following expression for quasi probability
distribution,

$P_{out}\left(\alpha,\kappa\right)=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\left(\left(N_{1}-\frac{(\kappa-1)}{2}\right)^{2}-4N_{2}N_{3}\right)^{-1/2}\\\
\times\exp{\left(\frac{-\left(N_{1}-\frac{(\kappa-1)}{2}\right)|\alpha|^{2}+N_{2}\alpha^{\ast
2}+N_{3}\alpha^{2}}{\left(\left(N_{1}-\frac{(\kappa-1)}{2}\right)^{2}-4N_{2}N_{3}\right)}\right)}\bigg{|}_{u=1}.$
(5.26)

In figure (6), we have plotted the quasi-probability distribution functions
for average number of thermal photon $n_{th}$, subtracted photons $m$,
squeezing parameter $\lambda$, and noise parameter $s$ for PSSTS (input and
output states). It can be seen that the PSSTS $Q$-function is similar to its
non-squeezed counterpart, i.e., PSTS. However, in contrast to the PSTS
scenario, figure (3), the $W$ and $P$ functions exhibit negative regions for
small values of $m$. This once again brings out the positive role of squeezing
in highlighting the quantumness. Furthermore, we have seen that $W$ and $P$
functions exhibit more negative regions when odd number of photons are
subtracted from the squeezed thermal state.

##  6 Photon Statistics

The intrinsic statistical properties of photons in a light source can be
ascertained by the experimental and theoretical study of photon statistics.
Broadly, three types of statistics can be obtained: Poissonian, super-
Poissonian, and sub-Poissonian. These are arrived at by an analysis of the
variance and average number of photon counts of the distribution. A semi-
classical theory of light can be used to describe both Poissonian and super-
Poissonian light, in which an electromagnetic wave models the light source and
atoms are modeled according to quantum mechanics. In contrast, sub-Poissonian
light requires the quantization of the electromagnetic field for a proper
description [76]. Here, we will highlight three facets of photon statistics,
viz., photon number distribution (PND), second-order correlation and Mandel’s
$Q_{M}$ parameter.

###  6.1 Photon Number Distribution

The photon-number distribution (PND) is a key characteristic of every optical
field [77]. The PND, i.e., the probability of finding $n$ photons in a quantum
state described by the density operator $\rho$ is

$P\left(n\right)=Tr[\rho|{n}\rangle\langle{n}|]=\langle{n}|\rho|{n}\rangle.$
(6.1)

The photon number distribution function sums up to one for all input and
output states.

####  6.1.1 PND for Thermal States

##### A. PATS

From Eqs. (2.2) and (6.1),

$\displaystyle\langle{n}|\rho|n\rangle$
$\displaystyle=N_{m}^{-1}\langle{n}|:\hat{a}^{{\dagger}m}e^{-A\hat{a}^{\dagger}\hat{a}}\hat{a}^{m}:|n\rangle=N_{m}^{-1}\frac{n!}{(n-m!)}\langle{n-m}|:e^{-A\hat{a}^{\dagger}\hat{a}}:|n-m\rangle$
$\displaystyle=N_{m}^{-1}\frac{n!}{(n-m!)}\sum_{k=0}\langle{n-m}|:(-A)^{k}\frac{\hat{a}^{{\dagger}k}\hat{a}^{k}}{k!}:|n-m\rangle$
$\displaystyle=N_{m}^{-1}\frac{n!}{(n-m)!}\sum_{k=0}(-A)^{k}\frac{(n-m)!}{k!(n-m-k)!}.$
(6.2)

From the binomial series,

$(1-x)^{n}=\sum_{k=0}{\frac{n!}{k!(n-k)!}(-x)^{k}}.$ (6.3)

Now, from Eqs. (6.1) and (6.3), we will get following expression of the the
photon number distribution for photon added thermal state
$P_{in}\left(n\right)$, used as input to the Gaussian channel, is seen to be,

$P_{in}\left(n\right)=N^{-1}\frac{n!}{\left(n-m\right)!}\left(1-A\right)^{n-m}.$
(6.4)

Figure 7: Photon number distributions for (a) PATS at the input with average
thermal photon $n_{th}=0.2$, and (b) noisy PATS at the output with average
thermal photon $n_{th}=0.2$ and noise parameter $s=0.4$.

Similarly, we can get following $P_{out}\left(n\right)$ of photon number
distribution for noisy-PATS at the output of the Gaussian channel as

$P_{out}(n)=N_{m}^{-1}\sum_{l=0}^{m}\frac{m!^{2}s^{l}}{l!\>(m-l)!^{2}\left(As+1\right)^{2m-l+1}}\frac{n!}{(n-m+l)!}\left(1-\frac{A}{As+1}\right)^{n-m+l}.$
(6.5)

##### B. PSTS

From Eqs. (2.4) and (6.1),

$\displaystyle P_{in}\left(n\right)$
$\displaystyle=\langle{n}|\rho_{PSTS}|{n}\rangle=N_{m^{-}}^{-1}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m}}e^{{-\frac{|\alpha|^{2}}{n_{th}}}}\langle{n}|\alpha\rangle\langle\alpha|{n}\rangle}$
$\displaystyle=N_{m^{-}}^{-1}\int{\frac{d^{2}\alpha}{\pi}\frac{(\alpha^{\ast}\alpha)^{m+n}}{n!}\exp{\bigg{[}-\frac{n_{th}+1}{n_{th}}|\alpha|^{2}\bigg{]}}}$
$\displaystyle=N_{m^{-}}^{-1}\frac{(m+n)!}{n!}\left(\frac{n_{th}}{n_{th}+1}\right)^{m+n+1}.$
(6.6)

Figure 8: Photon number distributions for (a) PSTS at the input with average
thermal photon $n_{th}=0.2$, and (b) noisy PSTS at the output with average
thermal photon $n_{th}=0.2$ and noise parameter $s=0.5$.

Similarly, PND for noisy PSTS at the outout $P_{out}(n)$ can be shown to be,

$P_{out}\left(n\right)=N_{m^{-}}^{-1}\partial_{u}^{n}\left[\frac{1}{n!\>s\left(\frac{1}{s}-u\right)}\frac{m!}{\left(\frac{1}{n_{th}}-u-\frac{u^{2}}{\frac{1}{s}-u}\right)^{m+1}}\right]\bigg{|}_{u=-1}.$
(6.7)

##### C. PAKFTS

From Eqs. (2.4), (6.1) and (6.4), we can get the following expression of
photon number distribution for PAKFTS,

$P_{in}(n)=N_{\text{km}}^{-1}\bigg{[}\frac{n!}{(n-m)!}(1-A)^{n-m}-\frac{e^{-\beta\hbar\omega
k}}{k!}\frac{n!}{(n-m-k)!}\left|\left\langle
n-m-k\middle|0\right\rangle\right|^{2}\bigg{]}.$ (6.8)

Figure 9: Photon number distributions for (a) PAKFTS at the input with average
thermal photon $n_{th}=0.2$, and (b) noisy PAKFTS at the output with average
thermal photon $n_{th}=0.2$ and noise parameter $s=0.4$.

Likewise, the photon number distribution for the output state of PAKFTS, comes
out to be

$\displaystyle P_{out}(n)$
$\displaystyle=N_{\text{km}}^{-1}\bigg{[}\sum_{l=0}^{m}{\frac{{m!}^{2}{n!s}^{l}}{l!\>((m-l)!)^{2}(n-m-l)!\left(\text{As}+1\right)^{2m-l+1}}\left(\frac{\text{As}+A-1}{\text{As}+1}\right)^{n-m-l}}$
$\displaystyle-\frac{e^{-\beta\hbar\omega
k}}{k!}\sum_{l=0}^{m+k}{\frac{{(m+k)!}^{2}n!s^{l}}{l!\>((m+k-l)!)^{2}\>(n-m-k+l)!\>(s+1)^{2(m+k)-l+1}}\left(\frac{s}{s+1}\right)^{n-m-k+l}}\bigg{]}.$
(6.9)

####  6.1.2 PND for Squeezed Thermal States

##### A. Photon Added Squeezed Thermal States

For photon added squeezed thermal state, the PND, using Eq. (6.1), is

$P_{in}(n)=\left\langle\left.n\right|\right.\rho_{PASTS}\left|\left.n\right\rangle\right.$
(6.10)

To facilitate the computations, we write Eq. (2.10) in terms of the coherent
states basis as:

$\rho_{PASTS}=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\alpha^{\ast}\beta)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}}{2}-\frac{|\beta|^{2}}{2}\right]}|\alpha\rangle\langle\beta|.$
(6.11)

Figure 10: Photon number distributions for (a) PASTS at the input with average
thermal photon $n_{th}=0.2$, squeezing parameter $\lambda=0.2$, and (b) noisy
PASTS at the output with average thermal photon $n_{th}=0.2$, squeezing
parameter $\lambda=0.2$ and noise parameter $s=0.4$.

