1. Introduction

Continuous variable (CV) quantum optical systems are well-established tools for both theoretical and experimental investigations of quantum information processing (QIP) [1, 2]. Entangled states represent key resources, both for quantum computers and for many communication schemes [1, 3], an can be realized with Gaussian two-mode states; these states are relatively easy to work with theoretically and are also commonly produced in a laboratory. It has been successfully applied to implement various important protocols, such as quantum teleportation [4–6], quantum dense coding [7–9], and entanglement swapping [10]. This advancement comes from the development of Gaussian optical operations, such as beam splitting, phase shifting, squeezing, displacement, and homodyne detection. Recently, it became evident that the understanding of entanglement behavior beyond Gaussian systems is a necessity [11–13]. Furthermore, recent theoretical investigations have shown some limits to the Gaussian operations. For example, the no-go theorem relating to the distillation of entanglement shard by distant parties using only Gaussian local operations and classical communications (LOCC) [14, 15]. Moreover, on the theoretical level, the study of entanglement in many-body systems has been limited to Gaussian states [16–19] where the quantification of quantum correlations (QC) reduces to the study of the covariance matrix, but the non-Gaussian entanglement doesn't have such a simplified approach.

The problem of quantifying entanglement in non-Gaussian systems, in a way that is independent of particular external parameters, hasn't solved yet; it is our main objective in this paper. An entanglement measure E of the state ρ should satisfy some criteria [20] to be an entanglement monotone. Many quantities have been proposed as a quantifier of entanglement in discrete variables (DV) and CV Gaussian states. Recently, however, two entanglement measures that are much

more amenable to evaluation have been proposed, the negativity of the partial transpose (NPT) and its logarithmic extension [21].

In this chapter, we are interested in establishing a direct measure of entanglement in non-Gaussian systems. This measure is based on the Wigner representation in the phase space of the non-Gaussian states. That is, they are defined in terms of the quantification of the degree of the negativity of Wigner function (NWF) [22, 23]. The most distinctive feature of this entanglement measure is the ease of calculated with a numerical integration.

### 2. Two-mode quasi-Bell state: an entangled non-Gaussian state

The simplest example of a non-Gaussian state is the single-photon state. There are also other examples that can be generated by excitations of Gaussian states [24, 25]. Here we are going to use quasi-classical state that has been extensively studied for its nonclassical proprieties and violation of Bell inequalities; it is the superposition of two-mode standard coherent states (SCS). Let us consider two modes of electromagnetic fields A and B with corresponding annihilation operators a^ and ^ b. Two-mode coherent states are defined by ∣α, βi ¼ Dað Þ α Dbð Þ β ∣0, 0i, where ∣0, 0i is the two-mode vacuum state and Dið Þ α is the displacement operator of the mode i ið Þ ¼ A; B . The state ∣α, βi can be expressed into the form

$$|a,\beta\rangle = e^{-\left(\left|a\right|^2 + \left|\beta\right|^2\right)/2} \sum\_{n,m}^{\infty} \frac{\alpha^n \beta^n}{\sqrt{n!m!}} |n,m\rangle,\tag{1}$$

RRRR

h dð Þ � , where

and d<sup>2</sup>

where

(α ¼ β); we find

49

� <sup>¼</sup> <sup>1</sup>

<sup>R</sup>^<sup>W</sup> <sup>R</sup>^; <sup>α</sup>; <sup>β</sup>; � �dR^ <sup>¼</sup> 1. Hence the doubled volume of the integrated negative part

QCS <sup>R</sup>^; <sup>α</sup>; <sup>β</sup> � � �

� � �

δNGð Þ¼ ρ Sð Þ ρ∥τ : (7)

δNGð Þ¼ ρ Sð Þ� τ Sð Þρ , (8)

are the symplectic eigenvalues of the

� �, (10)

2

� �: (11)

CCCA, (12)

CCCA, (13)

� � (9)

dR^ � <sup>1</sup>: (6)

W�

By definition, the quantity δ is equal to 0 for coherent and squeezed vacuum states, for which W is nonnegative. In this work we shall treat δ as a parameter

