Characterization of quantumness of non-Gaussian states under the influence of Gaussian channel

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 kth number filtrated thermal state could be further enhanced by adding photon(s).

Appendices

Appendix

Appendix A: Normalization constant

For any density, we have the following,

$$\begin{aligned} \textrm{Tr}[{\hat{\rho }}]=1. \end{aligned}$$

1.1 For PATS

$$\begin{aligned} N_m= & {} {\textrm{Tr}\left[ :{\hat{a}}^{\dag m} e^{-A{\hat{a}}^\dag {\hat{a}}}{\hat{a}}^m:\right] }=\int {\left\langle \alpha \big |\right. :{\hat{a}}^{\dag m} {e^{-A{\hat{a}}^{\dag }{\hat{a}}}}{\hat{a}}^{m}:\big |\left. \alpha \right\rangle }\frac{d^{2}\alpha }{\pi }\\= & {} \int {(\alpha \alpha ^{*})^{m} e^{-A|\alpha |^2}}\frac{d^{2}\alpha }{\pi }, \end{aligned}$$

let $\alpha =x+\iota y$, $\alpha \alpha ^{*}=x^2+y^2=r^2$ and $d^{2}\alpha =r d r d\theta$, now we have,

$$\begin{aligned} N_m=\int {r^{2m+1}e^{-A r^{2}}{}{d r} d\theta =m!/A^{(m+1)}}. \end{aligned}$$

1.2 For PSTS

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

1.3 For PAKFTS

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

We have,

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

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

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

(A.1)

1.4 For PASTS

We have $\textrm{tr}({\hat{\rho }}_{\textrm{PASTS}})=1.$

$$\begin{aligned} N_{a,m}&=\frac{1}{\sqrt{A}}\int \langle \alpha \big |:{\hat{a}}^{\dag m}\exp {\left[ \frac{C}{2}({\hat{a}}^{\dag 2}+{{\hat{a}}}^2)+(B-1){{\hat{a}}}^\dag {\hat{a}}\right] } {\hat{a}}^{m}:\big |\alpha \rangle \frac{d^2\alpha }{\pi }\\&=\frac{1}{\sqrt{A}}\int (\alpha ^*\alpha )^{m}\exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\alpha }^2)+(B-1){\alpha }^{*}\alpha \right] } \frac{d^2\alpha }{\pi }\\&\quad =\frac{1}{\sqrt{A}}\partial _{v}^{m}\int \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\alpha }^2)+v{\alpha }^{*}\alpha \right] } \frac{d^2\alpha }{\pi }\bigg |_{v=B}. \end{aligned}$$

Now, using Eq. (4.4)

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

(A.2)

1.5 For PSSTS

We have $\textrm{Tr}({\hat{\rho }}_{\textrm{PSSTS}})=1.$

$$\begin{aligned} N_{a,m^{-}}&=\textrm{Tr}\bigg [\frac{1}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\frac{|\alpha |^2+|\beta |^2}{2}\right] }{|\alpha \rangle \langle \beta |}\bigg ]\\&=\frac{1}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\frac{|\alpha |^2+|\beta |^2}{2}\right] } {\langle \beta |\alpha \rangle }\\&= \frac{1}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\alpha \beta ^*-{|\alpha |^2-|\beta |^2}\right] }\\&=\frac{1}{\sqrt{A}}\partial _{u}^{m}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }\\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +u \alpha \beta ^*-{|\alpha |^2-|\beta |^2}\right] }\bigg |_{u=1}\\&=\frac{1}{\sqrt{A}}\partial _{u}^{m}\int \frac{d^2\alpha }{\pi }\exp {\left[ \frac{C}{2}(\alpha ^{ *2}+u^2\alpha ^{ 2})-(1-Bu)|\alpha |^2\right] }\bigg |_{u=1}\\&=\frac{1}{\sqrt{A}}\partial _{u}^{m}\left[ (1-Bu)^2-C^2u^2\right] ^{-1/2}\bigg |_{u=1}. \end{aligned}$$

Appendix B: Characteristic function (CF)

1.1 CF For PASTS

$$\begin{aligned} \textrm{Tr}\left[ e^{\gamma {\hat{a}}^\dag }{\hat{\rho }} e^{-\gamma ^{*} {\hat{a}}}\right]&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int {\langle \alpha |}:\exp {\left( \gamma {\hat{a}}^\dag \right) }{\hat{a}}^{\dag m}\\&\quad \times \exp {\left[ \frac{C}{2}({\hat{a}}^{\dag 2}+{{\hat{a}}}^2)+(B-1){{\hat{a}}}^\dag {\hat{a}}\right] } {\hat{a}}^{m}\exp \left( -\gamma ^{*} {\hat{a}}\right) :{|\alpha \rangle }\frac{d^2\alpha }{\pi }\\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int \left( \alpha ^*\alpha \right) ^{m}\\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+\alpha ^{2})+(B-1) \alpha ^{*}\alpha +\gamma \alpha ^{*}-\gamma ^{*}\alpha \right] }\frac{d^2\alpha }{\pi }\\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial _X^{m}\int \exp {\left[ \frac{C}{2}(\alpha ^{*2}+\alpha ^{2})+X \alpha ^{*}\alpha +\gamma \alpha ^{*}-\gamma ^{*}\alpha \right] }\frac{d^2\alpha }{\pi }. \end{aligned}$$

Using Eq. (4.4)

$$\begin{aligned}{} & {} {\textrm{Tr}\left[ e^{\gamma {\hat{a}}^\dag }{\hat{\rho }} e^{-\gamma ^{*} {\hat{a}}}\right] } =\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial _X^{m}\\{} & {} \quad \times \left[ {(X^2- C^2)}^{-1/2}\exp \left[ \frac{\frac{C}{2}({{\gamma }^{*}}^2+\gamma ^{2}) - X {\left| \gamma \right| ^{2}}}{(X^2-C^2)}\right] \right] , \end{aligned}$$

where $X=(1-B)$.