From Eqs. (6.10) and (6.11),

$\displaystyle\left\langle\left.n\right|\right.\rho_{PASTS}\left|\left.n\right\rangle\right.$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\alpha^{\ast}\beta)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}}{2}-\frac{|\beta|^{2}}{2}\right]}\langle
n|\alpha\rangle\langle\beta|n\rangle$
$\displaystyle=\frac{N_{a,m}^{-1}}{n!\sqrt{A}}\partial_{x}^{m}\partial_{u}^{n}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+{x}\alpha^{\ast}\beta+u\alpha\beta^{\ast}-|\alpha|^{2}-|\beta|^{2}\right]}\bigg{|}_{u=1,x=B}$
$\displaystyle=\frac{N_{a,m}^{-1}}{n!\sqrt{A}}\partial_{x}^{m}\partial_{u}^{n}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+u^{2}\alpha^{2})-|\alpha|^{2}(1-xu)\right]}\bigg{|}_{u=1,x=B}.$

From the above and Eq. (6.10), we get the following PND for PASTS,

$P_{in}(n)=\frac{N_{a,m}^{-1}}{n!\sqrt{A}}\partial_{x}^{m}\partial_{u}^{n}\left[(1-xu)^{2}-C^{2}u^{2}\right]^{-1/2}\bigg{|}_{u=1,x=B}.$
(6.12)

Similarly, the photon number distribution $P_{out}(n)$ for the output of PASTS
can be shown to be

$P_{out}(n)=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\partial_{u}^{n}\left[{(Y^{2}-C^{2})}^{-1/2}\left(\left(u\left(Y_{0}-1+\frac{1}{s}\right)+1\right)^{2}-(C_{0}u)^{2}\right)^{-1/2}\right]\bigg{|}_{u=0}.$
(6.13)

##### B. PSSTS

From Eqs. (2.12) and (6.1),

$\displaystyle\left<n|\rho_{PSSTS}|n\right>$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}\langle
n|\alpha\rangle\langle\beta|n\rangle$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{n!\sqrt{A}}\partial_{v}^{m+n}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+v\alpha\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}\bigg{|}_{v=0}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{n!\sqrt{A}}\partial_{v}^{m+n}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+v^{2}\alpha^{2})-{(1-Bv)|\alpha|^{2}}\right]}\bigg{|}_{v=0}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{n!\sqrt{A}}\partial_{v}^{m+n}\left((1-Bv)^{2}-C^{2}v^{2}\right)^{-1/2}\bigg{|}_{v=0}.$

From the above and Eq. (6.1), we get the following PND for PSSTS,

$P_{in}\left(n\right)=\frac{N_{a,m^{-}}^{-1}}{n!\sqrt{A}}\partial_{v}^{m+n}\left[(1-Bv)^{2}-C^{2}v^{2}\right]^{-1/2}\bigg{|}_{v=0}.$
(6.14)

Figure 11: Photon number distributions for (a) PSSTS at the input with average
thermal photon $n_{th}=0.2$, squeezing parameter $\lambda=0.3$, and (b) noisy
PSSTS at the output with average thermal photon $n_{th}=0.2$, squeezing
parameter $\lambda=0.3$ and noise parameter $s=0.5$.

Similarly, the photon number distribution $P_{out}(n)$ for the output of PSSTS
is,

$P_{out}\left(n\right)=\frac{N_{a,m^{-}}^{-1}}{n!\sqrt{A}}\partial_{v}^{n}\partial_{u}^{m}\left[\frac{1}{s\left(1+\frac{1}{s}-v\right)}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\right],$
(6.15)

where, $u=\frac{(1-v)^{2}}{\left(1+\frac{1}{s}-v\right)}+v$ and $v=0$.

It can be seen for the thermal and squeezed thermal states that the peak for
the photon number distribution is at zero photon number $n=0$. With the
addition (or subtraction) of photons to (from) the thermal and squeezed
thermal states, the peak shifts from zero to non-zero photons $n\neq 0$ (see
figs. [7-11]). The position of the peak value of PND depends on the number of
photons $m$, added (subtracted), to (from) the states. For the case of photon
addition, the peak of the PND is located at $m=n$ exactly while this is not so
for the photon subtraction case. Furthermore, we have seen that in the photon
addition scenario (see Figs. 7 and 10) there is a condition for the
distribution of photons, viz. $m\leq n$. Additionally, for the PAKFTS it is
observed that at $n=m+k$ there is no photon distribution (see Fig. 9). It can
also be seen that as we increase the noise parameter $n_{th}$ and squeezing
parameter $\lambda$, a flatter and wider distribution is obtained.
At the output, we can see that the above restrictions to the photon number
distribution do not apply. This is an artifact of the randomization caused due
to the interaction with the Gaussian channel. Further increase in the noise
$s$ would eventually revert the states to their initial (Poissonian) form,
albeit with a wider spread.

###  6.2 Second Order Correlation and Mandel $Q_{M}$ Parameter

To study the statistical properties of the photon added and subtracted states,
we will now examine the second order correlation function $g^{2}(s)$ as well
as the Mandel $Q_{M}$-parameter. The second order correlation is defined as,

$g^{2}\left(s\right)=\frac{\left<\hat{a}^{{\dagger}2}\hat{a}^{2}\right>}{\left<\hat{a}^{{\dagger}}\hat{a}\right>^{2}}.$
(6.16)

In the classical regime, the second order correlation function lies in the
range $1\leq g^{2}(s)\leq 2$ for thermal light where $s$ is the coherence
time. For a coherent state, it can be shown that $g^{2}(s)=1$. Evidently
$g^{2}(s)<1$ is outside the allowed range of its classical counterpart and may
be interpreted as an indication of the quantum regime. When
$g^{2}(0)<g^{2}(s)$, it characterizes photon anti-bunching; its opposite is
photon bunching.

The Mandel’s $Q_{M}$-parameter is defined as follows

$Q_{M}=\frac{\left<\hat{a}^{{\dagger}2}\hat{a}^{2}\right>-{\left<\hat{a}^{\dagger}\hat{a}\right>}^{2}}{\left<\hat{a}^{\dagger}\hat{a}\right>},$
(6.17)

which implies the deviation of the variance of the photon number distribution
of the field state under consideration from the Poissonian distribution of the
coherent state. If $Q_{M}=0$ the field is said to have Poissonian photon
statistics while for $Q_{M}>0$, its super-Poissonian and $Q_{M}<0$ implies
sub-Poissonian statistics. It is well known that the negativity of the
$Q_{M}$-Parameter refers to sub-Poissonian statistics of the state. But a
state can be nonclassical even though $Q_{M}$ is positive.

####  6.2.1 For Thermal States

##### A. PATS

For $r^{th}$ moment,

$\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=Tr\left[\hat{\rho}\hat{a}^{{\dagger}r}\hat{a}^{r}\right]=\sum_{n=0}^{\infty}\left<n|\hat{a}^{r}\rho\hat{a}^{{\dagger}r}|n\right>=N_{m}^{-1}{\sum_{n=0}^{\infty}{\frac{\left(n+r\right)!^{2}}{n!\left(n+r-m\right)!}\left(1-A\right)^{n+r-m}}}.$
(6.18)

Figure 12: (a) Second ordered correlation function for output PATS with
average thermal photon $n_{th}=0.3$, (b) Mandel’s $Q_{M}$ function for PATS at
the input, and (c) Mandel’s $Q_{M}$ function for noisy PATS at the output with
noise parameter $s=0.3$.

