Non-Gaussian Entanglement and Wigner Function

Abstract

A measure of non-Gaussian entanglement in continuous variable (CV) systems based on the volume of the negative part of the Wigner function is proposed. We analyze comparatively this quantity with a numerical evaluation of the negativity of the partial transpose (NPT) considering a system of Bell states formed in the coherent state basis (quasi-Bell states).

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, 5, 6], quantum dense coding [7, 8, 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, 12, 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, 17, 18, 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 $\rho$ 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 $\widehat{a}$ and $\widehat{b}$ . Two-mode coherent states are defined by $\mid \alpha ,\beta ⟩=D_a\left(\alpha \right)D_b\left(\beta \right)\mid 0,0⟩$ , where $\mid 0,0⟩$ is the two-mode vacuum state and $D_i\left(\alpha \right)$ is the displacement operator of the mode $i\left(i=A,B\right)$ . The state $\mid \alpha ,\beta ⟩$ can be expressed into the form

$$\mid \alpha ,\beta ⟩=e^{-\left({\left|\alpha \right|}^2+{\left|\beta \right|}^2\right)/2}\sum _{n,m}^{\infty }\frac{{\alpha }^n{\beta }^n}{\sqrt{n!m!}}\mid n,m⟩,$$ E1

where $\mid n_1,n_2⟩$ are the two-mode Fock states. The quasi-Bell coherent states (QBS) are defined by the following superpositions of two-mode coherent states:

$$\mid {\psi }_{\pm }⟩=N_{\pm }\left(|\alpha ,\beta ⟩\pm |-\alpha ,-\beta ⟩\right),$$ E2

$$\mid {\varphi }_{\pm }⟩=N_{\pm }\left(|\alpha ,-\beta ⟩\pm |-\alpha ,\beta ⟩\right),$$ E3

where $N_{\pm }=1\sqrt{2\text{exp}\left(-2{\left|\alpha \right|}^2-2{\left|\beta \right|}^2\right)+2}$ is the normalization factor.

The Wigner function $\mathcal{W}\left(\widehat{R},\alpha ,\beta \right)$ of the state (1) is given by

$$\mathcal{W}\left(\widehat{R},\alpha ,\beta \right)=\frac{1}{\pi }\text{exp}\left(f\left(\alpha ,q_1,p_1\right)+f\left(\beta ,q_2,p_2\right)\right),$$ E4

where $\widehat{R}={\left(q_1,p_1,q_2,p_2\right)}^T$ is the quadrature operators vector and $f\left(x,y,z\right)=-2{\left|x\right|}^2+\sqrt{2}\left(x^{\ast }+x\right)y+i\sqrt{2}\left(x-x^{\ast }\right)z-y^2-z^2$ . For the quasi-Bell entangled coherent states Eq. (2), the Wigner function is given by [26, 27].

$$\begin{array}{ll}{\mathcal{W}}_{\mathit{QCS}}^{\pm }\left({\displaystyle \widehat{R}},\alpha ,\beta \right)& =N_{\alpha ,\beta ,\pm }^2\left[\mathcal{W}\left({{\displaystyle \widehat{R}}}_1,\alpha ,\alpha \right)\mathcal{W}\left({{\displaystyle \widehat{R}}}_2,\beta ,\beta \right)\right)\\ & \pm \mathcal{W}\left({{\displaystyle \widehat{R}}}_1,\alpha ,-\alpha \right)\mathcal{W}\left({{\displaystyle \widehat{R}}}_2,\beta ,-\beta \right)\\ & \pm \mathcal{W}\left({{\displaystyle \widehat{R}}}_1,-\alpha ,\alpha \right)\mathcal{W}\left({{\displaystyle \widehat{R}}}_2,-\beta ,\beta \right)\\ & \left(+\mathcal{W}\left({{\displaystyle \widehat{R}}}_1,-\alpha ,-\alpha \right)\mathcal{W}\left({{\displaystyle \widehat{R}}}_2,-\beta ,-\beta \right)\right],\end{array}$$ E5