Thus, the characteristic function for PASTS will be-

$$\begin{aligned}{} & {} \chi _{\textrm{in}}\left( \gamma ,\kappa \right) =\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial _X^{m}\\{} & {} \quad \times \left[ {(X^2- C^2)}^{-1/2} \exp \left[ \frac{\frac{C}{2}({{\gamma }^{*}}^2+\gamma ^{2}) - X {\left| \gamma \right| ^{2}}}{(X^2-C^2)}+\frac{\kappa +1}{2}{|\gamma |}^2\right] \right] \end{aligned}$$

Similarly, for noisy PASTS at the output,

$$\begin{aligned}&\textrm{Tr}\left[ e^{\gamma {{\hat{a}}}^\dag }{\hat{\rho }} e^{-\gamma ^*{\hat{a}}}\right] =\frac{N_{a,m}^{-1}}{s \sqrt{A}}\partial _Y^m\\&\qquad \times \left[ \int \frac{d^2\alpha }{\pi }\left\langle \alpha \Bigg | e^{\gamma {{\hat{a}}}^\dag }:e^{-\frac{{{\hat{a}}}^\dag {\hat{a}}}{s}}\left( Y^2-C^2\right) ^{-1/2} \exp \left[ \frac{\frac{C}{2}\left( {{\hat{a}}}^{{\dag 2}}+{{\hat{a}}}^2\right) -Y{{\hat{a}}}^\dag {\hat{a}}}{s^2\left( Y^2-C^2\right) }\right] :e^{-\gamma ^*{\hat{a}}}\Bigg |\alpha \right\rangle \right] \\&\quad =\frac{N_{a,m}^{-1}}{s \sqrt{A}}\partial _Y^m\bigg [\left( Y^2-C^2\right) ^{-1/2}\int \frac{d^2z}{\pi }\\&\qquad \times \exp \left[ -\left( Y_{0}+1/s\right) \alpha ^*\alpha +\left( C_{0}/2\right) \left( {\alpha ^*}^2+\alpha ^2\right) +\gamma \alpha ^*-\gamma ^*\alpha \right] \bigg ], \end{aligned}$$

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

Using Eq. (4.4)

$$\begin{aligned} \textrm{Tr}\left[ e^{\gamma {{\hat{a}}}^\dag }{\hat{\rho }} e^{-\gamma ^*{\hat{a}}}\right]&=\frac{N_{a,m}^{-1}}{s \sqrt{A}}\partial _Y^m\bigg [\left( Y^2-C^2\right) ^{-1/2}\left( \left( Y_{0}+1/s\right) ^2-{C_{0}}^2\right) ^{-1/2}\\&\left. \quad \times \exp \left[ \frac{\frac{C_{0}}{2}\left( {\gamma ^*}^2+\gamma ^2\right) -\left( Y_{0}+1/s\right) \gamma ^*\gamma }{\left( \left( Y_{0}+1/s\right) ^2-{C_{0}}^2\right) }\right] \right] . \end{aligned}$$

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

$$\begin{aligned}{} & {} \chi _{\textrm{out}}\left( \gamma .\kappa \right) =\frac{N_{a,m}^{-1}}{s \sqrt{A}}\partial _Y^{m}\left( Y^2-C^2\right) ^{-1/2}\left( \left( Y_{0}+1/s\right) ^2-{C_{0}}^2\right) ^{-1/2}\\{} & {} \quad \times \exp \left\{ \frac{\frac{C_{0}}{2}\left( {\gamma ^*}^2+\gamma ^2\right) }{\left( \left( Y_{0}+1/s\right) ^2-{C_{0}}^2\right) }-\left( \frac{\left( Y_{0}+1/s\right) }{\left( \left( Y_{0}+1/s\right) ^2-{C_{0}}^2\right) } -\frac{\beta +1}{2}\right) |\gamma |^2\right\} , \end{aligned}$$

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

$$\begin{aligned} \chi _{\textrm{out}}\left( \gamma ,\kappa \right)&=\frac{N_{a,m}^{-1}}{s \sqrt{A}}\partial _Y^{m} \bigg [\left( Y^2-C^2\right) ^{-1/2}\left( \left( Y_0+1/s\right) ^2-{C_0}^2\right) ^{-1/2}\nonumber \\&\left. \quad \times \exp \left[ \frac{C_{1}}{2}\left( {\gamma ^*}^2+\gamma ^2\right) -Y_1\gamma ^*\gamma \right] \right] . \end{aligned}$$

(A.3)