For the output PATS

$\displaystyle\langle\hat{a}^{{\dagger}r}\hat{a}^{r}\rangle=Tr\left[\phi_{s}(\rho)\hat{a}^{{\dagger}r}\hat{a}^{r}\right]$
$\displaystyle=N_{m}^{-1}\sum_{l=0}^{m}\frac{{m!}^{2}s^{l}}{l!{\left(m-l\right)!)}^{2}\left(As+1\right)^{2m-l+1}}\sum_{n=0}{\frac{\left(n+r\right)!^{2}}{n!\left(n+r-m+l\right)!}}\left(1-\frac{A}{As+1}\right)^{n+r-m+l}.$
(6.19)

##### B. PSTS

For $r^{th}$ moment,

$\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=N_{m^{-}}^{-1}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m+r}}e^{-\frac{\alpha^{\ast}\alpha}{n_{th}}}}=N_{m^{-}}^{-1}(m+r)!(n_{th})^{m+r+1}.$
(6.20)

Figure 13: (a) Second ordered correlation function for output PSTS with
average thermal photon $n_{th}=0.4$, (b) Mandel’s $Q_{M}$ function for PSTS at
the input, and (c) Mandel’s $Q_{M}$ function for noisy PSTS at the output with
noise parameter $s=0.2$.

Similarly, for the output PSTS

$\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=N_{m^{-}}^{-1}\partial_{u}^{r}\left[\frac{1}{s\left(\frac{1}{s}-u\right)}\frac{m!}{\left(\frac{1}{n_{th}}-u-\frac{u^{2}}{\frac{1}{s}-u}\right)^{m+1}}\right]\bigg{|}_{u=0}.$
(6.21)

##### C. PAKFTS

For $r^{th}$ moment,

$\left\langle{{\hat{a}}^{{\dagger}r}\hat{a}}^{r}\right\rangle=N_{\text{km}}^{-1}\sum_{n=0}\left[\frac{{((n+r)!)}^{2}}{n!(n+r-m)!}(1-A)^{n+r-m}-\frac{e^{-\beta\hbar\omega
k}}{k!}\frac{{((n+r)!)}^{2}}{n!\ (n+r-m-k)!}\left|\left\langle
n+r-m-k\middle|0\right\rangle\right|^{2}\right].$ (6.22)

Figure 14: (a) Second ordered correlation function for output PAKFTS with
average thermal photon $n_{th}=0.2$, (b) Mandel’s $Q_{M}$ function for PAKFTS
at the input, and (c) Mandel’s $Q_{M}$ function for noisy PAKFTS at the output
with noise parameter $s=0.4$.

For the output PAKFTS

$\displaystyle\left\langle{{\hat{a}}^{{\dagger}r}\hat{a}}^{r}\right\rangle$
$\displaystyle=N_{\text{km}}^{-1}\sum_{n=0}\bigg{[}\sum_{l=0}^{m}{\frac{{m!}^{2}{{((n+r)!)}^{2}s}^{l}}{l!\>n!\>((m-l)!)^{2}\>(n+r-m+l)!\left(\text{As}+1\right)^{2m-l+1}}\left(\frac{\text{As}-A+1}{\text{As}+1}\right)^{n+r-m+l}}$
$\displaystyle-\frac{e^{-\beta\hbar\omega
k}}{k!}\sum_{l=0}^{m+k}{\frac{{(m+k)!}^{2}{((n+r)!)}^{2}s^{l}}{l!\>n!\>((m+k-l)!)^{2}(n+r-m-k+l)!(s+1)^{2(m+k)-l+1}}\left(\frac{s}{s+1}\right)^{n+r-m-k+l}}\bigg{]}.$
(6.23)

####  6.2.2 For Squeezed Thermal States

##### A. PASTS

For $r^{th}$ moment,

$\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{x}^{m}\partial_{u}^{r}\left[(1-xu)^{2}-C^{2}u^{2}\right]^{-1/2}\bigg{|}_{u=1}.$
(6.24)

Figure 15: (a) Second ordered correlation function $g^{2}(s)$ for noisy PASTS
at the output with average thermal photon $n_{th}=0.2$ and squeezing parameter
$\lambda=0.2$, (b) $Q_{M}$ for PASTS at the input with average thermal photon
$n_{th}=0.2$, and (c) $Q_{M}$ for noisy PASTS at the output with average
thermal photon $n_{th}=0.1$a nd noise parameter $s=0.2$.

The corresponding output PASTS is

$\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\partial_{u}^{r}\left[{(Y^{2}-C^{2})}^{-1/2}\left(\left(u\left(Y_{0}-1+\frac{1}{s}\right)+1\right)^{2}-(C_{0}u)^{2}\right)^{-1/2}\right]\bigg{|}_{u=1}.$
(6.25)

##### B. PSSTS

For $r^{th}$ moment,

$\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=tr[\hat{a}^{{\dagger}r}\hat{a}^{r}\rho_{PSSTS}]=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{m+r}\left[(1-Bv)^{2}-C^{2}v^{2}\right]^{-1/2}\bigg{|}_{v=1}.$
(6.26)

The corresponding output PSSTS is

Figure 16: (a) Second ordered correlation function $g^{2}(s)$ for noisy PSSTS
at the output with average thermal photon $n_{th}=0.01$, squeezing parameter
$\lambda=0.2$, (b) $Q_{M}$ for PSSTS at the input with average thermal photon
$n_{th}=0.01$, and (c) $Q_{M}$ for noisy PSSTS at the output with average
thermal photon=0.01 and noise parameter $s=0.1$.

$\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{r}\partial_{u}^{m}\left[\frac{1}{s\left(1+\frac{1}{s}-v\right)}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\right],$
(6.27)

where, $u=\frac{(1-v)^{2}}{\left(1+\frac{1}{s}-v\right)}+v$ and $v=1$.

The $r^{th}$ moment for all input and output states are calculated in Appendix
C. After putting the values of $r^{th}$ moment in Eqs. (6.16) and (6.17) one
can get the second order correlation function $g^{2}(s)$ and Mandel’s
$Q_{M}$-parameter, respectively.

From the Figs. (12-16) we observe that with the addition of photons to the
initial thermal Gaussian state, the $g^{2}$ function’s value becomes less than
one, an indicator of quantum regime. With increase in noise however, the
output state’s $g^{2}$ tends to two, corresponding to the thermal state. This
reduction of $g^{2}$ to a value less than one is not observed for the case of
photon subtraction (see Fig. 13). However, for the photon subtracted squeezed
thermal state (Fig. 16), the reduction to less than one is observed for very
small temperatures. This suggest that the quantum features of the photon added
states are more robust, as compared to their photon subtracted counterparts.
Negative values of $Q_{M}$, indicative of sub-Poissonian behavior, is observed
for various parameters for input and output states, for cut-off values of
$n_{th}$ and squeezing parameter $\lambda$. It’s observed that the cut-off
values can be increased (decreased) by adding (subtracting) photons to the
thermal or the squeezed thermal states. For PSTS we see that $Q_{M}$ is only a
function of $n_{th}$ and not effected by the subtraction of photons (see Fig.
13(b)). For PSSTS case the negative $Q_{M}$ function is observed only for odd
subtracted photons (see Figs. 16(b) and 16(c)). A similar pattern was observed
for the corresponding $W$ and $P$ functions. However, here some negative
regions were observed for even subtracted photons (Fig. 6). As we increase the
value of the noise parameter in the output states, the negative $Q_{M}$ shifts
towards positive values and cut-off values for $\\{n_{th},\lambda\\}$, the
thermal and squeezed parameters, also decrease. For PSTS at the input and
output, negative value of Mandel’s $Q_{M}$ parameter was not observed (see
Fig. 13(b) and 13(c)).
For thermal and squeezed thermal states, photon bunching is observed. Apart
form this, some states belonging to the PAKFTS and PSSTS exhibit bunching at
$\\{m=0,\>k=1\\}$ and ${m=2}$, respectively (see Figs. 14(a) and 16(a)). Anti-
bunching is observed in other photon added and subtracted states. The states
exhibiting photon bunching do not show negative values of $Q_{M}$, which is
not so for the anti-bunching cases.

##  7 Conclusion

In this work, we have systematically studied the impact of noise, modelled by
a noisy Gaussian channel, on a wide range of non-Gaussian input states,
covering both photon added, subtracted and hole burning scenarios. The impact
of noise is presented as an input-output problem. Making use of the IWOP
technique allows for the derivation of analytical expressions for the various
output states. The quantum nature of the states, both input and output, is
developed by studying the corresponding photon statistics and quasi-
probability distributions. This further allows for gauging the impact of noise
(decoherence) on the non-Gaussian input states. It’s found that photon
addition has more robust quantum mechanical properties as compared to the case
of photon subtraction. It’s also seen that with increase in noise, there is a
tendency to lesser non-classicality as well as non-Gaussianity. The input
states considered (except PSTS) have highly non-classical properties. This is
depicted by the negative region at the center of the phase space. It was seen
that the threshold value of the noise parameter corresponding to the
transition of quasi-probability distribution functions from partial negative
($W$ and $P$) and zero ($Q$) to completely positive definite, at the center of
phase space, is dependent not only on the noise parameter, but also on the
average number of thermal photons in the state and the squeezing parameter.

