Entanglement versus negative domains of Wigner functions

We show that $s$ waves, that is wave functions that only depend on a hyperradius, are entangled if and only if the corresponding Wigner functions exhibit negative domains. We illustrate this feature using a special class of $s$ waves which allows us to perform the calculations analytically. This class includes a Gaussian, a maximally entangled as well as a “shell” state.

Figure 1

Comparison between three different measures of the entanglement contained in the elementary $s$ wave, Eq. (8), as a function of the interpolation parameter $a$ and number $D=2d$ of dimensions: The von Neumann entropy, Eq. (33) (top), the maximal value of the CHSH sum $S\left({\Psi }^{\left(D\right)}\right)$, Eq. (36) (middle), and the numerically calculated negative volume $V_{-}\left({\Psi }^{\left(D\right)}\right)$, Eq. (47) (bottom), of the Wigner function corresponding to the elementary $s$ wave. All three figures are similar in their characteristic features. In particular, all three measures vanish for a Gaussian, that is $a=0$, and reach their maximal values $E=1$, $S=2\sqrt{2}$ at the critical point $a=-2∕D$. The only exception to this behavior appears in two dimensions, where the negative volume $V_{-}\left({\Psi }^{\left(D\right)}\right)$ has a local minimum at $a=-1$ as shown on the left-hand inset. In all other dimensions we observe an absolute maximum of $V_{-}\left({\Psi }^{\left(D\right)}\right)$ corresponding to maximally entangled states as shown by the example of $D=4$ in the neighborhood of $a=-0.5$. On the left-hand inset the separation of the ticks on the vertical and horizontal axes is $1\times {10}^{-3}$ and $5\times {10}^{-2}$, respectively. On the right-hand inset the separation of the ticks on the vertical and horizontal axes is $3\times {10}^{-3}$ and $5\times {10}^{-3}$, respectively. The horizontal axes on three figures are identical.

Figure 2

Wigner functions $W_S^{\left(D\right)}$ (top left) and $W_{\max }^{\left(D\right)}$ (bottom left) given by Eqs. (43, 45) and corresponding to the “shell” wave function Eq. (11) and the maximally entangled state Eq. (13), respectively, multiplied by the appropriate factor $r^{D-1}{k}^{D-1}{\left(\sin u\right)}^{D-2}{S}^{\left(D\right)}{S}^{(D-1)}$ representing the volume element in $D$ dimensions. Both distributions display dominant positive maxima and a characteristic negative domain. The latter is clearly more pronounced at the expense of the positive contributions in the case of the Wigner function of the maximally entangled state. Contour lines on the bottom emphasize the positive regions (solid thin curves) and the negative regions (dotted curves) as well as the border line (fat curve) between them. For both figures in the left-hand column we have chosen the dimension $D=4$ and the angle $u$ between $\vec{r}$ and $\vec{k}$ equal to $5\pi ∕12$. Border lines separating positive regions of $W_S^{\left(D\right)}$ (top right) and $W_{\max }^{\left(D\right)}$ (bottom right) from negative ones for different angles $u$ but fixed dimension $D=4$. In the case of $W_S^{\left(D\right)}$ all border lines connect the same points on the $k$ axis. As the angle $u$ increases starting from $u=0$ in steps of $\pi ∕12$ the border lines circumnavigate increasing volumes in phase space. When $u=0$ the negative domain of the Wigner function is smallest. In the other extreme, that is, for $u=\pi ∕2$ we find an open canal. For angles $u$ smaller than $\pi ∕4$ the Wigner function $W_{\max }^{\left(D\right)}$ of the maximally entangled state is always positive.