1.2 CF For PSSTS

$$\begin{aligned}&{\textrm{Tr}\left[ e^{-{{\gamma }^{*}}{\hat{a}}}{\hat{\rho }}e^{\gamma {\hat{a}}^{\dag }}\right] }=\textrm{Tr}\bigg [\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \int \frac{d^{2}\alpha }{\pi }\frac{d^{2}\beta }{\pi }(\beta ^{*}\alpha )^{m}\\&\qquad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+\beta ^{2})+B \alpha ^{*}\beta -\frac{|\alpha |^{2}+|\beta |^{2}}{2}\right] } e^{-\gamma ^{*} {\hat{a}}}|\alpha \rangle \langle \beta |e^{\gamma {\hat{a}}^{\dag }}\bigg ]\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\\&\qquad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\gamma ^*\alpha +\gamma \beta ^*-\frac{|\alpha |^2+|\beta |^2}{2}\right] }\langle \beta |\alpha \rangle \\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\\&\qquad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\beta ^*\alpha -\gamma ^*\alpha +\gamma \beta ^*-{|\alpha |^2-|\beta |^2}\right] }\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }\\&\qquad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +u \beta ^*\alpha -\gamma ^*\alpha +\gamma \beta ^*-{|\alpha |^2-|\beta |^2}\right] }\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\int \frac{d^2\alpha }{\pi }\exp {\left[ \frac{C}{2}(\alpha ^{*2})-\gamma ^*\alpha -{|\alpha |^2}\right] }\\&\qquad \times \exp {\big [B\alpha ^*(u \alpha +\gamma )+\frac{C}{2}(u\alpha +\gamma )^{2}\big ]}\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\int \frac{d^2\alpha }{\pi }\\&\qquad \times \exp {\bigg [\frac{C}{2}(\alpha ^{*2}+u^2\alpha ^2)-(1-Bu)|\alpha |^2-\alpha (\gamma ^*-\gamma C u)+B\gamma \alpha ^*+\frac{C}{2}\gamma ^2\bigg ]}\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\bigg [\left( (1-Bu)^2-C^2u^2\right) ^{-1/2}\\&\qquad \times \exp {\bigg [\frac{-(1-Bu)(\gamma ^*-\gamma C u)B \gamma +\frac{C}{2}((\gamma ^*-\gamma C u)^2+u^2 B^2\gamma ^2)}{(\left( 1-Bu)^2-C^2u^2\right) }+\frac{C}{2}\gamma ^2\bigg ]}\bigg ]\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\bigg [\left( (1-Bu)^2-C^2u^2\right) ^{-1/2}\\&\qquad \times \exp \bigg [\frac{\frac{C}{2}\gamma ^{*2}+\gamma ^{2}\left( \frac{C(u^2 B^2+C^2 u^2)}{2}+C u B(1-Bu)\right) -|\gamma |^2(B(1-Bu)+C^2 u)}{(\left( 1-Bu)^2-C^2u^2\right) }\\&\qquad +\frac{C}{2}\gamma ^2\bigg ]\bigg ]\bigg |_{u=1}. \end{aligned}$$

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

$$\begin{aligned} \chi _{\textrm{PSSTS}}\left( \gamma ,\kappa \right)&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\big [\left( (1-Bu)^2-C^2u^2\right) ^{-1/2}\nonumber \\&\quad \times \exp {\left( A_1\gamma ^{*2}+A_2\gamma ^{ 2}-A_3|\gamma |^2\right) }\big ]\bigg |_{u=1}, \end{aligned}$$

(A.4)

where $A_1=\frac{\left( C/2\right) }{\left( (1-Bu)^2-C^2u^2\right) }$, $A_2=\frac{\left( \frac{C(u^2 B^2+C^2 u^2)}{2}+B C u(1-Bu)\right) }{\left( (1-Bu)^2-C^2u^2\right) }+\frac{C}{2}$, $A_3=\frac{(B(1-Bu)+C^2 u)}{\left( (1-Bu)^2-C^2u^2\right) }-\frac{\kappa -1}{2}.$

Similarly, for noisy PSSTS at the output

$$\begin{aligned}&{\textrm{Tr}\left[ e^{-\gamma ^{*} {\hat{a}}}\phi _{s}({\hat{\rho }})e^{\gamma {\hat{a}}^\dag }\right] }=\textrm{Tr}\bigg [\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2z}{s\pi }\frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } (\beta ^*\alpha )^m\\&\qquad \times \exp {\left( -\gamma ^{*} {\hat{a}}\right) }\big |{\alpha +z}\rangle \langle {\beta +z}\big |\exp {\left( \gamma {\hat{a}}^\dag \right) }\\&\qquad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\frac{1}{2}\{z(\alpha ^*-\beta ^*) -z^*(\alpha -\beta )\}\right. \\&\qquad \left. -\frac{|\alpha |^2+|\beta |^2}{2}-\frac{|z|^2}{s}\right] \bigg ]\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2z}{s\pi }\frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } (\beta ^*\alpha )^m \langle {\beta +z}\big |{\alpha +z}\rangle \\&\qquad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\frac{1}{2}\{(z(\alpha ^*-\beta ^*)-z^*(\alpha -\beta ))\}\right. \\&\qquad \left. +\gamma (z^*+\beta ^*) -\gamma ^*(z+\alpha )-\frac{|\alpha |^2+|\beta |^2}{2}-\frac{|z|^2}{s}\right] , \end{aligned}$$