� �

It is clear from expression (5) and the plot in Figure 1 that the Wigner function of the quasi-Bell state (2) is non-Gaussian. In order to characterize this non-Gaussianity, several measures of the degree of non-Gaussianity were proposed [28, 29]. According to [29], the degree of non-Gaussianity of state ρ is defined by

where S ρ1∥ρ<sup>2</sup> ð Þ is the quantum relative entropy between states ρ<sup>1</sup> and ρ2. Here τ is the reference Gaussian state with the same first and second moments of ρ.

where Sð Þρ is the Von Neumann entropy of the state ρ. Also Sð Þ¼ τ h dð Þþ <sup>þ</sup>

1 2 � �

covariance matrix σ of the reference Gaussian state τ. Here Δð Þ¼ δ I<sup>1</sup> þ I<sup>2</sup> þ 2I3, where I<sup>1</sup> ¼ detð Þ A , I<sup>2</sup> ¼ detð Þ B , I<sup>3</sup> ¼ detð Þ C , and I<sup>4</sup> ¼ detð Þ σ are the four local

> <sup>σ</sup> <sup>¼</sup> A C C† B

> > Ri; Rj

� � � � � h i Ri Rj

For the considered states (2), we suppose that the two fields have the same mode

u<sup>þ</sup> 0 r<sup>þ</sup> 0 0 v<sup>þ</sup> 0 s<sup>þ</sup> r<sup>þ</sup> 0 u<sup>þ</sup> 0 0 s<sup>þ</sup> 0 v<sup>þ</sup>

u� 0 r� 0 0 v� 0 s� r� 0 u� 0 0 s� 0 v� 1

1

� <sup>x</sup> � <sup>1</sup> 2 � � ln <sup>x</sup> � <sup>1</sup>

This property of reference state τ leads to Tr½ �¼ ρ ln τ Tr½ � τ ln τ , so that

of the Wigner function of the state (2) may be written as

ZZZZ

R^

characterizing the properties of the state under consideration.

δ ψð Þ¼ �

Non-Gaussian Entanglement and Wigner Function DOI: http://dx.doi.org/10.5772/intechopen.86426

h xð Þ¼ x þ

symplectic invariants of the covariance matrix:

� � q

<sup>2</sup> Δð Þ� δ

1 2 � � ln <sup>x</sup> <sup>þ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>Δ</sup>ð Þ<sup>δ</sup> <sup>2</sup> � <sup>4</sup>I<sup>4</sup>

> <sup>σ</sup>ij <sup>¼</sup> <sup>1</sup> 2

σψ<sup>þ</sup> ¼

σψ� ¼

0

BBB@

0

BBB@

where ∣n1, n2i are the two-mode Fock states. The quasi-Bell coherent states (QBS) are defined by the following superpositions of two-mode coherent states:

$$|\psi\_{\pm}\rangle = \mathcal{N}\_{\pm}(|a,\beta\rangle \pm |-a,-\beta\rangle),\tag{2}$$

$$|\phi\_{\pm}\rangle = \mathcal{N}\_{\pm}(|a, -\beta\rangle \pm |-a, \beta\rangle),\tag{3}$$

where N� ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 exp �2j j <sup>α</sup> <sup>2</sup> � <sup>2</sup>j j <sup>β</sup> <sup>2</sup> � � <sup>þ</sup> <sup>2</sup> r is the normalization factor.

The Wigner function W R^; α; β � � of the state (1) is given by

$$\mathcal{W}(\hat{R}, a, \beta) = \frac{1}{\pi} \exp\left(f(a, q\_1, p\_1) + f(\beta, q\_2, p\_2)\right),\tag{4}$$

where <sup>R</sup>^ <sup>¼</sup> <sup>q</sup>1; <sup>p</sup>1; <sup>q</sup>2; <sup>p</sup><sup>2</sup> � �<sup>T</sup> is the quadrature operators vector and f xð Þ¼� ; y; z 2j j x <sup>2</sup> <sup>þ</sup> ffiffi 2 <sup>p</sup> <sup>x</sup>ð Þ <sup>∗</sup> <sup>þ</sup> <sup>x</sup> <sup>y</sup> <sup>þ</sup> <sup>i</sup> ffiffi 2 <sup>p</sup> <sup>x</sup> � <sup>x</sup><sup>∗</sup> ð Þ<sup>z</sup> � <sup>y</sup><sup>2</sup> � <sup>z</sup>2. For the quasi-Bell entangled coherent states Eq. (2), the Wigner function is given by [26, 27].

