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##  Appendix A Proof of Lemma 2.1

We restate Lemma 2.1 here for convenience.

######  Lemma A.1 (Lemma 2.1, restated).

Let $X_{1},\dots,X_{N}$ be pairwise independent random variables over
$\\{0,1\\}$ such that $\Pr[X_{i}=1]=p>0$ for $i=1,\dots,N$. Then, it holds
that

$\Pr\left[\forall i\in[N],X_{i}=0\right]\leqslant\frac{1}{pN}.$

###### Proof.

Let $X=\sum_{i=1}^{N}X_{i}$ and note that $\operatorname*{\mathrm{E}}[X]=pN$.
We have

$\Pr\left[X=0\right]\leqslant\Pr\left[\left|X-\operatorname*{\mathrm{E}}[X]\right|\geqslant
pN\right]\leqslant\frac{\operatorname*{\mathrm{Var}}[X]}{(pN)^{2}}=\frac{(1-p)pN}{(pN)^{2}}\leqslant\frac{1}{pN},$

where the second inequality follows from Chebyshev’s inequality and the
equality holds due to the pairwise independence of the $X_{i}$’s. ∎

##  Appendix B Proof of Theorem 3.4

We restate Theorem 3.4 here for convenience.

######  Theorem B.1 (Restatement of Theorem 3.4, $q$-ary BCH codes).

Fix a prime power $q$. Then, given integers $m^{\prime}=q^{r}-1$ and
$1\leqslant d\leqslant m^{\prime}$ for some positive integer $r$, it is
possible to construct in time $\operatorname{poly}(m^{\prime})$ a generator
matrix $G_{\textnormal{BCH}}\in\mathbb{F}_{q}^{m^{\prime}\times n^{\prime}}$
such that
$\mathcal{C}_{\textnormal{BCH}}=\mathcal{C}(G_{\textnormal{BCH}})\subseteq\mathbb{F}_{q}^{m^{\prime}}$
has minimum distance at least $d$ and codimension

$\displaystyle m^{\prime}-n^{\prime}$
$\displaystyle\leqslant\lceil(d-1)(1-1/q)\rceil\log_{q}(m^{\prime}+1).$

###### Proof.

Let $\alpha$ be a primitive element of $\mathbb{F}_{q^{r}}$. For a given
blocklength $m^{\prime}=q^{r}-1$ and design distance $d\leqslant m^{\prime}$,
we define the (narrow-sense, primitive) $q$-ary BCH code
$\mathcal{C}_{\textnormal{BCH}}\subseteq\mathbb{F}_{q}^{m^{\prime}}$ as

$\mathcal{C}_{\textnormal{BCH}}=\left\\{(c_{0},\dots,c_{m^{\prime}-1})\in\mathbb{F}_{q}:f(x)=\sum_{i=0}^{m^{\prime}-1}c_{i}x^{i}\in\mathbb{F}_{q^{r}}[x]\textnormal{
satisfies }f(\alpha^{i})=0\textnormal{ for }i=1,\dots,d-1\right\\}.$ (13)

By [Gur10, Theorem 3], it holds that $\mathcal{C}_{\textnormal{BCH}}$ is a
subset of a Reed-Solomon code over $\mathbb{F}_{q^{r}}$ with blocklength
$m^{\prime}$ and minimum distance $d$. Therefore,
$\mathcal{C}_{\textnormal{BCH}}$ has minimum distance at least $d$.

To see the claim about the codimension, note that each constraint of the form
$f(\alpha^{i})=0$ over $\mathbb{F}_{q^{r}}$ corresponds to
$r=\log_{q}(m^{\prime}+1)$ linear constraints over $\mathbb{F}_{q}$. Moreover,
if $f(\gamma)=0$ for some $\gamma\in\mathbb{F}_{q^{r}}$, it follows that

$0=f(\gamma)^{q}=\sum_{i=0}^{m^{\prime}-1}c_{i}^{q}\left(\gamma^{i}\right)^{q}=\sum_{i=0}^{m^{\prime}-1}c_{i}\left(\gamma^{q}\right)^{i}=f(\gamma^{q}),$

where we have used the fact that
$(\beta_{1}+\beta_{2})^{q}=\beta_{1}^{q}+\beta_{2}^{q}$ for any
$\beta_{1},\beta_{2}\in\mathbb{F}_{q^{r}}$ and that $\beta^{q}=\beta$ for all
$\beta\in\mathbb{F}_{q}$ (with the natural embedding of $\mathbb{F}_{q}$ in
$\mathbb{F}_{q^{r}}$). Therefore, there are at least
$\left\lfloor\frac{d-1}{q}\right\rfloor$ redundant constraints over
$\mathbb{F}_{q^{r}}$ in Equation 13. Combining both observations above shows
that the codimension of $\mathcal{C}_{\textnormal{BCH}}$ is at most

$\lceil(d-1)(1-1/q)\rceil\log_{q}(m^{\prime}+1).$

Finally, since each constraint $f(\alpha^{i})=0$ over $\mathbb{F}_{q^{r}}$ can
be transformed into $r$ linear constraints over $\mathbb{F}_{q}$ in time
$\operatorname{poly}(q^{r})=\operatorname{poly}(m^{\prime})$, we can construct
the parity-check matrix (and hence the generator matrix
$G_{\textnormal{BCH}}$) of $\mathcal{C}_{\textnormal{BCH}}$ in time
$\operatorname{poly}(m^{\prime})$ as well. ∎

##  Appendix C The Haviv-Regev conditions for general $\ell_{p}$ norms

In this section, we prove a generalization of Haviv-Regev’s conditions which
allow tensoring of approximate $\mathrm{SVP}_{p}$ instances for all
$p\geqslant 1$. Our argument follows that of [HR12] for $p=2$ closely. For
convenience, we restate the key lemma here. See 5.1

We will require some auxiliary lemmas in order to prove Lemma 5.1, starting
with a version of Minkowski’s first theorem for (possibly non-full-rank)
lattices in the $\ell_{2}$ norm.