one can have,

$$\begin{aligned}&\langle {\beta +z}|{\alpha +z}\rangle =\exp {\bigg [-\frac{|\alpha +z|^2}{2}-\frac{|\beta +z|^2}{2}+(\beta ^{*}+z^*)(\alpha +z)\bigg ]}\\&\quad =\exp \bigg [-\frac{|\alpha |^2+|z|^2+\alpha z^*+z \alpha ^*}{2}-\frac{|\beta |^2+|z|^2+\beta z^*+z \beta ^*}{2}\\&\qquad +(\beta ^{*}\alpha +\beta ^{*}z+z^*\alpha +|z|^2)\bigg ]\\&\quad =\exp \left[ -\frac{|\alpha |^2}{2}-\frac{|\beta |^2}{2} -\frac{1}{2}\{z(\alpha ^*-\beta ^*)-z^*(\alpha -\beta )\}+\beta ^*\alpha \right] .\\&{\textrm{Tr}\left[ e^{-\gamma ^{*} {\hat{a}}}\phi _{s}({\hat{\rho }})e^{\gamma {\hat{a}}^\dag }\right] }=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2z}{s\pi }\frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } (\beta ^*\alpha )^m\\&\qquad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\beta ^*\alpha -z\gamma ^*\right. \\&\qquad \left. +z^*\gamma +\gamma \beta ^*-\gamma ^*\alpha -|\alpha |^2-|\beta |^2-\frac{|z|^2}{s}\right] \\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } (\beta ^*\alpha )^m\exp \bigg [\frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta \\&\qquad +\beta ^*\alpha +\gamma \beta ^*-\gamma ^*\alpha -|\alpha |^2-|\beta |^2-{s}{|\gamma |^2}\bigg ]\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\exp \bigg [\frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta \\&\qquad +\beta ^*\alpha -\gamma ^*\alpha +\gamma \beta ^*-{|\alpha |^2-|\beta |^2}-{s}{|\gamma |^2}\bigg ]\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }\\&\qquad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +u \beta ^*\alpha -\gamma ^*\alpha +\gamma \beta ^*-{|\alpha |^2-|\beta |^2}-{s}{|\gamma |^2}\right] \bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\int \frac{d^2\alpha }{\pi }\exp {\left[ \frac{C}{2}(\alpha ^{*2})-\gamma ^*\alpha -{|\alpha |^2}\right] }\\&\qquad \times \exp {\bigg [B\alpha ^*(u \alpha +\gamma )+\frac{C}{2}(u\alpha +\gamma )^{2}-{s}{|\gamma |^2}\bigg ]}\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\bigg [\int \frac{d^2\alpha }{\pi }\exp \bigg [\frac{C}{2}(\alpha ^{*2}+u^2\alpha ^2)-(1-Bu)|\alpha |^2-\alpha (\gamma ^*-\gamma C u)\\&\qquad +B\gamma \alpha ^*+\frac{C}{2}\gamma ^2-{s}{|\gamma |^2}\bigg ]\bigg ]\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\left( (1-Bu)^2-C^2u^2\right) ^{-\frac{1}{2}}\\&\qquad \times \exp \bigg [\frac{-(1-Bu)(\gamma ^*-\gamma C u)B \gamma +\frac{C}{2}((\gamma ^*-\gamma C u)^2+u^2 B^2\gamma ^2)}{(\left( 1-Bu)^2-C^2u^2\right) }\\&\qquad +\frac{C}{2}\gamma ^2-{s}{|\gamma |^2}\bigg ]\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\bigg [\left( (1-Bu)^2-C^2u^2\right) ^{-1/2}\\&\qquad \times \exp \bigg [\frac{\frac{C}{2}\gamma ^{*2}+\gamma ^{2}\left( \frac{C(u^2 B^2+C^2 u^2)}{2}+C u B(1-Bu)\right) -|\gamma |^2(B(1-Bu)+C^2 u)}{(\left( 1-Bu)^2-C^2u^2\right) }\\&\qquad +\frac{C}{2}\gamma ^2-{s}{|\gamma |^2}\bigg ]\bigg ]\bigg |_{u=1}\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\bigg [\left( (1-Bu)^2-C^2u^2\right) ^{-1/2}\exp {\left( N_1\gamma ^{*2}+N_2\gamma ^{ 2}-N_3|\gamma |^2\right) }\bigg ]\bigg |_{u=1}, \end{aligned}$$

where $N_1=\frac{\left( {C/2}\right) }{\left( (1-Bu)^2-C^2u^2\right) },N_2=\frac{\left( \frac{C(u^2 B^2+C^2 u^2)}{2}+B C u(1-Bu)\right) }{\left( (1-Bu)^2-C^2u^2\right) }+\frac{C}{2},N_3=\frac{(B(1-Bu)+C^2 u)}{\left( (1-Bu)^2-C^2u^2\right) }+s$.

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

$$\begin{aligned} \chi \left( \gamma ,\kappa \right)&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _{u}^{m}\bigg [\left( (1-Bu)^2-C^2u^2\right) ^{-1/2}\nonumber \\&\quad \times \exp {\bigg [-\left( N_1-\frac{(\kappa -1)}{2}\right) |\gamma |^2+N_2\gamma ^2+N_3\gamma ^{*2}\bigg ]}\bigg ]\bigg |_{u=1}. \end{aligned}$$

(A.5)

Appendix C: $r^{\textrm{th}}$ moment for input and output states

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

$$\begin{aligned} \left\langle {\hat{a}}^{\dag r}{\hat{a}}^r\right\rangle =\textrm{Tr}\left[ {\hat{\rho }}{\hat{a}}^{\dag r}{\hat{a}}^r\right] . \end{aligned}$$

(A.6)