For photon-added (thermal and squeezed thermal states) and photon-subtracted
squeezed states, by an odd number of photons, the negativity of the Mandel
${Q_{M}}$ parameter is noticeable. However, for non-classical states, Mandel’s
$Q_{M}$ parameter do not always indicate a negative value. In fact, for the
case where an even number of photons were subtracted from the squeezed states,
it was observed to be positive. It follows that the negativity of the $Q_{M}$
parameter is sufficient to distinguish the classical state from the
nonclassical state. The non-classicality of quantum states, could be also
explored via measures such as those based on the volume of the negative part
of the Wigner function [78], on the nonclassical depth [79]. Both the features
of photon bunching and antibunching were observed in the non-Gaussian states.
For the case of photon addition and filtration, we obtained conditions
indicative of hole burning in the input states. However, in the output these
holes get filled up due to the influence of noise generated due to the passage
through the Gaussian channel.

It was observed that photon addition was more robust than photon subtraction
in withstanding the impact of noise. This could be ascribed to the non-
Gaussianity (visualized by the phase space distributions) for photon added
states being greater than for the subtracted photon states. In addition, for
KFTS nonclassical behaviour was observed due to filtration on the thermal
state that could be further enhanced by adding photon(s). PAKFTS came out to
be most robust against noise that other states considered.
It would be of interest to evaluate the impact of more general quantum
channels on the non-Gaussian input states.

##  Appendix A Appendix: Normalization Constant

For any density, we have the following,

$Tr[\rho]=1.$

##### For PATS

$N_{m}={Tr\left[:\hat{a}^{{\dagger}m}e^{-A\hat{a}^{\dagger}\hat{a}}\hat{a}^{m}:\right]}=\int{\left\langle\left.\alpha\right|\right.:\hat{a}^{{\dagger}m}{e^{-A\hat{a}^{{\dagger}}\hat{a}}}\hat{a}^{m}:\left|\left.\alpha\right\rangle\right.}\frac{d^{2}\alpha}{\pi}=\int{(\alpha\alpha^{\ast})^{m}e^{-A|\alpha|^{2}}}\frac{d^{2}\alpha}{\pi},$

let $\alpha=x+\iota y$, $\alpha\alpha^{\ast}=x^{2}+y^{2}=r^{2}$ and
$d^{2}\alpha=rdrd\theta$,

now we have,

$N_{m}=\int{r^{2m+1}e^{-Ar^{2}}{}{dr}d\theta=m!/A^{\left(m+1\right)}}.$

Where we used the Integration formula
$\int_{0}^{\infty}x^{2m+1}\exp{(-ax^{2})}{dx}=\frac{m!}{2a^{m+1}}.$

##### For PSTS

$N_{m^{-}}=Tr\bigg{[}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m}}e^{-\frac{|\alpha|^{2}}{n_{th}}}\left|\left.\alpha\right\rangle\right.\left\langle\left.\alpha\right|\right.}\bigg{]}=\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m}}e^{-\frac{|\alpha|^{2}}{n_{th}}}}=m!{(n_{th})}^{\left(m+1\right)}.$

##### For PAKFTS

$\displaystyle N_{\text{km}}^{\ }$
$\displaystyle=\text{Tr}\bigg{[}:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:-\frac{e^{-\beta\hbar\omega
k}}{k!}:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:\bigg{]}$
$\displaystyle=\int{\frac{d^{2}\alpha}{\pi}\langle\alpha|\
:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:-\frac{e^{-\beta\hbar\omega
k}}{k!}:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:|\alpha\rangle}$
$\displaystyle=\int{\frac{d^{2}\alpha}{\pi}\langle\alpha|\
:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:|\alpha\rangle}-\frac{e^{-\beta\hbar\omega
k}}{k!}\int{\frac{d^{2}\alpha}{\pi}\langle\alpha|:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:|\alpha\rangle}.$

We have,

$\int{\frac{d^{2}\alpha}{\pi}\langle\alpha|:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:|\alpha\rangle}=\int{\frac{d^{2}\alpha}{\pi}\left(\alpha\alpha^{*}\right)^{m}}e^{-A|\alpha|^{2}}=\frac{m!}{A^{m+1}},$

using the above equation we can get following expression of normalization
constant For PAKFTS-

$N_{km}=\frac{m!}{A^{m+1}}-\frac{e^{-\beta\hbar\omega k}}{k!}(m+k)!.$ (A.1)

##### For PASTS

we have $tr(\rho_{PASTS})=1.$

$\displaystyle N_{a,m}$
$\displaystyle=\frac{1}{\sqrt{A}}\int\left\langle\left.\alpha\right|\right.:\hat{a}^{{\dagger}m}\exp{\left[\frac{C}{2}(\hat{a}^{{\dagger}2}+{\hat{a}}^{2})+(B-1){\hat{a}}^{\dagger}\hat{a}\right]}\hat{a}^{m}:\left|\left.\alpha\right\rangle\right.\frac{d^{2}\alpha}{\pi}$
$\displaystyle=\frac{1}{\sqrt{A}}\int(\alpha^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\alpha}^{2})+(B-1){\alpha}^{\ast}\alpha\right]}\frac{d^{2}\alpha}{\pi}$
$\displaystyle=\frac{1}{\sqrt{A}}\partial_{v}^{m}\int\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\alpha}^{2})+v{\alpha}^{\ast}\alpha\right]}\frac{d^{2}\alpha}{\pi}\bigg{|}_{v=B}.$

Now using the Eq. (4.4)

$N_{a,m}=\frac{1}{\sqrt{A}}\partial_{v}^{m}\left[(v^{2}-C^{2})^{-1/2}\right]\bigg{|}_{v=B}.$
(A.2)

##### For PSSTS

we have $Tr(\rho_{PSSTS})=1.$

$\displaystyle N_{a,m^{-}}$
$\displaystyle=Tr\bigg{[}\frac{1}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}{|\alpha\rangle\langle\beta|}\bigg{]}$
$\displaystyle=\frac{1}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}{\langle\beta|\alpha\rangle}$
$\displaystyle=\frac{1}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\alpha\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}$
$\displaystyle=\frac{1}{\sqrt{A}}\partial_{u}^{m}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+u\alpha\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}\bigg{|}_{u=1}$
$\displaystyle=\frac{1}{\sqrt{A}}\partial_{u}^{m}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+u^{2}\alpha^{2})-(1-Bu)|\alpha|^{2}\right]}\bigg{|}_{u=1}$
$\displaystyle=\frac{1}{\sqrt{A}}\partial_{u}^{m}\left[(1-Bu)^{2}-C^{2}u^{2}\right]^{-1/2}\bigg{|}_{u=1}.$

##  Appendix B Appendix: Characteristic Function (CF)

##### CF For PASTS

$\displaystyle{Tr\left[e^{\gamma\hat{a}^{\dagger}}\hat{\rho}e^{-\gamma^{\ast}\hat{a}}\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int{\langle\alpha|}:\exp{\left(\gamma\hat{a}^{\dagger}\right)}\hat{a}^{{\dagger}m}\exp{\left[\frac{C}{2}(\hat{a}^{{\dagger}2}+{\hat{a}}^{2})+(B-1){\hat{a}}^{\dagger}\hat{a}\right]}\hat{a}^{m}\exp\left(-\gamma^{\ast}\hat{a}\right):{|\alpha\rangle}\frac{d^{2}\alpha}{\pi}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int{\left(\alpha^{*}\alpha\right)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\alpha^{2})+(B-1)\alpha^{\ast}\alpha+\gamma\alpha^{\ast}-\gamma^{\ast}\alpha\right]}\frac{d^{2}\alpha}{\pi}}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{X}^{m}\int{\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\alpha^{2})+X\alpha^{\ast}\alpha+\gamma\alpha^{\ast}-\gamma^{\ast}\alpha\right]}\frac{d^{2}\alpha}{\pi}}.$

Using the Eq. (4.4)

${Tr\left[e^{\gamma\hat{a}^{\dagger}}\hat{\rho}e^{-\gamma^{\ast}\hat{a}}\right]}=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{X}^{m}\left[{(X^{2}-C^{2})}^{-1/2}\exp\bigg{[}\frac{\frac{C}{2}({{\gamma}^{\ast}}^{2}+\gamma^{2})-X{\left|\gamma\right|^{2}}}{(X^{2}-C^{2})}\bigg{]}\right],$

where $X=(1-B)$.