$$\begin{split} \mathcal{W}\_{\text{QCS}}^{\pm}(\hat{\boldsymbol{R}}, \boldsymbol{a}, \boldsymbol{\beta}) &= N\_{\boldsymbol{a}, \boldsymbol{\beta}, \pm}^{2} [\mathcal{W}(\hat{\boldsymbol{R}}\_{1}, \boldsymbol{a}, \boldsymbol{a}) \mathcal{W}(\hat{\boldsymbol{R}}\_{2}, \boldsymbol{\beta}, \boldsymbol{\beta})] \\ &\quad \pm \mathcal{W}(\hat{\boldsymbol{R}}\_{1}, \boldsymbol{a}, -\boldsymbol{a}) \mathcal{W}(\hat{\boldsymbol{R}}\_{2}, \boldsymbol{\beta}, -\boldsymbol{\beta}) \\ &\quad \pm \mathcal{W}(\hat{\boldsymbol{R}}\_{1}, -\boldsymbol{a}, \boldsymbol{a}) \mathcal{W}(\hat{\boldsymbol{R}}\_{2}, -\boldsymbol{\beta}, \boldsymbol{\beta}) \\ &\quad + \mathcal{W}(\hat{\boldsymbol{R}}\_{1}, -\boldsymbol{a}, -\boldsymbol{a}) \mathcal{W}(\hat{\boldsymbol{R}}\_{2}, -\boldsymbol{\beta}, -\boldsymbol{\beta}) \text{]}, \end{split} \tag{5}$$

where R^<sup>1</sup> and R^<sup>2</sup> are the quadrature operators vectors of the first and second modes and W R^i; x; y � � is the Wigner function of one-mode coherent state with i ¼ 1, 2; f g x; y ¼ �f g α; �β satisfies the normalization condition

more amenable to evaluation have been proposed, the negativity of the partial

In this chapter, we are interested in establishing a direct measure of entanglement in non-Gaussian systems. This measure is based on the Wigner representation in the phase space of the non-Gaussian states. That is, they are defined in terms of the quantification of the degree of the negativity of Wigner function (NWF) [22, 23]. The most distinctive feature of this entanglement

2. Two-mode quasi-Bell state: an entangled non-Gaussian state

The simplest example of a non-Gaussian state is the single-photon state. There are also other examples that can be generated by excitations of Gaussian states [24, 25]. Here we are going to use quasi-classical state that has been extensively studied for its nonclassical proprieties and violation of Bell inequalities; it is the superposition of two-mode standard coherent states (SCS). Let us consider two modes of electromagnetic fields A and B with corresponding annihilation operators

b. Two-mode coherent states are defined by ∣α, βi ¼ Dað Þ α Dbð Þ β ∣0, 0i, where

∞ n, <sup>m</sup>

þ 2

<sup>π</sup> exp <sup>f</sup> <sup>α</sup>; <sup>q</sup>1; <sup>p</sup><sup>1</sup>

2

entangled coherent states Eq. (2), the Wigner function is given by [26, 27].

� �<sup>T</sup> is the quadrature operators vector and

where R^<sup>1</sup> and R^<sup>2</sup> are the quadrature operators vectors of the first and second modes and W R^i; x; y � � is the Wigner function of one-mode coherent state with

α<sup>n</sup>β<sup>n</sup> ffiffiffiffiffiffiffiffiffi

∣ψ�i ¼ N�ðjα; βi �j� α; �βiÞ, (2) ∣ϕ�i ¼ N�ðjα; �βi �j� α; βiÞ, (3)

� � <sup>þ</sup> <sup>f</sup> <sup>β</sup>; <sup>q</sup>2; <sup>p</sup><sup>2</sup>

<sup>α</sup>, <sup>β</sup>,� <sup>W</sup> <sup>R</sup>^1; <sup>α</sup>; <sup>α</sup> � �<sup>W</sup> <sup>R</sup>^2; <sup>β</sup>; <sup>β</sup> � � �

<sup>þ</sup><sup>W</sup> <sup>R</sup>^1; �α; �<sup>α</sup> � �<sup>W</sup> <sup>R</sup>^2; �β; �<sup>β</sup> � ��,

� <sup>W</sup> <sup>R</sup>^1; <sup>α</sup>; �<sup>α</sup> � �<sup>W</sup> <sup>R</sup>^2; <sup>β</sup>; �<sup>β</sup> � � � <sup>W</sup> <sup>R</sup>^1; �α; <sup>α</sup> � �<sup>W</sup> <sup>R</sup>^2; �β; <sup>β</sup> � �

is the normalization factor.