1.1 For PATS

$r^{\textrm{th}}$-moment of PATS can be find using Eq. (A.6),

$$\begin{aligned} \langle {n}|{\hat{a}}^r{\hat{\rho }}{\hat{a}}^{\dag r}|n\rangle&=N_{m}^{-1}\langle {n}|{\hat{a}}^r:{\hat{a}}^{\dag m} e^{-A{\hat{a}}^\dag {\hat{a}}}{\hat{a}}^{m}:{\hat{a}}^{\dag r}|n\rangle \\&=N_{m}^{-1}\frac{(n+r)!}{n!}\langle {n+r}|:{\hat{a}}^{\dag m} e^{-A{\hat{a}}^\dag {\hat{a}}}{\hat{a}}^{ m}:|n+r\rangle \\&=N_{m}^{-1}\frac{(n+r)!}{n!}\sum _{l=0}{\frac{(n+r)!}{(n+r-m)!}}\langle {n+r-m}|: (-A)^{l}\frac{{\hat{a}}^{\dag k}{\hat{a}}^{l}}{l!}:|n+r-m\rangle \\&=N_{m}^{-1}\frac{(n+r)!}{n!}\sum _{l=0}{\frac{(n+r)!}{(n+r-m)!}}\frac{(-A)^{l}(n+r-m)}{l!\,(n+r-m-l)}\\&=N_{m}^{-1}\frac{(n+r)!}{n!}{\frac{(n+r)!}{(n+r-m)!}}(1-A)^{n+r-m}. \end{aligned}$$

Thus, one can see,

$$\begin{aligned} \langle {\hat{a}}^{\dag r}{\hat{a}}^r\rangle&=\textrm{Tr}\left[ {\hat{\rho }}{\hat{a}}^{\dag r}{\hat{a}}^r\right] =\sum _{n=0}^{\infty }\langle {n}|{\hat{a}}^r{\hat{\rho }}{\hat{a}}^{\dag r}|n \rangle \nonumber \\&=N_{m}^{-1} {\sum _{n=0}^{\infty } {\frac{\left( n+r\right) !^2}{n!\left( n+r-m\right) !} \left( 1-A\right) ^{n+r-m}}}. \end{aligned}$$

(A.7)

Similarly, for noisy PATS at the output,

$$\begin{aligned} \langle {\hat{a}}^{\dag r}{\hat{a}}^{r}\rangle&=\textrm{Tr}\left[ {{\Phi }_s\left( {\hat{\rho }}_{\textrm{PATS}}\right) }{\hat{a}}^{\dag r}{\hat{a}}^r\right] =\sum _{n=0}\langle {n}|{\hat{a}}^r{\Phi }_s\left( {\hat{\rho }}_{\textrm{PATS}}\right) {\hat{a}}^{\dag r}|n \rangle \\&=N_{m}^{-1}:\sum _{l=0}^{m}\frac{{m!}^2 (n+r)! \langle {n+r}|\left( {{\hat{a}}}^\dag \right) ^{m-l}\exp \left( -\frac{A{{\hat{a}}}^{\dag }{\hat{a}}}{As+1}\right) \left( {\hat{a}}\right) ^{m-l}|{n+r}\rangle s^l}{n!\,l!\,{\left( m-l\right) !)}^2\left( As+1\right) ^{2m-l+1}}: , \end{aligned}$$

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

$$\begin{aligned} \langle {\hat{a}}^{\dag r}{\hat{a}}^{r}\rangle&=\textrm{Tr}\left[ \phi _s({\hat{\rho }}){\hat{a}}^{\dag r}{\hat{a}}^{r}\right] \nonumber \\&=N_{m}^{-1} \sum _{l=0}^{m} \frac{{m!}^{2} s^{l}}{l!{\left( m-l\right) !)}^{2}\left( As+1\right) ^{2m-l+1}}\sum _{n=0}{\frac{\left( n+r\right) !^{2}}{n!\left( n+r-m+l\right) !}}\nonumber \\&\quad \times \left( 1-\frac{A}{As+1}\right) ^{n+r-m+l} . \end{aligned}$$

(A.8)

1.2 For PSTS

For $r^{\textrm{th}}$ moment

$$\begin{aligned} \langle {\hat{a}}^{\dag r}{\hat{a}}^r\rangle&=\textrm{Tr}[{\hat{a}}^{\dag r}{\hat{a}}^r{\hat{\rho }}_{\textrm{PSTS}}]=N_{m^-}^{-1}\textrm{Tr}\bigg [\int {\frac{d^2 \alpha }{\pi }{(\alpha ^{*}\alpha )^m}e^{-\frac{|\alpha |^2}{n_{\textrm{th}}}}{\hat{a}}^{\dag r}{\hat{a}}^r\big |\alpha \rangle \langle \alpha |\bigg ]}\nonumber \\&=N_{m^-}^{-1}\int {\frac{d^2 \alpha }{\pi }{(\alpha ^{*}\alpha )^{m+r}}e^{-\frac{|\alpha |^2}{n_{\textrm{th}}}}}=N_{m^-}^{-1} (m+r)! (n_{\textrm{th}})^{m+r+1}. \end{aligned}$$

(A.9)