Thus the characteristic function for PASTS will be-

$\chi_{in}\left(\gamma,\kappa\right)=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{X}^{m}\left[{(X^{2}-C^{2})}^{-1/2}\exp\bigg{[}\frac{\frac{C}{2}({{\gamma}^{\ast}}^{2}+\gamma^{2})-X{\left|\gamma\right|^{2}}}{(X^{2}-C^{2})}+\frac{\kappa+1}{2}{|\gamma|}^{2}\bigg{]}\right]$

Similarly, for noisy PASTS at the output,

$\displaystyle Tr\left[e^{\gamma{\hat{a}}^{\dagger}}\rho
e^{-\gamma^{\ast}\hat{a}}\right]$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\left[\int\frac{d^{2}\alpha}{\pi}\left\langle\alpha\middle|e^{\gamma{\hat{a}}^{\dagger}}:e^{-\frac{{\hat{a}}^{\dagger}\hat{a}}{s}}\left(Y^{2}-C^{2}\right)^{-1/2}\exp\left[\frac{\frac{C}{2}\left({\hat{a}}^{{{\dagger}2}}+{\hat{a}}^{2}\right)-Y{\hat{a}}^{\dagger}\hat{a}}{s^{2}\left(Y^{2}-C^{2}\right)}\right]:e^{-\gamma^{\ast}\hat{a}}\middle|\alpha\right\rangle\right]$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\bigg{[}\left(Y^{2}-C^{2}\right)^{-1/2}\int\frac{d^{2}z}{\pi}\exp\left[-\left(Y_{0}+1/s\right)\alpha^{\ast}\alpha+\left(C_{0}/2\right)\left({\alpha^{\ast}}^{2}+\alpha^{2}\right)+\gamma\alpha^{\ast}-\gamma^{\ast}\alpha\right]\bigg{]},$

where, $Y_{0}=\frac{Y}{\left(Y^{2}-C^{2}\right)}$ and
$C_{0}=\frac{C}{\left(Y^{2}-C^{2}\right)}.$

Using the Eq. (4.4)

$Tr\left[e^{\gamma{\hat{a}}^{\dagger}}\rho
e^{-\gamma^{\ast}\hat{a}}\right]=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\bigg{[}\left(Y^{2}-C^{2}\right)^{-1/2}\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)^{-1/2}\exp\bigg{[}\frac{\frac{C_{0}}{2}\left({\gamma^{\ast}}^{2}+\gamma^{2}\right)-\left(Y_{0}+1/s\right)\gamma^{\ast}\gamma}{\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)}\bigg{]}\bigg{]}.$

After putting the value of trace in Eq. (5.3), we get the following expression
for characteristic function for the output state of PASTS -

$\chi_{out}\left(\gamma.\kappa\right)=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\left(Y^{2}-C^{2}\right)^{-1/2}\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)^{-1/2}\\\
\times\exp\left\\{\frac{\frac{C_{0}}{2}\left({\gamma^{\ast}}^{2}+\gamma^{2}\right)}{\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)}-\left(\frac{\left(Y_{0}+1/s\right)}{\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)}-\frac{\beta+1}{2}\right)|\gamma|^{2}\right\\},$

let we denote
$Y_{1}=\frac{\left(Y_{0}+1/s\right)}{\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)}\
-\frac{\kappa+1}{2}$,
$C_{1}=\frac{C_{0}}{\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)}.$

$\chi_{out}\left(\gamma,\kappa\right)=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\bigg{[}\left(Y^{2}-C^{2}\right)^{-1/2}\left(\left(Y_{0}+1/s\right)^{2}-{C_{0}}^{2}\right)^{-1/2}exp\bigg{[}\frac{C_{1}}{2}\left({\gamma^{\ast}}^{2}+\gamma^{2}\right)-Y_{1}\gamma^{\ast}\gamma\bigg{]}\bigg{]}.$
(B.1)

##### CF For PSSTS

$\displaystyle{Tr\left[e^{-{{\gamma}^{\ast}}\hat{a}}\hat{\rho}e^{\gamma\hat{a}^{{\dagger}}}\right]}$
$\displaystyle=Tr\bigg{[}\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}e^{-\gamma^{\ast}\hat{a}}|\alpha\rangle\langle\beta|e^{\gamma\hat{a}^{{\dagger}}}\bigg{]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\gamma^{\ast}\alpha+\gamma\beta^{\ast}-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}\langle\beta|\alpha\rangle$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\beta^{\ast}\alpha-\gamma^{\ast}\alpha+\gamma\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+u\beta^{\ast}\alpha-\gamma^{\ast}\alpha+\gamma\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2})-\gamma^{\ast}\alpha-{|\alpha|^{2}}\right]}\exp{\big{[}B\alpha^{\ast}(u\alpha+\gamma)+\frac{C}{2}(u\alpha+\gamma)^{2}\big{]}}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\int\frac{d^{2}\alpha}{\pi}\exp{\bigg{[}\frac{C}{2}(\alpha^{\ast
2}+u^{2}\alpha^{2})-(1-Bu)|\alpha|^{2}-\alpha(\gamma^{\ast}-\gamma
Cu)+B\gamma\alpha^{\ast}+\frac{C}{2}\gamma^{2}\bigg{]}}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\exp{\bigg{[}\frac{-(1-Bu)(\gamma^{\ast}-\gamma
Cu)B\gamma+\frac{C}{2}((\gamma^{\ast}-\gamma
Cu)^{2}+u^{2}B^{2}\gamma^{2})}{(\left(1-Bu)^{2}-C^{2}u^{2}\right)}+\frac{C}{2}\gamma^{2}\bigg{]}}\bigg{]}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}$
$\displaystyle\times\exp{\bigg{[}\frac{\frac{C}{2}\gamma^{\ast
2}+\gamma^{2}(\frac{C(u^{2}B^{2}+C^{2}u^{2})}{2}+CuB(1-Bu))-|\gamma|^{2}(B(1-Bu)+C^{2}u)}{(\left(1-Bu)^{2}-C^{2}u^{2}\right)}+\frac{C}{2}\gamma^{2}\bigg{]}}\bigg{]}\bigg{|}_{u=1}.$

After putting the value of trace in Eq. (5.8), we get the following expression
for characteristic function,

$\displaystyle\chi_{PSSTS}\left(\gamma,\kappa\right)$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\big{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\exp{\left(A_{1}\gamma^{\ast
2}+A_{2}\gamma^{2}-A_{3}|\gamma|^{2}\right)}\big{]}\bigg{|}_{u=1},$ (B.2)

where $A_{1}=\frac{\left(C/2\right)}{(\left(1-Bu)^{2}-C^{2}u^{2}\right)}$,
$A_{2}=\frac{(\frac{C(u^{2}B^{2}+C^{2}u^{2})}{2}+BCu(1-Bu))}{(\left(1-Bu)^{2}-C^{2}u^{2}\right)}+\frac{C}{2}$,
$A_{3}=\frac{(B(1-Bu)+C^{2}u)}{\left(1-Bu)^{2}-C^{2}u^{2}\right)}-\frac{\kappa-1}{2}.\\\
$
Similarly, for noisy PSSTS at the output