######  Lemma C.1 (Minkowski’s first theorem).

Let $\mathcal{L}$ be a rank-$r$ lattice. Then, it holds that

$\det(\mathcal{L})\geqslant\left(\frac{\lambda_{1}^{(2)}(\mathcal{L})}{\sqrt{r}}\right)^{r}.$

We will also need to relate $\ell_{p}$ norms to the $\ell_{2}$ norm. When
$p\geqslant 2$, a standard application of Hölder’s inequality yields

$\|\boldsymbol{v}\|_{2}\geqslant\|\boldsymbol{v}\|_{p}\geqslant|\mathsf{supp}(\boldsymbol{v})|^{1/p-1/2}\|\boldsymbol{v}\|_{2},$

where $\mathsf{supp}(\boldsymbol{v})=\\{i\in[m]:\boldsymbol{v}_{i}\neq 0\\}$
is the support of $\boldsymbol{v}$, for any vector
$\boldsymbol{v}\in\mathbb{R}^{m}$. When $p<2$ we have that

$\|\boldsymbol{v}\|_{p}\geqslant\|\boldsymbol{v}\|_{2}.$

We can combine the inequalities above to conclude in particular that

$\|\boldsymbol{v}\|_{p}\geqslant|\mathsf{supp}(\boldsymbol{v})|^{1/\max(2,p)-1/2}\|\boldsymbol{v}\|_{2}$
(14)

and

$\|\boldsymbol{v}\|_{2}\geqslant|\mathsf{supp}(\boldsymbol{v})|^{-1/2}\|\boldsymbol{v}\|_{p}$
(15)

for any $p\geqslant 1$.

We prove a generalization of [HR12, Claim 3.5] for $\ell_{p}$ norms with
$p\geqslant 1$.

######  Lemma C.2 (Generalization of [HR12, Claim 3.5]).

Let $(B,k)$ be a NO instance of $\gamma$-$\mathrm{SVP}_{p}$ with the
properties outlined in Lemma 5.1, and let $\mathcal{L}$ be a sublattice of
$\mathcal{L}(B)$ of rank $r$. Then, at least one of the following properties
holds, where $d=\gamma k$:

  * •

Every basis matrix of $\mathcal{L}$ has more than $d^{p}$ nonzero rows;

  * •

Every basis matrix of $\mathcal{L}$ has only even entries and has more than
$(d/2)^{p}$ nonzero rows;

  * •

$\det(\mathcal{L})>d^{r(c+p/2)}$ and there is a basis matrix of $\mathcal{L}$
which has at most $(d/2)^{p}$ nonzero rows. In particular, every vector in
$\mathcal{L}$ has Hamming weight at most $(d/2)^{p}$.

###### Proof.

We assume that the first two properties do not hold and show that the third
property holds in that case. Since there is a basis matrix of $\mathcal{L}$
with at most $d^{p}$ nonzero rows, we conclude that $r\leqslant d^{p}$ and
that every vector in $\mathcal{L}$ has Hamming weight at most $d^{p}$. By the
properties of $(B,k)$, this implies that $\mathcal{L}\subseteq
2\mathbb{Z}^{m}$. Therefore, it must be the case that there is a basis matrix
of $\mathcal{L}$ that has at most $(d/2)^{p}$ nonzero rows, and so every
vector in $\mathcal{L}$ has Hamming weight at most $(d/2)^{p}$. As a result,
we also gather that all vectors $\boldsymbol{w}\in\mathcal{L}$ satisfy
$\|\boldsymbol{w}\|_{p}>d^{c+3p/2}$, and so

$\det(\mathcal{L})\geqslant\left(\frac{\lambda_{1}^{(2)}(\mathcal{L})}{\sqrt{r}}\right)^{r}\geqslant\left(\frac{d^{-p/2}\lambda_{1}^{(p)}(\mathcal{L})}{\sqrt{r}}\right)^{r}>d^{r(c+p/2)},$

where the first inequality follows from Lemma C.1, the second inequality holds
by Equation 15 and the fact that every vector in $\mathcal{L}$ has Hamming
weight at most $d^{p}$, and the third inequality uses the fact that
$r\leqslant d^{p}$. ∎

We will also need the following technical lemma from [HR12], which is specific
for the $\ell_{2}$ norm.

######  Lemma C.3 ([HR12, Claim 3.6]).

Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be integer lattices of rank
$r\geqslant 1$ generated by the bases
$U=(\boldsymbol{u}_{1},\dots,\boldsymbol{u}_{r})$ and
$W=(\boldsymbol{w}_{1},\dots,\boldsymbol{w}_{r})$, respectively. Consider the
vector
$\boldsymbol{v}=\sum_{i=1}^{r}\boldsymbol{u}_{i}\otimes\boldsymbol{w}_{i}\in\mathcal{L}_{1}\otimes\mathcal{L}_{2}$.
Then, it holds that

$\|\boldsymbol{v}\|_{2}\geqslant\sqrt{r}(\det(\mathcal{L}_{1})\cdot\det(\mathcal{L}_{2}))^{1/r}.$

We are now ready to prove Lemma 5.1.

###### Proof of Lemma 5.1.

It suffices to show that if $(B,k)$ is a NO instance of
$\gamma$-$\mathrm{SVP}_{p}$ satisfying the conditions from the lemma
statement, $\mathcal{L}_{1}=\mathcal{L}(B)$, and
$\mathcal{L}_{2}=\bigotimes_{i=1}^{c^{\prime}}\mathcal{L}$ for some integer
$1\leqslant c^{\prime}<c$ and such that
$\lambda_{1}^{(p)}(\mathcal{L}_{2})>d^{c^{\prime}}$, then

$\lambda_{1}^{(p)}(\mathcal{L}_{1}\otimes\mathcal{L}_{2})>d^{c^{\prime}+1}.$

Consider an arbitrary nonzero vector
$\boldsymbol{v}\in\mathcal{L}_{1}\otimes\mathcal{L}_{2}$. As shown in [HR12,
Proof of Lemma 3.4], we can write
$\boldsymbol{v}=B^{\prime}_{1}(B^{\prime}_{2})^{T}$ for full-column-rank
matrices $B^{\prime}_{1}$ and $B^{\prime}_{2}$ (note that $B^{\prime}_{1}$ and
$B^{\prime}_{2}$ have the same number of columns) such that
$\mathcal{L}^{\prime}_{i}=\mathcal{L}(B^{\prime}_{i})\subseteq\mathcal{L}_{i}$
for $i=1,2$. We now proceed by case analysis based on Lemma C.2 applied to
$\mathcal{L}^{\prime}_{1}$:

  * •

$B^{\prime}_{1}$ has more than $d^{p}$ nonzero rows: In this case, more than
$d^{p}$ rows of $B^{\prime}_{1}(B^{\prime}_{2})^{T}$ are nonzero vectors from
$\mathcal{L}^{\prime}_{2}$, and so