Similarly for noisy PSTS at the output, for $r^{\textrm{th}}$ moment

$$\begin{aligned} \left\langle {\hat{a}}^{\dag r}{\hat{a}}^r\right\rangle&=\textrm{Tr}[{\hat{a}}^{\dag r}{\hat{a}}^r\phi _{s}({\hat{\rho }})]\nonumber \\&=N_{m^-}^{-1}\textrm{Tr}\bigg [\int \int {\frac{d^2 \alpha }{\pi }\frac{d^2 z}{\pi s}{(\alpha ^{*}\alpha )^m}\exp {\bigg \{-\frac{|\alpha |^2}{n_{\textrm{th}}}-\frac{|z|^2}{s}\bigg \}}{\hat{a}}^r|\alpha +z\rangle \langle \alpha +z|{\hat{a}}^{\dag r}}\bigg ]\nonumber \\&=N_{m^-}^{-1}\int \int {\frac{d^2 \alpha }{\pi }\frac{d^2 z}{\pi s}{(\alpha ^{*}\alpha )^m}{|\alpha +z|^{2r}}\exp {\bigg [-\frac{|\alpha |^2}{n_{\textrm{th}}}-\frac{|z|^2}{s}\bigg ]}\langle \alpha +z|\alpha +z\rangle }\nonumber \\&=N_{m^-}^{-1}\partial _{u}^{r}\int \int {\frac{d^2 \alpha }{\pi }\frac{d^2 z}{\pi s}{(\alpha ^{*}\alpha )^m}\exp {\bigg [-\frac{|\alpha |^2}{n_{\textrm{th}}}-\frac{|z|^2}{s}+u|\alpha +z|^2\bigg ]}}\bigg |_{u=0}\nonumber \\&=N_{m^-}^{-1}\partial _{u}^{r}\int \int \frac{d^2 \alpha }{\pi }\frac{d^2 z}{\pi s}{(\alpha ^{*}\alpha )^m}\exp \bigg [-(1/n_{\textrm{th}}-u)|\alpha |^2-\left( \frac{1}{s}-u\right) |z|^2\nonumber \\&\quad +u(z\alpha ^{*}+z^*\alpha )\bigg ] \bigg |_{u=0}\nonumber \\&=N_{m^-}^{-1}\partial _{u}^{r}\left[ \frac{1}{\left( \frac{1}{s}-u\right) s}\int {\frac{d^2 \alpha }{\pi }{(\alpha ^{*}\alpha )^m}\exp {\bigg [-\left( \frac{1}{n_{\textrm{th}}}-u-\frac{u^2}{\frac{1}{s}-u}\right) |\alpha |^2\bigg ]}}\right] \bigg |_{u=0}\nonumber \\&=N_{m^-}^{-1}\partial _{u}^{r}\left[ \frac{1}{\left( \frac{1}{s}-u\right) s}\frac{m!}{\left( \frac{1}{n_{\textrm{th}}}-u-\frac{u^2}{\frac{1}{s}-u}\right) ^{m+1}}\right] \bigg |_{u=0}. \end{aligned}$$

(A.10)

1.3 For PAKFTS

We have density operator for PAKFTS

$$\begin{aligned} {\hat{\rho }}_{\text {PAKFT}} = N_{\text {km}}^{- 1}\left[ :{{{\hat{a}}}^{\dag m}e^{- A{{\hat{a}}}^{\dag }{\hat{a}}}{\hat{a}}}^{m}: - \frac{e^{- \beta \hbar \omega k}}{k!}:{{{\hat{a}}}^{\dag (m + k)}e}^{- {{\hat{a}}}^{\dag }{\hat{a}}}{{\hat{a}}}^{(m + k)}: \right] , \end{aligned}$$

using the above and Eq. (A.6), $r^{\textrm{th}}$ moment-

$$\begin{aligned} \left\langle {{{\hat{a}}}^{\dag r}{\hat{a}}^{r}} \right\rangle&= N_{\text {km}}^{- 1}\, \text {Tr}\left[{{{\hat{a}}}^{r}:{{{\hat{a}}}^{\dag m}e^{- A{{\hat{a}}}^{\dag }{\hat{a}}}{\hat{a}}}^{m}:{\hat{a}}}^{\dag r} \right]\nonumber \\&\quad - N_{\text {km}}^{- 1}\ \frac{e^{- \beta \hbar \omega k}}{k!} \text {Tr}\left[{{{\hat{a}}}^{r}:{{{\hat{a}}}^{\dag (m + k)}e}^{- {{\hat{a}}}^{\dag }{\hat{a}}}{{\hat{a}}}^{(m + k)}:{\hat{a}}}^{\dag r} \right]\end{aligned}$$

(A.11)

$$\begin{aligned}&= N_{\text {km}}^{- 1}\sum _{n = 0}^{}\left[ \frac{{((n + r)!)}^{2}}{n!(n + r - m)!}(1 - A)^{n + r - m} \right. \nonumber \\&\left. \quad - \frac{e^{- \beta \hbar \omega k}}{k!}\frac{{((n + r)!)}^{2}}{n!\ (n + r - m - k)!}\left| \left\langle n + r - m - k \Bigg | 0 \right\rangle \right| ^{2} \right] . \end{aligned}$$

(A.12)

Similarly, for noisy PAKFTS the output,

$$\begin{aligned} \left\langle {{{\hat{a}}}^{\dag r}{\hat{a}}^{r}} \right\rangle&= N_{\text {km}}^{- 1}\sum _{n = 0}\left[ \sum _{l = 0}^{m}\frac{{m!}^{2}{{((n + r)!)}^{2}s}^{l}}{l!\,n!\,((m - l)!)^{2}\,(n + r - m + l)!\left( \text {As} + 1 \right) ^{2m - l + 1}}\right. \nonumber \\&\quad \times \left( \frac{\text {As} -A + 1}{\text {As} + 1} \right) ^{n + r - m +l} - \frac{e^{- \beta \hbar \omega k}}{k!}\sum _{l = 0}^{m + k}\nonumber \\&\quad \times \frac{{(m + k)!}^{2}{((n + r)!)}^{2}s^{l}}{l!\,n!\,((m + k - l)!)^{2}(n + r - m - k + l)!(s + 1)^{2(m + k) - l + 1}}\nonumber \\&\left. \quad \times \left( \frac{s}{s + 1} \right) ^{n + r - m - k + l} \right] . \end{aligned}$$