$\displaystyle{Tr\left[e^{-\gamma^{\ast}\hat{a}}\phi_{s}(\hat{\rho})e^{\gamma\hat{a}^{\dagger}}\right]}$
$\displaystyle=Tr\bigg{[}\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left(-\gamma^{\ast}\hat{a}\right)}\left|\right.{\alpha+z}\left\rangle\right.\left\langle\right.{\beta+z}\left|\right.\exp{\left(\gamma\hat{a}^{\dagger}\right)}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\frac{1}{2}\\{z(\alpha^{\ast}-\beta^{\ast})-z^{\ast}(\alpha-\beta)\\}-\frac{|\alpha|^{2}+|\beta|^{2}}{2}-\frac{|z|^{2}}{s}\right]}\bigg{]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\left\langle\right.{\beta+z}\left|\right.{\alpha+z}\left\rangle\right.$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\frac{1}{2}\\{(z(\alpha^{\ast}-\beta^{\ast})-z^{\ast}(\alpha-\beta))\\}+\gamma(z^{\ast}+\beta^{\ast})-\gamma^{\ast}(z+\alpha)-\frac{|\alpha|^{2}+|\beta|^{2}}{2}-\frac{|z|^{2}}{s}\right]},$

one can have,

$\displaystyle\left\langle\right.{\beta+z}\left|\right.{\alpha+z}\left\rangle\right.$
$\displaystyle=\exp{\bigg{[}-\frac{|\alpha+z|^{2}}{2}-\frac{|\beta+z|^{2}}{2}+(\beta^{\ast}+z^{\ast})(\alpha+z)\bigg{]}}$
$\displaystyle=\exp{\bigg{[}-\frac{|\alpha|^{2}+|z|^{2}+\alpha
z^{\ast}+z\alpha^{\ast}}{2}-\frac{|\beta|^{2}+|z|^{2}+\beta
z^{\ast}+z\beta^{\ast}}{2}+(\beta^{\ast}\alpha+\beta^{\ast}z+z^{\ast}\alpha+|z|^{2})\bigg{]}}$
$\displaystyle=\exp{\bigg{[}-\frac{|\alpha|^{2}}{2}-\frac{|\beta|^{2}}{2}-\frac{1}{2}\\{z(\alpha^{\ast}-\beta^{\ast})-z^{\ast}(\alpha-\beta)\\}+\beta^{\ast}\alpha\bigg{]}}.$
$\displaystyle{Tr\left[e^{-\gamma^{\ast}\hat{a}}\phi_{s}(\hat{\rho})e^{\gamma\hat{a}^{\dagger}}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\beta^{\ast}\alpha-z\gamma^{\ast}+z^{\ast}\gamma+\gamma\beta^{\ast}-\gamma^{\ast}\alpha-|\alpha|^{2}-|\beta|^{2}-\frac{|z|^{2}}{s}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\beta^{\ast}\alpha+\gamma\beta^{\ast}-\gamma^{\ast}\alpha-|\alpha|^{2}-|\beta|^{2}-{|\gamma|^{2}}{s}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\beta^{\ast}\alpha-\gamma^{\ast}\alpha+\gamma\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+u\beta^{\ast}\alpha-\gamma^{\ast}\alpha+\gamma\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}-{|\gamma|^{2}}{s}\right]}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2})-\gamma^{\ast}\alpha-{|\alpha|^{2}}\right]}\exp{\bigg{[}B\alpha^{\ast}(u\alpha+\gamma)+\frac{C}{2}(u\alpha+\gamma)^{2}-{|\gamma|^{2}}{s}\bigg{]}}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\int\frac{d^{2}\alpha}{\pi}\exp{\bigg{[}\frac{C}{2}(\alpha^{\ast
2}+u^{2}\alpha^{2})-(1-Bu)|\alpha|^{2}-\alpha(\gamma^{\ast}-\gamma
Cu)+B\gamma\alpha^{\ast}+\frac{C}{2}\gamma^{2}-{|\gamma|^{2}}{s}\bigg{]}}\bigg{]}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-\frac{1}{2}}\exp{\bigg{[}\frac{-(1-Bu)(\gamma^{\ast}-\gamma
Cu)B\gamma+\frac{C}{2}((\gamma^{\ast}-\gamma
Cu)^{2}+u^{2}B^{2}\gamma^{2})}{(\left(1-Bu)^{2}-C^{2}u^{2}\right)}+\frac{C}{2}\gamma^{2}-{|\gamma|^{2}}{s}\bigg{]}}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}$
$\displaystyle\times\exp{\bigg{[}\frac{\frac{C}{2}\gamma^{\ast
2}+\gamma^{2}(\frac{C(u^{2}B^{2}+C^{2}u^{2})}{2}+CuB(1-Bu))-|\gamma|^{2}(B(1-Bu)+C^{2}u)}{(\left(1-Bu)^{2}-C^{2}u^{2}\right)}+\frac{C}{2}\gamma^{2}-{|\gamma|^{2}}{s}\bigg{]}}\bigg{]}\bigg{|}_{u=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\exp{\left(N_{1}\gamma^{\ast
2}+N_{2}\gamma^{2}-N_{3}|\gamma|^{2}\right)}\bigg{]}\bigg{|}_{u=1},$

where
$N_{1}=\frac{\left({C/2}\right)}{\left((1-Bu)^{2}-C^{2}u^{2}\right)},N_{2}=\frac{(\frac{C(u^{2}B^{2}+C^{2}u^{2})}{2}+BCu(1-Bu))}{\left((1-Bu)^{2}-C^{2}u^{2}\right)}+\frac{C}{2},N_{3}=\frac{(B(1-Bu)+C^{2}u)}{\left((1-Bu)^{2}-C^{2}u^{2}\right)}+s$.

After putting the value of trace in Eq. (5.8). we get following expression for
characteristic function

$\chi\left(\gamma,\kappa\right)=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{u}^{m}\bigg{[}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2}\exp{\bigg{[}-\left(N_{1}-\frac{(\kappa-1)}{2}\right)|\gamma|^{2}+N_{2}\gamma^{2}+N_{3}\gamma^{\ast
2}\bigg{]}}\bigg{]}\bigg{|}_{u=1}.$ (B.3)

##  Appendix C Appendix: $r^{th}$ Moment For Input And Output States

For the derivation of $r^{th}$ moment for input and output states, we consider
the following

$\displaystyle\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=Tr\left[\hat{\rho}\hat{a}^{{\dagger}r}\hat{a}^{r}\right].$
(C.1)

##### For PATS

$r^{th}$-moment of PATS can be find using above Eq. (C.1),

$\displaystyle\langle{n}|\hat{a}^{r}\rho\hat{a}^{{\dagger}r}|n\rangle$
$\displaystyle=N_{m}^{-1}\langle{n}|\hat{a}^{r}:\hat{a}^{{\dagger}m}e^{-A\hat{a}^{\dagger}\hat{a}}\hat{a}^{m}:\hat{a}^{{\dagger}r}|n\rangle=N_{m}^{-1}\frac{(n+r)!}{n!}\langle{n+r}|:\hat{a}^{{\dagger}m}e^{-A\hat{a}^{\dagger}\hat{a}}\hat{a}^{m}:|n+r\rangle$
$\displaystyle=N_{m}^{-1}\frac{(n+r)!}{n!}\sum_{l=0}{\frac{(n+r)!}{(n+r-m)!}}\langle{n+r-m}|:(-A)^{l}\frac{\hat{a}^{{\dagger}k}\hat{a}^{l}}{l!}:|n+r-m\rangle$
$\displaystyle=N_{m}^{-1}\frac{(n+r)!}{n!}\sum_{l=0}{\frac{(n+r)!}{(n+r-m)!}}\frac{(-A)^{l}(n+r-m)}{l!\>(n+r-m-l)}$
$\displaystyle=N_{m}^{-1}\frac{(n+r)!}{n!}{\frac{(n+r)!}{(n+r-m)!}}(1-A)^{n+r-m}.$

Thus one can see,

$\displaystyle\langle\hat{a}^{{\dagger}r}\hat{a}^{r}\rangle=Tr\left[\hat{\rho}\hat{a}^{{\dagger}r}\hat{a}^{r}\right]=\sum_{n=0}^{\infty}\langle{n}|\hat{a}^{r}\rho\hat{a}^{{\dagger}r}|n\rangle=N_{m}^{-1}{\sum_{n=0}^{\infty}{\frac{\left(n+r\right)!^{2}}{n!\left(n+r-m\right)!}\left(1-A\right)^{n+r-m}}}.$
(C.2)