$\|\boldsymbol{v}\|_{p}>d\cdot\lambda_{1}^{(p)}(\mathcal{L}^{\prime}_{2})\geqslant
d\cdot\lambda_{1}^{(p)}(\mathcal{L}_{2})>d^{c^{\prime}+1}.$

  * •

$B^{\prime}_{1}$ has only even entries and has more than $(d/2)^{p}$ nonzero
rows: In this case, more than $(d/2)^{p}$ rows of
$B^{\prime}_{1}(B^{\prime}_{2})^{T}$ are even multiples of nonzero vectors
from $\mathcal{L}^{\prime}_{2}$, and so

$\|\boldsymbol{v}\|_{p}>2((d/2)^{p})^{1/p}\lambda_{1}^{(p)}(\mathcal{L}^{\prime}_{2})\geqslant
d\cdot\lambda_{1}^{(p)}(\mathcal{L}_{2})>d^{c^{\prime}+1}.$

  * •

$\det(\mathcal{L}^{\prime}_{1})>d^{r(c+p/2)}$ and $B^{\prime}_{1}$ has at most
$(d/2)^{p}$ nonzero rows. In particular, $r\leqslant(d/2)^{p}$ and every
vector in $\mathcal{L}^{\prime}_{1}$ has Hamming weight at most $(d/2)^{p}$:
In this case, we have

$\displaystyle\|\boldsymbol{v}\|_{p}$ $\displaystyle\geqslant
d^{\frac{p}{\max(2,p)}-\frac{p}{2}}\|\boldsymbol{v}\|_{2}$
$\displaystyle\geqslant
d^{\frac{p}{\max(2,p)}-\frac{p}{2}}\sqrt{r}(\det(\mathcal{L}^{\prime}_{1})\cdot\det(\mathcal{L}^{\prime}_{2}))^{1/r}$
$\displaystyle\geqslant
d^{\frac{p}{\max(2,p)}-\frac{p}{2}}\sqrt{r}\det(\mathcal{L}^{\prime}_{1})^{1/r}$
$\displaystyle>d^{\frac{p}{\max(2,p)}-\frac{p}{2}}\cdot d^{c+p/2}$
$\displaystyle\geqslant d^{c^{\prime}+1}\ .$

The first inequality follows from Equation 14 and the fact that
$\boldsymbol{v}$ has support size at most $d^{p}$. The second inequality holds
via Lemma C.3 applied to $\mathcal{L}^{\prime}_{1}$ and
$\mathcal{L}^{\prime}_{2}$, which are both rank-$r$ lattices. The third
inequality uses the fact that $\det(\mathcal{L}^{\prime}_{2})\geqslant 1$,
which holds because $\mathcal{L}^{\prime}_{2}$ is a non-trivial integer
lattice (this in turn holds by the definition of $\mathcal{L}^{\prime}_{2}$
and the fact that $\boldsymbol{v}\neq\boldsymbol{0}$.). The fourth inequality
holds by the lower bound on $\det(\mathcal{L}^{\prime}_{1})$ and the fact that
$c^{\prime}\leqslant c-1$.∎


# Detecting nonclassicality and non-Gaussianity of a coherent superposed
quantum state

Deepak<EMAIL_ADDRESS>Department of Mathematics, J. C. Bose
University of Science and Technology,
YMCA, Faridabad 121006, India  Arpita Chatterjee<EMAIL_ADDRESS>Department of Mathematics, J. C. Bose University of Science and Technology,
YMCA, Faridabad 121006, India

###### Abstract

In this paper, we investigate the nonclassicality and non-Gaussianity of a
coherent superposed quantum state (CSQS) which is obtained by applying a
coherent superposition of field annihilation ($a$) and creation
($a^{\dagger}$) operators, $N(ta+ra^{\dagger})$ to a classical coherent state
$\ket{\alpha}$, where $t$ and $r$ are scalars with $t^{2}+r^{2}=1$. Such an
operation, when applied on states having classical characters, introduces
strong nonclassicality. We use different criteria to check the nonclassicality
and non-Gaussianity of the considered quantum state. We first compute the
Wigner function of CSQS. To study the nonclassicality of the considered state
we further use (i) linear entropy (LE) (ii) Wigner logarithmic negativity
(WLN) and (iii) skew information based measure. Relative entropy based measure
is considered to analyze the variation in non-Gaussianity of CSQS. Finally,
the dynamics of the Wigner function evolving under the photon loss channel is
addressed to probe the effect of noise on nonclassicality as well as non-
Gaussianity of CSQS.

creation operator, annihilation operator, coherent superposed quantum state,
nonclassicality, non-Gaussianity

##  I Introduction

In recent years, the nonclassicality and non-Gaussianity of a quantum state
have received much importance as the investigation of fundamental structure of
quantum states is crucial for current quantum technologies. The classification
of a state as a classical or nonclassical one [1] is an excellent example of
such characteristics. This attention is well-deserved, as it has long been
recognized that nonclassicality and quantum non-Gaussianity are two of the
most important features of the quantum world, both of which can lead to
quantum advantage [2]. The quantification of nonclassicality provided by
various quantum measures is also of great interest for experimental
implementation of quantum optical technologies. Another state property,
quantum non-Gaussianity [3, 4, 5, 6] is particularly important for various
quantum technology based applications [7, 8, 9]. Several distinct measures of
nonclassicality and non-Gaussianity are applied to different quantum states to
study these basic natures. P. Malpani et. al. [10] performed a quantitative
analysis of the nonclassical and non-Gaussian features for photon-added
displaced Fock state, as a test case, using a set of measures like
entanglement potential, Wigner Yanese skew information, Wigner logarithmic
negativity and relative entropy of non-Gaussianity. Kühn et. al. [11] proved
the usefulness of nonclassicality quasiprobabilities for certifying both
quantum non-Gaussianity and the degree of nonclassicality. However, there is
no unique approach for approximating the amount of nonclassicality or non-
Gaussianity. In this work, we begin by defining the nonclassicality and non-
Gaussianity, then discuss their significance in quantum technology and the
urgency for quantifying these characteristics. We also check how the
nonclassicality dominant parameters affect the non-Gaussianity.