(A.13)

1.4 For PASTS

For $r^{\textrm{th}}$ moment,

$$\begin{aligned} \left\langle {\hat{a}}^{\dag r}{\hat{a}}^{r}\right\rangle&=\textrm{Tr}{\left[ {\hat{a}}^{\dag r}{\hat{a}}^{r}{\hat{\rho }}_{\textrm{PASTS}}\right] }\nonumber \\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\textrm{Tr}\bigg [\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\alpha ^*\beta )^{ m}\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+\beta ^2)+B\alpha ^*\beta -\frac{|\alpha |^2}{2}-\frac{|\beta |^2}{2}\right] }{\hat{a}}^{\dag r}{\hat{a}}^r|\alpha \rangle \langle \beta |\bigg ] \nonumber \\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\alpha ^*\beta )^{ m}\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+\beta ^2)+B\alpha ^*\beta -\frac{|\alpha |^2}{2}-\frac{|\beta |^2}{2}\right] }\langle \beta |{\hat{a}}^{\dag r}{\hat{a}}^r|\alpha \rangle \nonumber \\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\alpha ^*\beta )^{m}(\alpha \beta ^*)^{r}\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+\beta ^2)+B\alpha ^*\beta +\alpha \beta ^*-|\alpha |^2-|\beta |^2\right] }\nonumber \\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial _{x}^{m}\partial _{u}^{r}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+\beta ^2)+x\alpha ^*\beta +u\alpha \beta ^*-|\alpha |^2-|\beta |^2\right] }\bigg |_{x=B, u=1}\nonumber \\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial _{x}^{m}\partial _{u}^{r}\int \frac{d^2\alpha }{\pi }\exp {\left[ \frac{C}{2}\alpha ^{*2}-|\alpha |^2+( x u |\alpha |^{2}+\frac{C}{2}u^2\alpha ^2)\right] }\bigg |_{x=B, u=1}\nonumber \\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial _{x}^{m}\partial _{u}^{r}\int \frac{d^2\alpha }{\pi }\exp {\left[ \frac{C}{2}(\alpha ^{*2}+u^2\alpha ^2)-|\alpha |^2(1-x u)\right] }\bigg |_{x=B, u=1}\nonumber \\&=\frac{N_{a,m}^{-1}}{\sqrt{A}}\partial _{x}^{m}\partial _{u}^{r}\left[ (1-x u)^2-C^2u^2\right] ^{-1/2}\bigg |_{x=B, u=1} . \end{aligned}$$

(A.14)

Similarly, for noisy PASTS at the output,

$$\begin{aligned} \left\langle {\hat{a}}^{\dag r}{\hat{a}}^{r}\right\rangle&=\textrm{tr}{\left[ {\hat{a}}^{\dag k}{\hat{a}}^{r}\phi _s({\hat{\rho }})\right] }\nonumber \\&=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\textrm{Tr}\bigg [\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }\partial _Y^{m}\nonumber \\&\quad \times \left[ {(Y^2- C^2)}^{-1/2} \exp \left[ \frac{C_0}{2}({{\alpha }^{*2}}+\beta ^2) - (Y_0-1+1/s) {\alpha ^*\beta }\right. \right. \nonumber \\&\quad \left. \left. -\frac{|\alpha |^2+|\beta |^2}{2}\right] \right] {{\hat{a}}^{\dag r}{\hat{a}}^{r}}|\alpha \rangle \langle \beta |\bigg ]\nonumber \\&=\frac{N_{a,m}^{-1}}{s\sqrt{A}}\int \int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }\partial _Y^{m}\nonumber \\&\quad \times \left[ {(Y^2- C^2)}^{-1/2} \exp \left[ \frac{C_0}{2}({{\alpha }^{*2}}+\beta ^2) - (Y_0-1+1/s) {\alpha ^*\beta }\right. \right. \nonumber \\&\quad \left. \left. -\frac{|\alpha |^2+|\beta |^2}{2}\right] \right] \langle \beta |{\hat{a}}^{\dag r}{\hat{a}}^r|\alpha \rangle \nonumber \\&=\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 ^*)^r \right. \nonumber \\&\quad \times \left. \exp \left[ \frac{C_0}{2}({{\alpha }^{*2}}+\beta ^2) - \left( Y_0-1+\frac{1}{s}\right) {\alpha ^*\beta }+\alpha \beta ^*-{|\alpha |^2-|\beta |^2}\right] \right] \nonumber \\&=\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}\nonumber \\&\quad \times \left[ {(Y^2- C^2)}^{-1/2} \exp \left[ \frac{C_0}{2}({{\alpha }^{*2}}+\beta ^2) - \left( Y_0-1+\frac{1}{s}\right) \right. \right. \nonumber \\&\quad \times {\alpha ^*\beta }+u{\alpha \beta ^*}-{|\alpha |^2-|\beta |^2}\bigg ]\bigg ]\nonumber \\&=\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}\right. \nonumber \\&\quad \left. \times \exp \left[ \frac{C_0}{2}{{\alpha }^{*2}}+\frac{C_0}{2}u^2{{\alpha }^{*2}}-{\left( u\left( Y_0-1+\frac{1}{s}\right) +1\right) |\alpha |^2}\right] \right] \nonumber \\&=\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] , \end{aligned}$$

(A.15)

where $u=1$.