� � � � , (4)

(5)

<sup>p</sup> <sup>x</sup> � <sup>x</sup><sup>∗</sup> ð Þ<sup>z</sup> � <sup>y</sup><sup>2</sup> � <sup>z</sup>2. For the quasi-Bell

<sup>n</sup>!m! <sup>p</sup> <sup>∣</sup>n, mi, (1)

∣0, 0i is the two-mode vacuum state and Dið Þ α is the displacement operator of the

þj j <sup>β</sup> <sup>2</sup> ð Þ<sup>=</sup><sup>2</sup> ∑

where ∣n1, n2i are the two-mode Fock states. The quasi-Bell coherent states (QBS) are defined by the following superpositions of two-mode coherent states:

transpose (NPT) and its logarithmic extension [21].

Advances in Quantum Communication and Information

a^ and ^

where N� ¼ 1

f xð Þ¼� ; y; z 2j j x

48

r

where <sup>R</sup>^ <sup>¼</sup> <sup>q</sup>1; <sup>p</sup>1; <sup>q</sup>2; <sup>p</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> ffiffi 2

W�

measure is the ease of calculated with a numerical integration.

mode i ið Þ ¼ A; B . The state ∣α, βi can be expressed into the form

� j j <sup>α</sup> <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 exp �2j j <sup>α</sup> <sup>2</sup> � <sup>2</sup>j j <sup>β</sup> <sup>2</sup> � �

The Wigner function W R^; α; β � � of the state (1) is given by

<sup>W</sup> <sup>R</sup>^; <sup>α</sup>; <sup>β</sup> � � <sup>¼</sup> <sup>1</sup>

<sup>p</sup> <sup>x</sup>ð Þ <sup>∗</sup> <sup>þ</sup> <sup>x</sup> <sup>y</sup> <sup>þ</sup> <sup>i</sup> ffiffi

i ¼ 1, 2; f g x; y ¼ �f g α; �β satisfies the normalization condition

QCS <sup>R</sup>^; <sup>α</sup>; <sup>β</sup> � � <sup>¼</sup> <sup>N</sup><sup>2</sup>

∣α, βi ¼ e

RRRR <sup>R</sup>^<sup>W</sup> <sup>R</sup>^; <sup>α</sup>; <sup>β</sup>; � �dR^ <sup>¼</sup> 1. Hence the doubled volume of the integrated negative part of the Wigner function of the state (2) may be written as

$$\delta(\boldsymbol{\Psi}\_{\pm}) = \iiint\limits\_{\hat{\mathbb{R}}} \int\limits\_{\hat{\mathbb{R}}} \left| \mathcal{W}\_{\text{QCS}}^{\pm}(\hat{\mathbb{R}}, a, \boldsymbol{\beta}) \right| d\hat{\mathbb{R}} - \mathbf{1}.\tag{6}$$

By definition, the quantity δ is equal to 0 for coherent and squeezed vacuum states, for which W is nonnegative. In this work we shall treat δ as a parameter characterizing the properties of the state under consideration.