Any quantum state $\rho$ can be represented in terms of the Glauber-Sudarshan
$P$ function as [12, 13]

$\rho=\int d(P(\alpha))\ket{\alpha}\bra{\alpha}$ (1)

The state is nonclassical if $P(\alpha)$ fails to be a probability
distribution function. In other words, a nonclassical state cannot be
represented in terms of a statistical mixture of coherent states. Nowadays the
nonclassical states are most significant in the domain of quantum metrology,
quantum computing, and quantum information processing [14]. Major demand for
enhancing device performance has raised, and it has been discovered that the
desired enhancement can be achieved only by using quantum resources. Non-
Gaussian states are those states which cannot be defined as a probabilistic
combination of Gaussian states [15, 11]. Again, all the Gaussian states can be
characterized by the initial two moments, which implies that the mean and
covariance matrix of the Gaussian states provide their complete information.
We can define a complex hull $\mathcal{G}$ with the classical probability
distribution $P_{cl}(\lambda)$

$\rho=\int{d(P_{cl}(\lambda))}\ket{\psi_{\mathcal{G}}(\lambda)}\bra{\psi_{\mathcal{G}}(\lambda)}$
(2)

in the Hilbert space $\mathcal{H}$. This set contains all the Gaussian states
and a few non-Gaussian states. Interestingly, the non-Gaussian states obtained
as the statistical mixtures of Gaussian states in the form (2) have limited
applications due to their origin in classical noise [11, 16]. Thus quantum
non-Gaussian states $\rho$ in $\mathcal{H}$ are the states which do not belong
to the complex hull ($\rho\notin\mathcal{G}$). It is worth concentrating on
the quantification of quantum non-Gaussianity in such a state as these states
can be applied as more robust resources compared to the Gaussian states ([17,
18, 19], and references therein). In particular, there are no-go theorems
limiting the use of Gaussian operations and Gaussian states in entanglement
distillation [20], quantum error correction [21], quantum computing [17], and
quantum bit commitment [22]. Use of non-Gaussian operations provides
advantages in quantum computation [17], quantum communication [23], quantum
metrology [18], and so on. These promising applications motivated people to
study about resource theory of non-Gaussianity [19], and a number of measures
of quantum non-Gaussianity ([24, 25, 26], and references therein). It is worth
to note that earlier quantum gravity and quantum information processing have
been developed as two independent branches, but recently different significant
and exciting results connecting these two fields have been reported ([27], and
references therein). It is observed that non-Gaussianity present in a state
can be utilized to identify the signature of quantum gravity as only quantum
theory of gravity can lead to non-Gaussianity [27].

In present study, we have considered a coherent superposed quantum states
defined as a specific combination of bosonic creation and annihilation
operators applied on the usual coherent state as
$(ta+ra^{\dagger})\ket{\alpha}$ with the standard relation $t^{2}+r^{2}=1$
[28]. It is transformed to photon added (subtracted) coherent state for $t=0$
$(1)$ and superposed coherent state for $t=\frac{1}{\sqrt{2}}$ (see Fig. 1).

Figure 1: Diagram for various limiting cases of CSQS i.e. for different values
of $t$, CSQS reduces to different quantum states as shown in the figure.

This coherent superposition $(ta+ra^{\dagger})$ acting on continuous variable
system was first proposed by Lee and Nha for quantum state engineering [28].
They have investigated how the superposed operation transforms a classical
state to a nonclassical one together with emerging nonclassical effects. They
have also discussed the experimental feasibility of such a system by using
beam splitters, parametric down converter and a nondegenerate parametric
amplifier [29]. They have studied the case of generating an arbitrary
superposition of number states involving up to two photons,
$C_{0}\ket{0}+C_{1}\ket{1}+C_{2}\ket{2}$, which can be used for quantum
information processing. In a previous work, we have considered a generalized
superposition of products of field annihilation ($a$) and creation
($a^{\dagger}$) operators of the type, $A=saa^{\dagger}+t{a^{\dagger}}a$ with
$s^{2}+t^{2}=1$, applied this SUP operator on coherent and thermal quantum
states, and the states thus produced are referred as SUP-operated coherent
state (SOCS) and SUP-operated thermal state (SOTS), respectively [30]. A
comparative study between the higher-order nonclassical properties of SOCS and
SOTS by using a set of nonclassicality witnesses (e.g., higher-order
antiubunching, higher-order sub-Poissonian photon statistics, higher-order
squeezing, Agarwal-Tara parameter, Klyshko’s condition) was reported. Both of
these work focused on the nonclassicality of the superposition states. In the
present study, our aim is to investigate the non-Gaussian properties of the
superposed state and to check if the nonclassicality enhancing parameters have
any mere effect on the non-Gaussianity.

The structure of the paper is as follows: Sect. II shows the detailed
description of the CSQS state and its Wigner function. Sect. III describes
different measures of nonclassicality and non-Gaussianity. The next section
reports the Wigner function undergoing via a photon lossy channel. The article
ends with a conclusion in Sect. V.

##  II Detailed Description of CSQS and Its Wigner Function

Given a coherent state $\ket{\alpha}$ as an initial state, the output state
after operating the superposition operation is defined by
$\ket{\psi}=N(ta+ra^{\dagger})\ket{\alpha}$, where $N$ is the normalization
constant as
$N=\left\\{|\alpha|^{2}+rt(\alpha^{2}+\alpha^{*2})+r^{2}\right\\}^{-\frac{1}{2}}$.
The Wigner function of the CSQS is derived as [28]

$W(\gamma,\gamma^{*})=N^{2}[|t\alpha+r(2\gamma^{*}-\alpha^{*})|^{2}-|r|^{2}]W_{0}(\gamma)$
(3)

where $W_{0}(\gamma)=\frac{2}{\pi}\exp(-2|\gamma-\alpha|^{2})$ is the Wigner
function of the initial coherent state $\ket{\alpha}$. This Wigner function is
plotted with respect to real and imaginary $\gamma$ for different values of
the state parameters $\alpha$ and $t$ in Fig. 2.

Figure 2: A comparison of Wigner function with respect to real and imaginary
$\gamma$ for $\alpha=0.5$ and (a) $t=1$, (b) $t=1/\sqrt{2}$, (c) $t=0$ and for
$t=0.5$ with (d) $\alpha=1$, (e) $\alpha=1.5$, (f) $\alpha=1.75$, (g) to (l)
are the corresponding contour plots. This figure shows non-Gaussian as well as
non-classical behavior.