1.5 For PSSTS

For $r^{\textrm{th}}$ moment,

$$\begin{aligned} \left\langle {\hat{a}}^{\dag r}{\hat{a}}^r\right\rangle&=\textrm{Tr}[{\hat{a}}^{\dag k}{\hat{a}}^k{\hat{\rho }}_{\textrm{PSSTS}}]\nonumber \\&=\textrm{Tr}\bigg [\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\frac{|\alpha |^2+|\beta |^2}{2}\right] }{\hat{a}}^{\dag r}{\hat{a}}^r|\alpha \rangle \langle \beta |\bigg ]\nonumber \\&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\frac{|\alpha |^2+|\beta |^2}{2}\right] }\langle \beta |{\hat{a}}^{\dag r}{\hat{a}}^r|\alpha \rangle \nonumber \\&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^{m+r}\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\frac{|\alpha |^2+|\beta |^2}{2}\right] }\langle \beta |\alpha \rangle \nonumber \\&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^{m+r}\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\alpha \beta ^*-{|\alpha |^2-|\beta |^2}\right] }\nonumber \\&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _v^{m+r}\int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }\nonumber \\&\quad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +v\alpha \beta ^*-{|\alpha |^2-|\beta |^2}\right] } \bigg |_{v=1}\nonumber \\&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _v^{m+r}\int \frac{d^2\alpha }{\pi }\exp {\left[ \frac{C}{2}(\alpha ^{*2}+v^2\alpha ^2)-{(1-B v)|\alpha |^2}\right] }\bigg |_{v=1}.\nonumber \\&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _v^{m+r}\left( (1-B v)^2-C^2 v^2\right) ^{-1/2}\bigg |_{v=1}. \end{aligned}$$

(A.16)

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

$$\begin{aligned} \left\langle {\hat{a}}^{\dag r}{\hat{a}}^r\right\rangle&=\textrm{Tr}[{\hat{a}}^{\dag k}{\hat{a}}^k\phi _s({\hat{\rho }}_{\textrm{PSTS}})]=\textrm{Tr}\bigg [ \frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2 z}{s\pi }\frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi }(\beta ^*\alpha )^m\\&\quad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\frac{1}{2}\{z(\alpha ^*-\beta ^*)-z^*(\alpha -\beta )\}\right. \\&\left. \quad -\frac{|\alpha |^2+|\beta |^2}{2}-\frac{|z|^2}{s}\right] {\hat{a}}^{\dag r}{\hat{a}}^r|{\alpha +z} \rangle \langle {\beta +z}|\bigg ]\\&=\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\int \frac{d^2 z}{\pi s}\frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } (\beta ^*\alpha )^m((\alpha +z)(\beta ^*+z^*))^r\\&\quad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +\frac{1}{2}\{z(\alpha ^*-\beta ^*)-z^*(\alpha -\beta )\}\right. \\&\left. \quad -\frac{|\alpha |^2+|\beta |^2}{2}-\frac{|z|^2}{s}\right] \langle {\beta +z}|{\alpha +z}\rangle , \end{aligned}$$

using above, one can see,

$$\begin{aligned}&\left\langle {\hat{a}}^{\dag r}{\hat{a}}^r\right\rangle =\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 ^*\alpha )^m((\alpha +z)(\beta ^*+z^*))^r\\&\qquad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\beta ^*z-z^*\alpha -{|\alpha |^2+|\beta |^2}-\left( 1+\frac{1}{s}\right) |z|^2\right] }\\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _v^r\int \frac{d^2z}{s\pi }\frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } (\beta ^*\alpha )^m\\&\qquad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -\beta ^*z-z^*\alpha +v(\alpha +z)(\beta ^*+z^*)-{|\alpha |^2+|\beta |^2}\right. \\&\qquad \left. -\left( 1+\frac{1}{s}\right) |z|^2\right] \\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _v^r\int \frac{d^2z}{s\pi }\frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } (\beta ^*\alpha )^m\\&\qquad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +v\alpha \beta ^*-{|\alpha |^2+|\beta |^2}\right. \\&\qquad \left. -\left( 1+\frac{1}{s}-v\right) |z|^2-z^*\alpha (1-v)-\beta ^*z(1-v)\right] \\&\quad =\frac{N_{a,m^{-}}^{-1}}{\sqrt{A}}\partial _v^r\int \frac{d^2\alpha }{\pi }\frac{d^2\beta }{\pi } \frac{(\beta ^*\alpha )^m}{s\left( 1+\frac{1}{s}-v\right) }\\&\qquad \times \exp \left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta +v\alpha \beta ^*-{|\alpha |^2+|\beta |^2}+\frac{(1-v)^2}{\left( 1+\frac{1}{s}-v\right) }\alpha \beta ^*\right] \\&\quad =\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) }\\&\qquad \times \exp {\left[ \frac{C}{2}(\alpha ^{*2}+{\beta }^2)+B\alpha ^*\beta -{|\alpha |^2+|\beta |^2}+u\alpha \beta ^*\right] }\\&\quad =\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 ^{*2}+u^2\alpha ^2)-(1-B u)|\alpha |^2\right) }\\&\quad =\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-B u)^2-C^2 u^2\right) ^{-1/2}, \end{aligned}$$

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