It is clear from expression (5) and the plot in Figure 1 that the Wigner function of the quasi-Bell state (2) is non-Gaussian. In order to characterize this non-Gaussianity, several measures of the degree of non-Gaussianity were proposed [28, 29]. According to [29], the degree of non-Gaussianity of state ρ is defined by

$$\delta\_{\rm NG}(\rho) = \mathbb{S}(\rho \| \pi). \tag{7}$$

where S ρ1∥ρ<sup>2</sup> ð Þ is the quantum relative entropy between states ρ<sup>1</sup> and ρ2. Here τ is the reference Gaussian state with the same first and second moments of ρ. This property of reference state τ leads to Tr½ �¼ ρ ln τ Tr½ � τ ln τ , so that

$$\delta\_{\rm NG}(\rho) = \mathbb{S}(\pi) - \mathbb{S}(\rho), \tag{8}$$

where Sð Þρ is the Von Neumann entropy of the state ρ. Also Sð Þ¼ τ h dð Þþ <sup>þ</sup> h dð Þ � , where

$$h(\mathbf{x}) = \left(\mathbf{x} + \frac{1}{2}\right) \ln\left(\mathbf{x} + \frac{1}{2}\right) - \left(\mathbf{x} - \frac{1}{2}\right) \ln\left(\mathbf{x} - \frac{1}{2}\right) \tag{9}$$

and d<sup>2</sup> � <sup>¼</sup> <sup>1</sup> <sup>2</sup> Δð Þ� δ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>Δ</sup>ð Þ<sup>δ</sup> <sup>2</sup> � <sup>4</sup>I<sup>4</sup> � � q are the symplectic eigenvalues of the covariance matrix σ of the reference Gaussian state τ. Here Δð Þ¼ δ I<sup>1</sup> þ I<sup>2</sup> þ 2I3, where I<sup>1</sup> ¼ detð Þ A , I<sup>2</sup> ¼ detð Þ B , I<sup>3</sup> ¼ detð Þ C , and I<sup>4</sup> ¼ detð Þ σ are the four local symplectic invariants of the covariance matrix:

$$
\sigma = \begin{pmatrix} \mathbf{A} & \mathbf{C} \\ \mathbf{C}^{\dagger} & \mathbf{B} \end{pmatrix}, \tag{10}
$$

where

$$
\sigma\_{\vec{\eta}} = \frac{1}{2} \left\langle \left\{ R\_i, R\_{\vec{\jmath}} \right\} \right\rangle - \left\langle R\_i \right\rangle \left\langle R\_{\vec{\jmath}} \right\rangle. \tag{11}
$$

For the considered states (2), we suppose that the two fields have the same mode (α ¼ β); we find

$$
\sigma\_{\mathbb{V}\_{+}} = \begin{pmatrix} u\_{+} & \mathbf{0} & r\_{+} & \mathbf{0} \\ \mathbf{0} & v\_{+} & \mathbf{0} & s\_{+} \\ r\_{+} & \mathbf{0} & u\_{+} & \mathbf{0} \\ \mathbf{0} & s\_{+} & \mathbf{0} & v\_{+} \end{pmatrix}, \tag{12}
$$

$$
\sigma\_{\mathbb{V}\_{-}} = \begin{pmatrix} u\_{-} & \mathbf{0} & r\_{-} & \mathbf{0} \\ \mathbf{0} & v\_{-} & \mathbf{0} & s\_{-} \\ r\_{-} & \mathbf{0} & u\_{-} & \mathbf{0} \\ \mathbf{0} & s\_{-} & \mathbf{0} & v\_{-} \end{pmatrix}, \tag{13}
$$

where we have defined

<sup>v</sup>� <sup>¼</sup> <sup>N</sup><sup>2</sup>

Figure 2.

<sup>u</sup>� <sup>¼</sup> <sup>N</sup><sup>2</sup>

with <sup>Γ</sup> <sup>¼</sup> h i <sup>α</sup>j � <sup>α</sup> <sup>¼</sup> Exp � j j <sup>α</sup> <sup>2</sup>

<sup>α</sup>, <sup>β</sup>,� <sup>∓</sup>4α<sup>2</sup>

Non-Gaussianity versus j j α . (a) for states ψ<sup>þ</sup> and (b) for state ψ�.

Non-Gaussian Entanglement and Wigner Function DOI: http://dx.doi.org/10.5772/intechopen.86426

> <sup>N</sup>ð Þ¼ <sup>ρ</sup>^ <sup>1</sup> 2 Tr

where ∥:∥ denotes the trace norm [21].