where ${\widehat{R}}_1$ and ${\widehat{R}}_2$ are the quadrature operators vectors of the first and second modes and $\mathcal{W}\left({\widehat{R}}_i,x,y\right)$ is the Wigner function of one-mode coherent state with $i=1,2$ ; $\left\{x,y\right\}=\left\{\pm \alpha ,\pm \beta \right\}$ satisfies the normalization condition ${\int \int \int \int }_{\widehat{R}}\mathcal{W}\left(\widehat{R},\alpha ,\beta ,\right)d\widehat{R}=1$ . Hence the doubled volume of the integrated negative part of the Wigner function of the state (2) may be written as

$$\delta \left({\psi }_{\pm }\right)=\underset{\widehat{R}}{\int \int \int \int }\left|{\mathcal{W}}_{\mathit{QCS}}^{\pm }\left(\widehat{R},\alpha ,\beta \right)\right|d\widehat{R}-1.$$ E6

By definition, the quantity $\delta$ is equal to 0 for coherent and squeezed vacuum states, for which $\mathcal{W}$ is nonnegative. In this work we shall treat $\delta$ 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 $\rho$ is defined by

Figure 1.

Wigner function of quasi-Bell states (2). (a) Wigner function of ${\psi }_{+}$ for $\left|\mathit{\alpha }\right|=\mathit{0}$ . (b) Wigner function of ${\psi }_{+}$ for $\left|\mathit{\alpha }\right|=\mathit{2}$ . (c) Wigner function of ${\psi }_{+}$ for $\left|\mathit{\alpha }\right|=\mathit{1}$ . (d) Wigner function of ${\psi }_{+}$ for $\left|\mathit{\alpha }\right|=\mathit{2}$ . (e) Wigner function of ${\psi }_{-}$ for $\left|\mathit{\alpha }\right|=\mathit{0.5}$ . (f) Wigner function of ${\psi }_{-}$ for $\left|\mathit{\alpha }\right|=\mathit{1}$ . (g) Wigner function of ${\psi }_{-}$ for $\left|\mathit{\alpha }\right|=\mathit{1.5}$ . (h) Wigner function of ${\psi }_{-}$ for $\left|\mathit{\alpha }\right|=\mathit{2}$ .

$${\delta }_{\mathit{NG}}\left(\rho \right)=S\left(\rho \parallel \tau \right).$$ E7

where $S\left({\rho }_1\parallel {\rho }_2\right)$ is the quantum relative entropy between states ${\rho }_1$ and ${\rho }_2$ . Here $\tau$ is the reference Gaussian state with the same first and second moments of $\rho$ . This property of reference state $\tau$ leads to $\mathrm{Tr}\left[\rho \text{ln}\tau \right]=\mathrm{Tr}\left[\tau \text{ln}\tau \right]$ , so that

$${\delta }_{\mathit{NG}}\left(\rho \right)=S\left(\tau \right)-S\left(\rho \right),$$ E8

where $S\left(\rho \right)$ is the Von Neumann entropy of the state $\rho$ . Also $S\left(\tau \right)=h\left(d_{+}\right)+h\left(d_{-}\right)$ , where

$$h\left(x\right)=\left(x+\frac{1}{2}\right)\text{ln}\left(x+\frac{1}{2}\right)-\left(x-\frac{1}{2}\right)\text{ln}\left(x-\frac{1}{2}\right)$$ E9

and $d_{\pm }^2=\frac{1}{2}\left(\mathrm{\Delta }\left(\delta \right)\pm \sqrt{\mathrm{\Delta }{\left(\delta \right)}^2-4I_4}\right)$ are the symplectic eigenvalues of the covariance matrix $\sigma$ of the reference Gaussian state $\tau$ . Here $\mathrm{\Delta }\left(\delta \right)=I_1+I_2+2I_3$ , where $I_1=\text{det}\left(\mathbf{A}\right)$ , $I_2=\text{det}\left(\mathbf{B}\right)$ , $I_3=\text{det}\left(\mathbf{C}\right)$ , and $I_4=\text{det}\left(\sigma \right)$ are the four local symplectic invariants of the covariance matrix:

