The negativity of Wigner function as a measure of quantum correlations

Abstract

In this paper, we study comparatively the behaviors of Wigner function and quantum correlations for two quasi-Werner states formed with two general bipartite superposed coherent states. We show that the Wigner function can be used to detect and quantify the quantum correlations. However, we show that it is in fact not sensitive to all kinds of quantum correlations but only to entanglement. Then, we analyze the measure of non-classicality of quantum states based on the volume occupied by the negative part of the Wigner function.

Appendix

Let us consider the Gisin states [35] based on the bipartite SCS defined as

$$\begin{aligned} \rho _{Gisin}= \frac{(1-a)}{2} \left( \left| ++\right\rangle \left\langle ++\right\rangle + \left| --\right| \left\langle -- \right| \right) + a \left| \psi \right\rangle \left\langle \psi \right| \end{aligned}$$

(28)

where $\left| \psi \right\rangle$ is the state written in the Eq. (17)

The Wigner function of the state $\rho _{Gisin}$ is expressed as:

$$\begin{aligned} W_{2mcsG}:= a W_{1} + ((1 - a)/2) W_{2} \end{aligned}$$

(29)

where

$$\begin{aligned} W_{1}= (n^{-})^{2} (\pi ^{ \frac{-3}{2}}) \Big ( W_{\alpha \; \alpha }\; W_{-\beta \; -\beta }-W_{\alpha \; -\alpha }\; W_{\beta \; -\beta } -W_{-\alpha \; \alpha }\; W_{-\beta \; \beta } +W_{-\alpha \; -\alpha }\; W_{\beta \; \beta } \Big ) \end{aligned}$$

and

$$\begin{aligned} W_{2}= W_{(++)_{\alpha }} W_{(++)_{\beta }} + W_{(--)_{\alpha }} W_{(--)_{\beta }} \end{aligned}$$

with,

$$\begin{aligned} \begin{aligned} W_{(++)_{\alpha }}&= \frac{(N^{+})^2}{(\pi ^{(3/2)})} W_{\alpha \; \alpha } + W_{\alpha \; -\alpha } +W_{-\alpha \; \alpha }+ W_{-\alpha \; -\alpha }\\ W_{(++)_{\beta }}&= \frac{(N^{+})^2}{(\pi ^{(3/2)})} W_{\beta \; \beta } + W_{\beta \; -\beta } +W_{-\beta \; \beta }+ W_{-\beta \; -\beta } \\ W_{(--)_{\alpha }}&= \frac{(N^{-})^2}{(\pi ^{(3/2)})} W_{\alpha \; \alpha } - W_{\alpha \; -\alpha } - W_{-\alpha \; \alpha }+ W_{-\alpha \; -\alpha }\\ W_{(--)_{\beta }}&= \frac{(N^{-})^2}{(\pi ^{(3/2)})} W_{\beta \; \beta } - W_{\beta \; -\beta } - W_{-\beta \; \beta }+ W_{-\beta \; -\beta } \end{aligned} \end{aligned}$$

(30)

We plot the Wigner function $\mathcal {W}$ and the quantum correlations (quantum discord $\mathcal {D}$, and entanglement E) as a function of the parameter $\alpha$ (Fig. 6a) and as a function of the mixing parameter a (Fig. 6b).

Fig. 6

Quantum correlations and Wigner function for the state $\rho _{Gisin}$ for fixed values of $p_1$ and $q_2$ ($p_1=q_2=0$) with respect to coherent amplitude $\alpha$ (a) and mixing parameter a (b). a $a=0.6, \beta =0.99, q_1=0.01$ and $p_2=0.51$, b $\alpha =0.06, \beta =0.18, q_1=0.01$ and $p_2=0.71$

The Figures do confirm the previous conclusions derived in the main text. As a matter of fact, we clearly see from both figures that:

• Quantum discord exists in the states even before quantum entanglement appears. This confirms the, now, obvious fact that there exist quantum correlations other than entanglement.

• Wigner function starts being negative exactly at the same point when entanglement appears in the states. This in term confirms the main result of this paper that the Wigner function is not sensitive to all kinds of quantum correlations but only to entanglement.

It is worthwhile, in addition to these remarks, in contrast to the figures obtained for the Werner states, that the behavior of the quantum discord is not smooth and shows some sudden change at some point. This sudden change is due to the fact that at this specific point there is a change in the value on the basis that maximizes the mutual information (9) (or equivalently minimizes the conditional entropy). This sudden change in the behavior of quantum discord of the Gisin states is the subject of a work under progress.