Similarly, for noisy PATS at the output,

$\displaystyle\langle\hat{a}^{{\dagger}r}\hat{a}^{r}\rangle$
$\displaystyle=Tr\left[\hat{{\Phi}_{s}\left(\rho_{PATS}\right)}\hat{a}^{{\dagger}r}\hat{a}^{r}\right]=\sum_{n=0}\langle{n}|\hat{a}^{r}{\Phi}_{s}\left(\rho_{PATS}\right)\hat{a}^{{\dagger}r}|n\rangle$
$\displaystyle=N_{m}^{-1}:\sum_{l=0}^{m}\frac{{m!}^{2}(n+r)!\left\langle\left.{n+r}\right|\right.\left({\hat{a}}^{\dagger}\right)^{m-l}\exp\left(-\frac{A{\hat{a}}^{{\dagger}}\hat{a}}{As+1}\right)\left(\hat{a}\right)^{m-l}\left|\left.{n+r}\right\rangle\right.s^{l}}{n!\>l!\>{\left(m-l\right)!)}^{2}\left(As+1\right)^{2m-l+1}}:,$

we get the following (making use of the Eq. (6.3)),

$\displaystyle\langle\hat{a}^{{\dagger}r}\hat{a}^{r}\rangle=Tr\left[\phi_{s}(\rho)\hat{a}^{{\dagger}r}\hat{a}^{r}\right]$
$\displaystyle=N_{m}^{-1}\sum_{l=0}^{m}\frac{{m!}^{2}s^{l}}{l!{\left(m-l\right)!)}^{2}\left(As+1\right)^{2m-l+1}}\sum_{n=0}{\frac{\left(n+r\right)!^{2}}{n!\left(n+r-m+l\right)!}}\left(1-\frac{A}{As+1}\right)^{n+r-m+l}.$
(C.3)

##### For PSTS

For $r^{th}$ moment

$\displaystyle\langle\hat{a}^{{\dagger}r}\hat{a}^{r}\rangle$
$\displaystyle=Tr[\hat{a}^{{\dagger}r}\hat{a}^{r}\rho_{PSTS}]=N_{m^{-}}^{-1}Tr\bigg{[}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m}}e^{-\frac{|\alpha|^{2}}{n_{th}}}\hat{a}^{{\dagger}r}\hat{a}^{r}\left|\left.\alpha\right\rangle\right.\left\langle\left.\alpha\right|\right.\bigg{]}}$
$\displaystyle=N_{m^{-}}^{-1}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m+r}}e^{-\frac{|\alpha|^{2}}{n_{th}}}}=N_{m^{-}}^{-1}(m+r)!(n_{th})^{m+r+1}.$
(C.4)

Similarly for noisy PSTS at the output,

for $r^{th}$ moment

$\displaystyle\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>$
$\displaystyle=Tr[\hat{a}^{{\dagger}r}\hat{a}^{r}\phi_{s}(\rho)]$
$\displaystyle=N_{m^{-}}^{-1}Tr\bigg{[}\int\int{\frac{d^{2}\alpha}{\pi}\frac{d^{2}z}{\pi
s}{(\alpha^{\ast}\alpha)^{m}}\exp{\bigg{\\{}-\frac{|\alpha|^{2}}{n_{th}}-\frac{|z|^{2}}{s}\bigg{\\}}}\hat{a}^{r}\left|\left.\alpha+z\right\rangle\right.\left\langle\left.\alpha+z\right|\right.\hat{a}^{{\dagger}r}}\bigg{]}$
$\displaystyle=N_{m^{-}}^{-1}\int\int{\frac{d^{2}\alpha}{\pi}\frac{d^{2}z}{\pi
s}{(\alpha^{\ast}\alpha)^{m}}{|\alpha+z|^{2r}}\exp{\bigg{[}-\frac{|\alpha|^{2}}{n_{th}}-\frac{|z|^{2}}{s}\bigg{]}}\left\langle\left.\alpha+z\right|\left.\alpha+z\right\rangle\right.}$
$\displaystyle=N_{m^{-}}^{-1}\partial_{u}^{r}\int\int{\frac{d^{2}\alpha}{\pi}\frac{d^{2}z}{\pi
s}{(\alpha^{\ast}\alpha)^{m}}\exp{\bigg{[}-\frac{|\alpha|^{2}}{n_{th}}-\frac{|z|^{2}}{s}+u|\alpha+z|^{2}\bigg{]}}}\bigg{|}_{u=0}$
$\displaystyle=N_{m^{-}}^{-1}\partial_{u}^{r}\int\int{\frac{d^{2}\alpha}{\pi}\frac{d^{2}z}{\pi
s}{(\alpha^{\ast}\alpha)^{m}}\exp{\bigg{[}-(1/n_{th}-u)|\alpha|^{2}-\left(\frac{1}{s}-u\right)|z|^{2}+u(z\alpha^{\ast}+z^{\ast}\alpha)\bigg{]}}}\bigg{|}_{u=0}$
$\displaystyle=N_{m^{-}}^{-1}\partial_{u}^{r}\left[\frac{1}{\left(\frac{1}{s}-u\right)s}\int{\frac{d^{2}\alpha}{\pi}{(\alpha^{\ast}\alpha)^{m}}\exp{\bigg{[}-\left(\frac{1}{n_{th}}-u-\frac{u^{2}}{\frac{1}{s}-u}\right)|\alpha|^{2}\bigg{]}}}\right]\bigg{|}_{u=0}$
$\displaystyle=N_{m^{-}}^{-1}\partial_{u}^{r}\left[\frac{1}{\left(\frac{1}{s}-u\right)s}\frac{m!}{\left(\frac{1}{n_{th}}-u-\frac{u^{2}}{\frac{1}{s}-u}\right)^{m+1}}\right]\bigg{|}_{u=0}.$
(C.5)

##### For PAKFTS

We have density operator for PAKFTS

$\displaystyle\rho_{\text{PAKFT}}=N_{\text{km}}^{-1}\left[:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:-\frac{e^{-\beta\hbar\omega
k}}{k!}:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:\right],$

using the above and Eq. (C.1), $r^{th}$ moment-

$\displaystyle\left\langle{{\hat{a}}^{{\dagger}r}\hat{a}}^{r}\right\rangle$
$\displaystyle=N_{\text{km}}^{-1}\
\text{Tr}\left[{{\hat{a}}^{r}:{{\hat{a}}^{{\dagger}m}e^{-A{\hat{a}}^{{\dagger}}\hat{a}}\hat{a}}^{m}:\hat{a}}^{{\dagger}r}\right]-\
N_{\text{km}}^{-1}\ \frac{e^{-\beta\hbar\omega k}}{k!}\
\text{Tr}\left[{{\hat{a}}^{r}:{{\hat{a}}^{{\dagger}(m+k)}e}^{-{\hat{a}}^{{\dagger}}\hat{a}}{\hat{a}}^{(m+k)}:\hat{a}}^{{\dagger}r}\right]$
(C.6)
$\displaystyle=N_{\text{km}}^{-1}\sum_{n=0}\left[\frac{{((n+r)!)}^{2}}{n!(n+r-m)!}(1-A)^{n+r-m}-\frac{e^{-\beta\hbar\omega
k}}{k!}\frac{{((n+r)!)}^{2}}{n!\ (n+r-m-k)!}\left|\left\langle
n+r-m-k\middle|0\right\rangle\right|^{2}\right].$ (C.7)

Similarly, for noisy PAKFTS the output,

$\displaystyle\left\langle{{\hat{a}}^{{\dagger}r}\hat{a}}^{r}\right\rangle$
$\displaystyle=N_{\text{km}}^{-1}\sum_{n=0}\bigg{[}\sum_{l=0}^{m}{\frac{{m!}^{2}{{((n+r)!)}^{2}s}^{l}}{l!\>n!\>((m-l)!)^{2}\>(n+r-m+l)!\left(\text{As}+1\right)^{2m-l+1}}\left(\frac{\text{As}-A+1}{\text{As}+1}\right)^{n+r-m+l}}$
$\displaystyle-\frac{e^{-\beta\hbar\omega
k}}{k!}\sum_{l=0}^{m+k}{\frac{{(m+k)!}^{2}{((n+r)!)}^{2}s^{l}}{l!\>n!\>((m+k-l)!)^{2}(n+r-m-k+l)!(s+1)^{2(m+k)-l+1}}\left(\frac{s}{s+1}\right)^{n+r-m-k+l}}\bigg{]}.$
(C.8)

##### For PASTS

For $r^{th}$ moment,

$\displaystyle\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>$
$\displaystyle=Tr{\left[\hat{a}^{{\dagger}r}\hat{a}^{r}\rho_{PASTS}\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}Tr\bigg{[}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\alpha^{\ast}\beta)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}}{2}-\frac{|\beta|^{2}}{2}\right]}\hat{a}^{{\dagger}r}\hat{a}^{r}|\alpha\rangle\langle\beta|\bigg{]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\alpha^{\ast}\beta)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}}{2}-\frac{|\beta|^{2}}{2}\right]}\left\langle\left.\beta\right|\right.\hat{a}^{{\dagger}r}\hat{a}^{r}\left|\left.\alpha\right\rangle\right.$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\alpha^{\ast}\beta)^{m}(\alpha\beta^{\ast})^{r}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+B\alpha^{\ast}\beta+\alpha\beta^{\ast}-|\alpha|^{2}-|\beta|^{2}\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{x}^{m}\partial_{u}^{r}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+\beta^{2})+x\alpha^{\ast}\beta+u\alpha\beta^{\ast}-|\alpha|^{2}-|\beta|^{2}\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{x}^{m}\partial_{u}^{r}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}\alpha^{\ast
2}-|\alpha|^{2}+(xu|\alpha^{2}|+\frac{C}{2}u^{2}\alpha^{2})\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{x}^{m}\partial_{u}^{r}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+u^{2}\alpha^{2})-|\alpha|^{2}(1-xu)\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial_{x}^{m}\partial_{u}^{r}\left[(1-xu)^{2}-C^{2}u^{2}\right]^{-1/2},$
(C.9)

where $u=1$ and $x=B$.