The existence of nonclassical and non-Gaussian features in the coherent
superposed quantum state can be studied in terms of the Wigner function. The
negative region of the Wigner function indicates the presence of
nonclassicality. In addition to that, it was proven by Hudson [31] that any
pure quantum state having a non-negative Wigner function is necessarily
Gaussian. It may be noted from (3) that the Wigner function has a Gaussian
form for $t=1$ as it corresponds to the Wigner function of the photon-
subtracted coherent state. It is clear from Fig. 2 that the non-zero values of
the displacement parameter can influence the position of the non-Gaussian peak
of the state for all values of $t$. Moreover, switching $t$ from 1 to 0 leads
to the non-Gaussianity of the Wigner function having a Gaussian factor. The
negative region of the Wigner function in Fig. 2 exhibits the nonclassical as
well as the non-Gaussian features of the CSQS. It is also observed that with a
decrease in values of $t$ keeping $\alpha$ fixed at 0.5, the negativity of the
Wigner function increases.

##  III Different Measures of nonclassicality and non-Gaussianity

In this section, we show how the non-Gaussianity and nonclassicality present
in CSQS vary with respect to different state parameters (i.e. displacement
($\alpha$), superposition ($t$) parameters). This study is very much relevant
in the context that such an exercise may help to identify the suitable state
parameters for an engineered quantum state which is to be used to perform a
quantum computation, communication or metrology task that requires
nonclassical and/or non-Gaussian states. For the quantification of
nonclassicality, the closed form analytical expressions of an entanglement
potential (to be referred as linear entropy potential), skew information based
measure, and Wigner logarithmic negativity are obtained in this section.
Further, Wigner logarithmic negativity and relative entropy of non-Gaussianity
are computed as measures of non-Gaussianity.

###  III.1 Linear Entropy Potential

Asboth [32] proposed a new measure of nonclassicality based on the fact that
if a single-mode nonclassical (classical) state is inserted from one input
port of a beam-splitter (BS) and a vacuum state is embedded from the other
end, the output two-mode state must be entangled (separable). As a result, the
nonclassicality of the input single-mode state (other than the vacuum state
inserted in the BS) can be indirectly measured by using an entanglement
measure. There are a variety of quantitative entanglement measures available
which can be used to estimate the nonclassicality of the input single-mode
state. When a measure of entanglement following Asboth’s approach is used to
assess the single-mode nonclassicality, the corresponding entanglement measure
is referred to as the entanglement potential in correspondence with Asboth’s
terminology. In particular, if concurrence (linear entropy) is used to
quantify the nonclassicality of the single-mode input state by measuring the
entanglement of the two-mode output state departing from the BS, the
nonclassicality criteria is called concurrence potential (linear entropy
potential) [33].

The linear entropy [10] for a bipartite state $\rho_{AB}$ is defined in terms
of the reduced subsystem as

$L_{E}=1-\text{Tr}(\rho_{B}^{2})$ (4)

where $\rho_{B}$ is the partial trace of $\rho_{AB}$ over the subsystem $A$.
Therefore, for a maximally entangled (separable) state, $L_{E}=1$ $(0)$ and
obtains a non-zero value for an entangled state in general [34, 35]. To
compute the linear entropy potential, we would require the post-BS state
$\rho_{AB}$ that originated if CSQS and vacuum states are mixed at a beam-
splitter. The expression for the two-mode output state with Fock state at one
port and vacuum at the other is as follows:

$\ket{n}\otimes\ket{0}=\ket{n,0}\,\xrightarrow{BS}\,\frac{1}{2^{n/2}}\sum_{j=0}^{n}\sqrt{{n\choose
j}}\ket{j,n-j}$ (5)

This equation can be used to find the post-BS density matrix
$\rho_{AB}=\ket{\phi}_{AB}{}_{AB}{\bra{\phi}}$ as follows: the coherent
superposed quantum state in terms of number state basis is expressed as

$\ket{\psi}=\sum_{p=0}^{\infty}\left(c^{\prime}_{1p}\ket{p}+c^{\prime}_{2p}\ket{p+1}\right)$

with
$c^{\prime}_{1p}=N\,te^{-\frac{|\alpha|^{2}}{2}}\frac{\alpha^{p+1}}{\sqrt{p!}}$
and
$c^{\prime}_{2p}=N\,re^{-\frac{|\alpha|^{2}}{2}}\sqrt{p+1}\frac{\alpha^{p}}{\sqrt{p!}}$.
Then using (5)

$\displaystyle{\ket{\phi}}_{AB}$ $\displaystyle=\ket{\psi}\otimes\ket{0}$
$\displaystyle\,\xrightarrow{BS}\,\sum_{p=0}^{\infty}\left(c_{1p}\sum_{j=0}^{p}\sqrt{p\choose
j}\ket{j,p-j}\right.$
$\displaystyle+\left.c_{2p}\sum_{j=0}^{p+1}\sqrt{p+1\choose
j}\ket{j,p+1-j}\right)$ (6)

where $c_{1p}=\frac{c^{\prime}_{1p}}{2^{p/2}}$ and
$c_{2p}=\frac{c^{\prime}_{2p}}{2^{(p+1)/2}}$. The simplified form of LE can be
obtained as (see Appendix A for detailed calculation)

$\displaystyle L_{E}$
$\displaystyle=1-N^{4}\left[t^{4}|\alpha|^{4}+t^{3}r\left\\{2\alpha^{2}|\alpha|^{2}+2\alpha^{*2}|\alpha|^{2}\right\\}\right.$
$\displaystyle+\left.t^{2}r^{2}\left\\{2|\alpha|^{2}(1+2|\alpha|^{2})+\alpha^{4}+\alpha^{*4}\right\\}\right.$
$\displaystyle+\left.tr^{3}\left\\{2\alpha^{2}(1+|\alpha|^{2})+2\alpha^{*2}(1+|\alpha|^{2})\right\\}\right.$
$\displaystyle+\left.r^{4}(1+5|\alpha|^{2}+2|\alpha|^{4})/2\right]$ (7)

This LE is plotted in Fig. 3 with respect to the state parameter $\alpha$ and
for different values of $r$.

