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, 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

$$N(\hat{\rho}) = \frac{1}{2}Tr\left(\sqrt{\left(\hat{\rho}^{PT}\right)^2} - \hat{\rho}^{PT}\right) = \frac{\|\hat{\rho}^{PT}\| - 1}{2},\tag{16}$$

where ∥:∥ 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:

$$\hat{\rho}\_{\pm} = \sum\_{n\_1, n\_2, m\_1, m\_2} \rho\_{n\_1, n\_2, m\_1, m\_2}^{\pm} |m\_1\rangle \left\langle n\_1 | \otimes | m\_2 \right\rangle \langle n\_2 |,\tag{17}$$

where

$$\rho\_{n\_1,n\_2,m\_1,m\_2}^{\pm} = N\_0^2 e^{-2|a|} \left( \frac{a^{(n\_1+n\_2)}}{\sqrt{n\_1!n\_2!}} + \frac{(-a)^{(m\_1+m\_2)}}{\sqrt{m\_1!m\_2!}} \right). \tag{18}$$

Figure 1.

50

function of ψ� for j j α ¼ 2.

Advances in Quantum Communication and Information

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

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

$$\hat{\rho}\_{\pm} = \sum\_{n\_{1\circ}n\_{2\circ}m\_{1\circ}m\_{2}} \rho\_{n\_{1\circ}m\_{2\circ}m\_{1\circ}n\_{2}}^{\pm} |m\_{1}\rangle\langle n\_{1}| \otimes |m\_{2}\rangle\langle n\_{2}|.\tag{19}$$

The plot in Figure 3c and d shows the behavior of the NWF as a function of j j α for the non-Gaussian system (2). These two plots show that the NWF δWF 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

For our measure, 1≥δWF ≥ 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

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 quanti-

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

values of α where the state (2) is maximally entangled (Bell state).

an entanglement quantifier [21].

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

fier of non-Gaussian bipartite entanglement.

entanglement in bipartite systems.

Mustapha Ziane\* and Morad El Baz

provided the original work is properly cited.

Faculté des Sciences, Université Mohammed 5, Rabat, Morocco

\*Address all correspondence to: mustapha.ziane@um5s.net.ma

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Author details

53

5. Conclusion

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.

### 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 j j α .

Figure 2 shows the variation of the degree of non-Gaussianity for the states in Eq. (2) as a function of coherent state amplitude j j α . We see that the non-Gaussianity δNG measured by (8) equal to 0 for small values of j j α increases with increasing values of the parameter j j α 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 j j α ¼ 0 and increase with increasing values of the parameter j j α to reach its maximum value that is, equal to 1 for j j α ≳ 1:3. Furthermore, it is seen that the entanglement for large values of α reaches its maximum value. It is worthwhile noting that, at the limit of large values of the parameter α, the coherent states ∣αi and ∣ � αi become orthogonal; thus the behavior of quasi-Bell state (2) is, as expected, exactly that of the Bell state.

Figure 3.

Negativity of the partial transpose versus j j α : for the quasi-Bell state ψ<sup>þ</sup> (a) and ψ� (b). Negativity of the Wigner function versus j j α : for the quasi-Bell state ψ<sup>þ</sup> (c) and ψ� (d).

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

The plot in Figure 3c and d shows the behavior of the NWF as a function of j j α for the non-Gaussian system (2). These two plots show that the NWF δWF 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≥δWF ≥ 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 α where the state (2) is maximally entangled (Bell state).
