5. Conclusion

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

ρ� n1,m2,m1,n<sup>2</sup>

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.

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

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.

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).

∣m1ih i n1j⊗jm<sup>2</sup> hn2∣: (19)

<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>

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considered early in terms of the coherent state amplitude j j α .

4. Discussion

Figure 3.

52

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.