Figure 3: Variation of LE with respect to the state parameter $\alpha$ for
different values of $r$. It shows that nonclassicality is increasing with $r$
and decreasing with $\alpha.$

The variation in the amount of nonclassicality with respect to the state
parameters is shown in Fig. 3. The nonclassicality is detected through the
linear entropy potential. It is clear that LE increases as $r$ increases from
0 to 1 and decreases with increasing $\alpha$. The maximum value of LE
obtained for the coherent superposed quantum state is 0.5 approximately.

###  III.2 Criteria Based on Skew Information

S. Luo et. al. [36] introduced the skew information based measure of
nonclassicality in the context of Wigner Yanase skew information [37] in 2019.
For a pure state $\rho$, this measure is defined as

$N(\rho)=\frac{1}{2}+\langle a^{\dagger}a\rangle-\langle
a^{\dagger}\rangle\langle a\rangle$ (8)

$N(\rho)$ denotes the quantum coherence of $\rho$ with respect to the
annihilation and creation operators, and is relatively easy to calculate. This
measure is based on averages, which takes numerical value $\frac{1}{2}$ for
classical coherent state and $n+\frac{1}{2}$ for mostly nonclassical $n$
photon Fock state. Thus any state $\rho$ with $N(\rho)>\frac{1}{2}$ is
nonclassical. However, this is only a one-sided condition for nonclassicality
as it fails for some Gaussian mixed states [36].

In the case of CSQS, $N(\rho)$ can be calculated using the general expectation
of $a^{\dagger p}a^{q}$ as follows:

$\displaystyle\langle{a^{\dagger m}a^{n}}\rangle$
$\displaystyle=N^{2}\alpha^{*m-1}\alpha^{n-1}\Big{[}|\alpha|^{4}+rt\Big{\\{}(m+|\alpha|^{2})\alpha^{2}$
(9)
$\displaystyle+(n+|\alpha|^{2})\alpha^{*2}\Big{\\}}+r^{2}\Big{\\{}mn+(m+n+1)|\alpha|^{2}\Big{\\}}\Big{]},$

The skew information based measure for CSQS can be simplified as

$\displaystyle N(\rho)$
$\displaystyle=\frac{1}{2}+N^{2}\Big{[}|\alpha|^{4}+rt(1+|\alpha|^{2})\Big{\\{}\alpha^{2}+\alpha^{*2}\Big{\\}}$
$\displaystyle+r^{2}\Big{\\{}1+3|\alpha|^{2}\Big{\\}}\Big{]}-N^{4}|\alpha|^{-2}\Big{[}|\alpha|^{4}+rt\Big{\\{}(|\alpha|^{2})\alpha^{2}\Big{.}$
$\displaystyle\Big{.}+(1+|\alpha|^{2})\alpha^{*2}\Big{\\}}+r^{2}\Big{\\{}2|\alpha|^{2}\Big{\\}}\Big{]}\Big{[}|\alpha|^{4}+rt\Big{\\{}(1+|\alpha|^{2})\alpha^{2}\Big{.}$
$\displaystyle\Big{.}+(|\alpha|^{2})\alpha^{*2}\Big{\\}}+r^{2}\Big{\\{}2|\alpha|^{2}\Big{\\}}\Big{]}$
(10)

and is plotted in Fig. 4.

Figure 4: Variation of $N(\rho)$ with respect to the state parameter $\alpha$
and for distinct values of $r$. It also shows behavior similar to linear
entropy.

It can be observed that the skew information based criteria exhibits
nonclassicality of CSQS as the value of $N(\rho)~{}\geq~{}1/2$ for distinct
parametric values. Further $N(\rho)$ is increasing (decreasing) with an
increase in $r$ ($\alpha$). That means skew information based measure behaves
similar to linear entropy. Also, the value of $N(\rho)$ lies between 0.5 and
1.5 approximately. Figure 4 shows that the nonclassicality expressed by the
skew information based measure cannot be enhanced by increasing the coherent
state parameter. It can also be seen that with an increase in values of $r$,
nonclassicality increases.

###  III.3 Logarithmic Negativity of Wigner Function

We have seen in Fig. 2 that the Wigner distribution diagnoses the nonclassical
as well as the non-Gaussian nature of the CSQS. This motivated us to quantify
the amount of nonclassicality and non-Gaussianity using the volume of the
negative part of the Wigner function. A measure of nonclassicality, named
Wigner logarithmic negativity, estimates the amount of non-Gaussianity present
in the CSQS as the negative values of the Wigner function also witness the
non-Gaussianity of CSQS. The Wigner logarithmic negativity is defined as [19]

$W=\log_{2}\left(\int{d^{2}\gamma}|W(\gamma,\gamma^{*})|\right)$ (11)

where the integration is performed over the complete region $\mathcal{R}^{2n}$
of phase-space.

The integration in (11) is computed to study the effect of different
parameters on the Wigner logarithmic negativity of CSQS as (see Appendix B for
details)

$W=\log_{2}[N^{2}((t+r)^{2}\alpha^{2}-5r^{2})]$ (12)

Figure 5: Variation of $W$ with respect to state parameter $\alpha$ for
distinct values of $r$. It also shows behavior similar to previous criteria
and its negative value shows that the volume of non-classicality and non-
Gaussianity shown by the Wigner function is less than one.

Fig. 5 shows that $W$ has a negative value in some region of $\alpha$.
Further, it can be noticed that $W$ first decreases up to a fixed value of
$\alpha$ and then increases. That means the amount of both nonclassicality and
non-Gaussianity decreases up to a fixed value of $\alpha$ and then increases
for all values of $r$ and $\alpha$.

Therefore, all these criteria show that CSQS is nonclassical as well as non-
Gaussian, and the amount of nonclassicality increases as $\alpha$ increases in
a specific region. In the next section, we find another criteria to determine
the amount of non-Gaussianity.