<sup>ρ</sup>^� <sup>¼</sup> <sup>∑</sup> <sup>n</sup>1, <sup>n</sup>2, <sup>m</sup>1, <sup>m</sup><sup>2</sup>

<sup>n</sup>1,n2,m1,m<sup>2</sup> <sup>¼</sup> <sup>N</sup><sup>2</sup>

quasi-Bell state (2) in the Fock basis:

ρ�

where

51

<sup>α</sup>, <sup>β</sup>,� <sup>4</sup>α<sup>2</sup> � <sup>Γ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> � �, r� <sup>¼</sup> <sup>4</sup>α<sup>2</sup>

<sup>Γ</sup><sup>2</sup> <sup>∓</sup> <sup>Γ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> � �, s� <sup>¼</sup> <sup>∓</sup>4Γ<sup>2</sup>

Gaussianity of states (2) in terms of j j α . These figures show that non-Gaussianity increases with increasing j j α (this behavior will be discussed in the fourth section).

In this section, we briefly review the NPT as a computable entanglement measure that possesses the proprieties of an entanglement monotone given in [21]. The NPT, Nð Þ ρ^ of a state ρ^ is defined as the absolute value of the sum of the negative eigenvalues of the partial transpose of ρ^ denoted ρ^PT. We may write it as

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ρ</sup>^PT � �<sup>2</sup> <sup>q</sup>

> > ρ� n1,n2,m1,m<sup>2</sup>

0e

� <sup>ρ</sup>^PT � � <sup>¼</sup> <sup>∥</sup>ρ^PT<sup>∥</sup> � <sup>1</sup>

The quasi-Bell coherent state (2) is defined in a non-orthonormal basis, and it is typically not possible to obtain an analytical expression for the negativity. However, as shown in the following, one can compute it numerically. First, we expand the

�2j j <sup>α</sup> <sup>α</sup>ð Þ <sup>n</sup>1þn<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>n</sup>1!n2! <sup>p</sup> <sup>þ</sup> ð Þ �<sup>α</sup> ð Þ <sup>m</sup>1þm<sup>2</sup>

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>m</sup>1!m2! <sup>p</sup>

2

3. Numerical evaluation of negativities (NPT and NWF)

N2

α2 N2

� �. Figure 2 shows the behavior of non-

<sup>α</sup>, <sup>β</sup>,�, (14)

<sup>α</sup>, <sup>β</sup>,�, (15)

<sup>2</sup> , (16)

: (18)

∣m1ih i n1j⊗jm<sup>2</sup> hn2∣, (17)

#### Figure 1.

Wigner function of quasi-Bell states (2). (a) Wigner function of ψ<sup>þ</sup> for j j α ¼ 0. (b) Wigner function of ψ<sup>þ</sup> for j j α ¼ 2. (c) Wigner function of ψ<sup>þ</sup> for j j α ¼ 1. (d) Wigner function of ψ<sup>þ</sup> for j j α ¼ 2. (e) Wigner function of ψ� for j j α ¼ 0:5. (f) Wigner function of ψ� for j j α ¼ 1. (g) Wigner function of ψ� for j j α ¼ 1:5. (h) Wigner function of ψ� for j j α ¼ 2.

Non-Gaussian Entanglement and Wigner Function DOI: http://dx.doi.org/10.5772/intechopen.86426

Figure 2. Non-Gaussianity versus j j α . (a) for states ψ<sup>þ</sup> and (b) for state ψ�.

where we have defined

$$u\_{\pm} = N\_{a,\beta,\pm}^2 (4a^2 \pm \Gamma^2 + 1), \qquad \qquad r\_{\pm} = 4a^2 N\_{a,\beta,\pm}^2,\tag{14}$$

$$
v\_{\pm} = N\_{a,\beta,\pm}^2 \left( \mp 4a^2 \Gamma^2 \mp \Gamma^2 + 1 \right), \qquad \qquad \varsigma\_{\pm} = \mp 4\Gamma^2 a^2 N\_{a,\beta,\pm}^2,\tag{15}$$

with <sup>Γ</sup> <sup>¼</sup> h i <sup>α</sup>j � <sup>α</sup> <sup>¼</sup> Exp � j j <sup>α</sup> <sup>2</sup> 2 � �. Figure 2 shows the behavior of non-Gaussianity of states (2) in terms of j j α . These figures show that non-Gaussianity increases with increasing j j α (this behavior will be discussed in the fourth section).