Similarly, for noisy PASTS at the output,

$\displaystyle\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=tr{\left[\hat{a}^{{\dagger}k}\hat{a}^{r}\phi_{s}(\rho)\right]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}Tr\bigg{[}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\partial_{Y}^{m}\left[{(Y^{2}-C^{2})}^{-1/2}\exp\left[\frac{C_{0}}{2}({{\alpha}^{\ast
2}}+\beta^{2})-(Y_{0}-1+1/s){\alpha^{\ast}\beta}-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]\right]{\hat{a}^{{\dagger}r}\hat{a}^{r}}|\alpha\rangle\langle\beta|\bigg{]}$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\partial_{Y}^{m}\left[{(Y^{2}-C^{2})}^{-1/2}\exp\left[\frac{C_{0}}{2}({{\alpha}^{\ast
2}}+\beta^{2})-(Y_{0}-1+1/s){\alpha^{\ast}\beta}-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]\right]\left\langle\left.\beta\right|\right.\hat{a}^{{\dagger}r}\hat{a}^{r}\left|\left.\alpha\right\rangle\right.$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\partial_{Y}^{m}\left[{(Y^{2}-C^{2})}^{-1/2}(\alpha\beta^{\ast})^{r}\exp\left[\frac{C_{0}}{2}({{\alpha}^{\ast
2}}+\beta^{2})-(Y_{0}-1+\frac{1}{s}){\alpha^{\ast}\beta}+\alpha\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]\right]$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\int\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\partial_{Y}^{m}\partial_{u}^{r}\left[{(Y^{2}-C^{2})}^{-1/2}\exp\left[\frac{C_{0}}{2}({{\alpha}^{\ast
2}}+\beta^{2})-(Y_{0}-1+\frac{1}{s}){\alpha^{\ast}\beta}+u{\alpha\beta^{\ast}}-{|\alpha|^{2}-|\beta|^{2}}\right]\right]$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\int\frac{d^{2}\alpha}{\pi}\partial_{Y}^{m}\partial_{u}^{r}\left[{(Y^{2}-C^{2})}^{-1/2}\exp\left[\frac{C_{0}}{2}({{\alpha}^{\ast
2}})+\frac{C_{0}}{2}u^{2}{{\alpha}^{\ast
2}}-{\left(u\left(Y_{0}-1+\frac{1}{s}\right)+1\right)|\alpha|^{2}}\right]\right]$
$\displaystyle=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\partial_{Y}^{m}\partial_{u}^{r}\left[{(Y^{2}-C^{2})}^{-1/2}\left(\left(u\left(Y_{0}-1+\frac{1}{s}\right)+1\right)^{2}-(C_{0}u)^{2}\right)^{-1/2}\right],$
(C.10)

where $u=1$.

##### For PSSTS

For $r^{th}$ moment,

$\displaystyle\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>$
$\displaystyle=Tr[\hat{a}^{{\dagger}k}\hat{a}^{k}\rho_{PSSTS}]$
$\displaystyle=Tr\bigg{[}\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}\hat{a}^{{\dagger}r}\hat{a}^{r}|\alpha\rangle\langle\beta|\bigg{]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}\langle\beta|\hat{a}^{{\dagger}r}\hat{a}^{r}|\alpha\rangle$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m+r}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\frac{|\alpha|^{2}+|\beta|^{2}}{2}\right]}\langle\beta|\alpha\rangle$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m+r}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\alpha\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{m+r}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+v\alpha\beta^{\ast}-{|\alpha|^{2}-|\beta|^{2}}\right]}\bigg{|}_{v=1}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{m+r}\int\frac{d^{2}\alpha}{\pi}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+v^{2}\alpha^{2})-{(1-Bv)|\alpha|^{2}}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{m+r}\left((1-Bv)^{2}-C^{2}v^{2}\right)^{-1/2}\bigg{|}_{v=1}.$
(C.11)

Similarly, for noisy PSSTS at the output, for $r^{th}$ moment,

$\displaystyle\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=Tr[\hat{a}^{{\dagger}k}\hat{a}^{k}\phi_{s}(\rho_{PSTS})]$
$\displaystyle=Tr\bigg{[}\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\frac{1}{2}{z(\alpha^{\ast}-\beta^{\ast})-z^{\ast}(\alpha-\beta)}-\frac{|\alpha|^{2}+|\beta|^{2}}{2}-\frac{|Z|^{2}}{s}\right]}\hat{a}^{{\dagger}r}\hat{a}^{r}|{\alpha+z}\rangle\langle{\beta+z}|\bigg{]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{\pi
s}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}((\alpha+z)(\beta^{\ast}+z^{\ast}))^{r}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+\frac{1}{2}\\{z(\alpha^{\ast}-\beta^{\ast})-z^{\ast}(\alpha-\beta)\\}-\frac{|\alpha|^{2}+|\beta|^{2}}{2}-\frac{|Z|^{2}}{s}\right]}\langle{\beta+z}|{\alpha+z}\rangle,$

using above, one can see,

$\displaystyle\left<\hat{a}^{{\dagger}r}\hat{a}^{r}\right>=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}((\alpha+z)(\beta^{\ast}+z^{\ast}))^{r}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\beta^{\ast}z-z^{\ast}\alpha-{|\alpha|^{2}+|\beta|^{2}}-(1+\frac{1}{s})|z|^{2}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{r}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-\beta^{\ast}z-z^{\ast}\alpha+v(\alpha+z)(\beta^{\ast}+z^{\ast})-{|\alpha|^{2}+|\beta|^{2}}-(1+\frac{1}{s})|z|^{2}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{r}\int\frac{d^{2}z}{s\pi}\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}(\beta^{\ast}\alpha)^{m}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+v\alpha\beta^{\ast}-{|\alpha|^{2}+|\beta|^{2}}-(1+\frac{1}{s}-v)|z|^{2}-z^{\ast}\alpha(1-v)-\beta^{\ast}z(1-v)\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{r}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\frac{(\beta^{\ast}\alpha)^{m}}{s\left(1+\frac{1}{s}-v\right)}$
$\displaystyle\times\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta+v\alpha\beta^{\ast}-{|\alpha|^{2}+|\beta|^{2}}+\frac{(1-v)^{2}}{\left(1+\frac{1}{s}-v\right)}\alpha\beta^{\ast}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{r}\partial_{u}^{m}\int\frac{d^{2}\alpha}{\pi}\frac{d^{2}\beta}{\pi}\frac{1}{s\left(1+\frac{1}{s}-v\right)}\exp{\left[\frac{C}{2}(\alpha^{\ast
2}+{\beta}^{2})+B\alpha^{\ast}\beta-{|\alpha|^{2}+|\beta|^{2}}+u\alpha\beta^{\ast}\right]}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{r}\partial_{u}^{m}\int\frac{d^{2}\alpha}{\pi}\frac{1}{s\left(1+\frac{1}{s}-v\right)}\exp{\left(\frac{C}{2}(\alpha^{\ast
2}+u^{2}\alpha^{2})-(1-Bu)|\alpha|^{2}\right)}$
$\displaystyle=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial_{v}^{r}\partial_{u}^{m}\frac{1}{s\left(1+\frac{1}{s}-v\right)}\left((1-Bu)^{2}-C^{2}u^{2}\right)^{-1/2},$

where $u=\frac{(1-v)^{2}}{\left(1+\frac{1}{s}-v\right)}+v$ and $v=1$.

## References

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