###  III.4 Relative Entropy for Measuring the Amount of non-Gaussianity

In the previous section, the non-Gaussianity is quantified by using Wigner
logarithmic negativity and it is found that in case of CSQS, variations in the
amount of non-Gaussianity with the state parameters of interest is analogous
to that of nonclassicality as quantified through different measures. This is
justified as the set of Wigner negative states is a subset of non-Gaussian
states [19]. In what follows, we reinvestigate this feature by using relative
entropy of non-Gaussianity. The relative entropy of non-Gaussianity is defined
in terms of its covariance matrix consisting of the first and second-order
moments as [10]

$\sigma=\left[\begin{matrix}\sigma_{pp}&\sigma_{qp}\\\
\sigma_{qp}&\sigma_{pp}\end{matrix}\right]$ (13)

where $\sigma_{pq}=\langle{pq+qp}\rangle-2\langle{p}\rangle\langle q\rangle$
and $p=\frac{a+a^{\dagger}}{\sqrt{2}},\,q=\frac{a-a^{\dagger}}{i\sqrt{2}}$.
Using (9), all the elements of the covariance matrix $\sigma$ can be
calculated, and thus the relative entropy measure of non-Gaussianity reduces
to [10]

$\delta[\ket{\psi}]=S(\tau_{G})=h(\det(\sqrt{\sigma}))$ (14)

with
$h(x)=\frac{x+1}{2}\log_{2}\left(\frac{x+1}{2}\right)-\frac{x-1}{2}\log_{2}\left(\frac{x-1}{2}\right)$.
The relative entropy of non-Gaussianity is plotted in Fig. 6.

Figure 6: Variation of $\delta$ with respect to $\alpha$ for different values
of $r$. The behavior of $\delta$ is similar to $W$.

Fig. 6 shows similar nature of the relative entropy of non-Gaussianity for
coherent superposed quantum state as shown by Wigner logarithmic negativity.
There is a region of $\alpha$ where $\delta$ is decreasing first then it is
increasing. Hence, we can conclude that amount of both nonclassicality and
non-Gaussianity first decreases and then increases for all values of
parameters. Further, all these parameters show that nonclassicality and non-
Gaussianity increase with the increase in $r$ for keeping $\alpha$ fixed.

##  IV Wigner Function of CSQS evolving under Photon Loss Channel

The interaction between a quantum system and its surroundings motivates
quantum to classical transition. So, the observed nonclassical and non-
Gaussian features are expected to decay due to the evolution of CSQS under the
lossy channel. In particular, the temporal evolution of a quantum state $\rho$
over the lossy channel can be studied using the LGKS master equation [38]
given by

$\frac{\partial\rho}{\partial t}=\kappa(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a)$ (15)

where $\kappa$ is the rate of decay. Analogously, the time evolution of the
Wigner function at time $t$ in terms of the initial Wigner function of the
state evolving under a lossy channel [39] can be defined as

$W(\zeta,t)=\frac{2}{T}\int\frac{\partial^{2}\gamma}{\pi}\exp[-\frac{2}{T}|\zeta-\gamma
e^{-\kappa t}|^{2}]W(\gamma,\gamma^{*},0)$ (16)

with $T=1-\exp(-2\kappa t)$, $W(\gamma,\gamma^{*},0)$ is the Wigner function
at initial time $t=0$ which is given in (3). The time evolution of the Wigner
function (16) models dissipation due to interaction with a vacuum reservoir as
well as inefficient detectors with efficiency $\eta=1-T$ . The detailed
analytic expression for (16) is obtained in Appendix C.

Figure 7: A comparison of Wigner function w.r.t. real and imaginary $\zeta$
for $\alpha=1.5,\,t=1/\sqrt{2}$ and (a) $\kappa t=0.1$, (b) $\kappa t=0.3$,
(c) $\kappa t=0.5$; for $\kappa t=0.3,\,\alpha=0.5$ and (d) $t=1$, (e)
$t=1/\sqrt{2}$, (f) $t=0$; for $\kappa t=0.3,\,t=1/\sqrt{2}$ and (g)
$\alpha=1$, (h) $\alpha=1.5$, (i) $\alpha=1.75$, (j) to (r) are the
corresponding contour plots.

It is clear from Fig. 7 that with an increase in the rescaled time $\kappa t$,
non-Gaussianity decreases for fixed values of $\alpha$ and $t$ and
nonclassicality nature is not revealed at all. Similarly, with an increase
(decrease) in $r$ $(t)$, non-Gaussianity nature increases, and nonclassicality
is not detected. Also, by increasing $\alpha$ with fixed $t$ and $\kappa t$
there are merely any changes in non-Gaussianity.

##  V Conclusion

From the above discussion, we can conclude that the Wigner function detects
the non-Gaussianity as well as the nonclassicality of CSQS. But if the
coherent superposed quantum state is evolved via a photon loss channel, the
Wigner function shows the non-Gaussianity but fails to detect the
nonclassicality. Linear entropy decreases with an increase in coherent state
parameter $\alpha$ and increases with increasing $r$. $N(\rho)$ also has a
similar behavior as linear entropy. Wigner logarithmic negativity decreases
with $\alpha$ up to a fixed value of $\alpha$ and then increases. Also, it is
negative in most of the regions and increases with an increase in the value of
$r$. The relative entropy measure first decreases with $\alpha$ up to a fixed
value and then increases with $r$. Thus the considered coherent superposed
quantum state is nonclassical as well as non-Gaussian.

The results reported here seem to be experimentally realizable, as photon-
added coherent states are prepared in laboratory by Bellini et. al. [40] and
non-Gaussianity present in this state is also measured experimentally [3].
Keeping this in mind, we end the article with an expectation that the current
result will be of further use for performing quantum optical and computing
tasks.

ACKNOWLEDGEMENT

Deepak’s work is supported by the Council of Scientific and Industrial
Research (CSIR), Govt. of India (Award no. 09/1256(0006)/2019-EMR-1).