$$\sigma =\left(\begin{array}{ll}\mathbf{A}& \mathbf{C}\\ {\mathbf{C}}^{†}& \mathbf{B}\end{array}\right),$$ E10

where

$${\sigma }_{\mathit{ij}}=\frac{1}{2}⟨\left\{R_i,R_j\right\}⟩-⟨R_i⟩⟨R_j⟩.$$ E11

For the considered states (2), we suppose that the two fields have the same mode ( $\alpha =\beta$ ); we find

$${\sigma }_{{\psi }_{+}}=\left(\begin{array}{llll}{u}_{+}& 0& r_{+}& 0\\ 0& v_{+}& 0& s_{+}\\ r_{+}& 0& u_{+}& 0\\ 0& s_{+}& 0& v_{+}\end{array}\right),$$ E12

$${\sigma }_{{\psi }_{-}}=\left(\begin{array}{llll}{u}_{-}& 0& r_{-}& 0\\ 0& v_{-}& 0& s_{-}\\ r_{-}& 0& u_{-}& 0\\ 0& s_{-}& 0& v_{-}\end{array}\right),$$ E13

where we have defined

$$u_{\pm }=N_{\alpha ,\beta ,\pm }^2\left(4{\alpha }^2\pm {\mathrm{\Gamma }}^2+1\right),r_{\pm }=4{\alpha }^2{N}_{\alpha ,\beta ,\pm }^2,$$ E14

$$v_{\pm }=N_{\alpha ,\beta ,\pm }^2\left(\mp 4{\alpha }^2{\mathrm{\Gamma }}^2 \mp {\mathrm{\Gamma }}^2+1\right),s_{\pm }=\mp 4{\mathrm{\Gamma }}^2{\alpha }^2{N}_{\alpha ,\beta ,\pm }^2,$$ E15

with $\mathrm{\Gamma }=⟨\alpha |-\alpha ⟩=\mathit{Exp}\left(-\frac{{\left|\alpha \right|}^2}{2}\right)$ . Figure 2 shows the behavior of non-Gaussianity of states (2) in terms of $\left|\alpha \right|$ . These figures show that non-Gaussianity increases with increasing $\left|\alpha \right|$ (this behavior will be discussed in the fourth section).

Figure 2.

Non-Gaussianity versus $\left|\mathit{\alpha }\right|$ . (a) for states ${\psi }_{+}$ and (b) for state ${\psi }_{-}$ .

3. Numerical evaluation of negativities (NPT and NWF)

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, $N\left(\widehat{\rho }\right)$ of a state $\widehat{\rho }$ is defined as the absolute value of the sum of the negative eigenvalues of the partial transpose of $\widehat{\rho }$ denoted ${\widehat{\rho }}^{\mathit{PT}}$ . We may write it as

$$N\left(\widehat{\rho }\right)=\frac{1}{2}\mathit{Tr}\left(\sqrt{{\left({\widehat{\rho }}^{\mathit{PT}}\right)}^2}-{\widehat{\rho }}^{\mathit{PT}}\right)=\frac{\parallel {\widehat{\rho }}^{\mathit{PT}}\parallel -1}{2},$$ E16

where $\parallel .\parallel$ denotes the trace norm [21].