##  Appendix A Linear Entropy

The density matrix using (III.1) is given by

$\displaystyle\rho_{AB}$ $\displaystyle=$
$\displaystyle\ket{\phi}_{AB}{}_{AB}\bra{\phi}$ $\displaystyle=$
$\displaystyle\sum_{p=0}^{\infty}\left(c_{1p}\sum_{j=0}^{p}\sqrt{p\choose
j}\ket{j,p-j}+c_{2p}\sum_{j=0}^{p+1}\sqrt{p+1\choose
j}\ket{j,p+1-j}\right)\times$
$\displaystyle\sum_{q=0}^{\infty}\left(c_{1q}^{*}\sum_{k=0}^{q}\sqrt{q\choose
k}\bra{k,q-k}+c_{2q}^{*}\sum_{k=0}^{q+1}\sqrt{q+1\choose
k}\bra{k,q+1-k}\right)$ $\displaystyle\Rightarrow\rho_{B}$ $\displaystyle=$
$\displaystyle\sum_{p,q=0}^{\infty}\left(c_{1p}c_{1q}^{*}\sum_{j=0}^{p}\sqrt{{p\choose
j}{q\choose
j}}\ket{p-j}\bra{q-j}+c_{1p}c_{2q}^{*}\sum_{j=0}^{p}\sqrt{{p\choose
j}{q+1\choose j}}\ket{p-j}\bra{q+1-j}\right.$
$\displaystyle+\left.c_{2p}c_{1q}^{*}\sum_{j=0}^{p+1}\sqrt{{p+1\choose
j}{q\choose
j}}\ket{p+1-j}\bra{q-j}+c_{2p}c_{2q}^{*}\sum_{j=0}^{p+1}\sqrt{{p+1\choose
j}{q+1\choose j}}\ket{p+1-j}\bra{q+1-j}\right)$
$\displaystyle\Rightarrow\text{Tr}(\rho_{B}^{2})$ $\displaystyle=$
$\displaystyle
N^{4}\left[t^{4}|\alpha|^{4}+t^{3}s\left\\{2\alpha^{2}|\alpha|^{2}+2\alpha^{*2}|\alpha|^{2}\right\\}+t^{2}s^{2}\left\\{2|\alpha|^{2}(1+2|\alpha|^{2})+\alpha^{4}+\alpha^{*4}\right\\}\right.$
$\displaystyle+\left.ts^{3}\left\\{2\alpha^{2}(1+|\alpha|^{2})+2\alpha^{*2}(1+|\alpha|^{2})\right\\}+s^{4}(1+5|\alpha|^{2}+2|\alpha|^{4})/2\right]$

##  Appendix B Wigner Logarithmic Negativity

$\displaystyle W$
$\displaystyle=log_{2}\left(\int{d^{2}\gamma}|W(\gamma,\gamma^{*})|\right)$
$\displaystyle=log_{2}\left(\int{d^{2}\alpha}|W_{0}(\alpha)N^{2}[|t\alpha_{0}+r(2\alpha^{*}-\alpha_{0}^{*})|^{2}-|r|^{2}]|\right)$
$\displaystyle=log_{2}\left(\frac{2N^{2}}{\pi}\int{d^{2}\alpha}\exp(-2|\alpha-k|^{2})\left||tk+r(2\alpha^{*}-k)|^{2}-|r|^{2}\right|\right)~{}~{}~{}(\text{assuming
$\alpha_{0}=k$, a real number for simplicity})$
$\displaystyle=log_{2}\left(\frac{2N^{2}}{\pi}\int{d^{2}\alpha}\exp(-2|\alpha-k|^{2})\left|(t-r)^{2}k^{2}+4r^{2}|\alpha|^{2}+2r(t-r)k(\alpha+\alpha^{*})-r^{2}\right|\right)~{}(\text{putting
$\alpha=x+iy$})$
$\displaystyle=log_{2}\left(\frac{2N^{2}}{\pi}\int{dxdy}\exp(-2(x-k)^{2}-2y^{2})\left|(t-r)^{2}k^{2}+4r^{2}(x^{2}+y^{2})+4r(t-r)kx-r^{2}\right|\right)$

Let $x_{1}$ and $x_{2}$ be the two solutions of
$4r^{2}x^{2}+4r(t-r)kx+(t-r)^{2}k^{2}+4r^{2}y^{2}-r^{2}$ with $x_{1}\leq
x_{2}=0$ and hence we get

$x_{1}=(-(t-r)k-r\sqrt{r^{2}-4y^{2}})/2r$
$x_{2}=(-(t-r)k+r\sqrt{r^{2}-4y^{2}})/2r$

Thus the modulus inside the integration is positive on the intervals
$(-\infty,x_{1})$ and $(x_{2},\infty)$ and negative in $(x_{1},x_{2})$. Using
$\int{dx}\exp(-2x^{2})=\sqrt{\frac{\pi}{2}}$, $\int{dx}\exp(-2x^{2})x=0$,
$\int{dx}\exp(-2x^{2})x^{2}=\frac{3}{4}\sqrt{\frac{\pi}{2}}$, we can calculate
as

$\displaystyle W$
$\displaystyle=log_{2}\left[N^{2}\left((t+r)^{2}k^{2}-5r^{2}\right)\right]$

##  Appendix C Wigner function evolving under the photon loss channel

$\displaystyle W(\zeta,t)$
$\displaystyle=\frac{2}{T}\int\frac{d^{2}\alpha}{\pi}\exp[\frac{-2}{T}|\zeta-\alpha
e^{-\kappa
t}|^{2}]\frac{2N^{2}}{\pi}\left[|\alpha_{0}|^{2}+r^{2}\left(-1+4|\alpha|^{2}-2\alpha\alpha_{0}^{*}-2\alpha^{*}\alpha_{0}\right)\right.$
$\displaystyle+\left.rt\left(2\alpha\alpha_{0}+2\alpha^{*}\alpha_{0}^{*}-\alpha_{0}^{*2}-\alpha_{0}^{2}\right)\right]\exp(-2|\alpha-\alpha_{0}|^{2})$
$\displaystyle=\frac{4N^{2}}{T\pi}\int\frac{d^{2}\alpha}{\pi}\exp[\frac{-2}{T}\left(|\zeta|^{2}+T|\alpha_{0}|^{2}+|\alpha|^{2}\\{T+e^{-2\kappa
t}\\}-\alpha(\zeta^{*}e^{-\kappa t}+T\alpha_{0}^{*})-\alpha^{*}(\zeta
e^{-\kappa t}+T\alpha_{0})\right)]$
$\displaystyle\times\left[|\alpha_{0}|^{2}+r^{2}\left(-1+4|\alpha|^{2}-2\alpha\alpha_{0}^{*}-2\alpha^{*}\alpha_{0}\right)+rt\left(2\alpha\alpha_{0}+2\alpha^{*}\alpha_{0}^{*}-\alpha_{0}^{*2}-\alpha_{0}^{2}\right)\right]$
$\displaystyle(\text{using $T+e^{-2\kappa t}=1$ and letting $\zeta e^{-\kappa
t}+T\alpha_{0}=\eta$})$
$\displaystyle=\frac{2N^{2}}{\pi}\exp[\frac{2}{T}\left(|\eta|^{2}-|\zeta|^{2}-T|\alpha_{0}|^{2}\right)]\left[t^{2}|\alpha_{0}|^{2}+r^{2}\left(-1+2T+|2\eta-\alpha_{0}|^{2}\right)+rt\left(2\text{Re}[(2\eta-\alpha_{0})\alpha_{0}]\right)\right]$

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