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

$${\widehat{\rho }}_{\pm }=\sum _{n_1,n_2,m_1,m_2}{\rho }_{n_1,n_2,m_1,m_2}^{\pm }\mid m_1⟩⟨n_1|\otimes |m_2⟩⟨n_2\mid ,$$ E17

where

$${\rho }_{n_1,n_2,m_1,m_2}^{\pm }=N_0^2{e}^{-2\left|\alpha \right|}\left(\frac{{\alpha }^{\left(n_1+n_2\right)}}{\sqrt{n_1!n_2!}}+\frac{{\left(-\alpha \right)}^{\left(m_1+m_2\right)}}{\sqrt{m_1!m_2!}}\right).$$ E18

The partial transpose of this state with respect to mode two is

$${\widehat{\rho }}_{\pm }=\sum _{n_1,n_2,m_1,m_2}{\rho }_{n_1,m_2,m_1,n_2}^{\pm }\mid m_1⟩⟨n_1|\otimes |m_2⟩⟨n_2\mid .$$ E19

The eigenvalues are obtained by numerical diagonalization of the partial transpose density matrix (19). With this result, we can obtain the NPT straightforwardly using Eq. (16), and Figure 3a and b shows the numerical values of this NPT.

Figure 3.

Negativity of the partial transpose versus $\left|\mathit{\alpha }\right|$ : for the quasi-Bell state ${\psi }_{+}$ (a) and ${\psi }_{-}$ (b). Negativity of the Wigner function versus $\left|\mathit{\alpha }\right|$ : for the quasi-Bell state ${\psi }_{+}$ (c) and ${\psi }_{-}$ (d).

4. Discussion

In this section, we will discuss the different behaviors of the non-Gaussian entanglement and the variation of the negativity of the WF for the bipartite system considered early in terms of the coherent state amplitude $\left|\alpha \right|$ .

Figure 2 shows the variation of the degree of non-Gaussianity for the states in Eq. (2) as a function of coherent state amplitude $\left|\alpha \right|$ . We see that the non-Gaussianity ${\delta }_{\mathit{NG}}$ measured by (8) equal to 0 for small values of $\left|\alpha \right|$ increases with increasing values of the parameter $\left|\alpha \right|$ to larger values much higher than 1 and does not establish in a maximum value. On the other hand, the NPT plots are shown in Figure 3a and b for the state (2) equal to 0 for $\left|\alpha \right|=0$ and increase with increasing values of the parameter $\left|\alpha \right|$ to reach its maximum value that is, equal to 1 for $\left|\alpha \right| \gtrsim 1.3$ . Furthermore, it is seen that the entanglement for large values of $\alpha$ reaches its maximum value. It is worthwhile noting that, at the limit of large values of the parameter $\alpha$ , the coherent states $\mid \alpha ⟩$ and $\mid -\alpha ⟩$ become orthogonal; thus the behavior of quasi-Bell state (2) is, as expected, exactly that of the Bell state.

The plot in Figure 3c and d shows the behavior of the NWF as a function of $\left|\alpha \right|$ for the non-Gaussian system (2). These two plots show that the NWF ${\delta }_{\mathit{WF}}$ has the same behavior as the NPT. This allows to show that they behave identically and they have the same inflection points. Which confirms that the NWF is a direct computable measure of non-Gaussian bipartite entanglement that posses the proprieties of an entanglement quantifier [21].

For our measure, $1\ge {\delta }_{\mathit{WF}}\ge 0$ , equal to zero when α became null and the state in Eq. 2 is now nothing but a two-vacuum product state, and it is maximal for large values of $\alpha$ where the state (2) is maximally entangled (Bell state).

5. Conclusion

In this work, we have evaluated the negativity of Wigner function and the negativity of the partial transpose in non-Gaussian states formed by two modes of field coherent states. We have shown that the negative parts of the Wigner function can be used as a detector of non-Gaussian entanglement. Interestingly, as used in this work, the degree of Wigner function negativity can be used as a direct quantifier of non-Gaussian bipartite entanglement.

This work allows us to describe the best characterization of the non-Gaussian Wigner function and the important use of its negativity in bipartite non-Gaussian systems, which gives more efficiency in CV quantum information theory, particularly in quantum computing [30], because the Wigner function can be measured experimentally [31, 32], including the measurements of its negative values [33]. The interest put on such experiments has triggered a search for operational definitions of the Wigner functions, based on the experimental setup [34, 35]. It does represent a major step forward in the detection and the quantification of non-Gaussian entanglement in bipartite systems.