Genuine multipartite nonlocality of permutationally invariant Gaussian states

We investigate genuine multipartite nonlocality of pure permutationally invariant multimode Gaussian states of continuous variable systems, as detected by the violation of Svetlichny inequality. We identify the phase space settings leading to the largest violation of the inequality when using displaced parity measurements, distinguishing our results between the cases of even and odd total number of modes. We further consider pseudospin measurements and show that, for three-mode states with asymptotically large squeezing degree, particular settings of these measurements allow one to approach the maximum violation of Svetlichny inequality allowed by quantum mechanics. This indicates that the strongest manifestation of genuine multipartite quantum nonlocality is in principle verifiable on Gaussian states.

Figure 1

Plots of the Svetlichny parameter $S_n\left(a\right)$ for pure permutationally invariant $n$-mode Gaussian states using displaced parity measurements with $q_0=0=q_1$ and variable settings $p_0,p_1$, for $a=1.5$ and representative choices of $n$. Top row: (a) $n=3$, (b) $n=5$, (c) $n=15$, and (d) $n=23$. Bottom row: (e) $n=4$, (f) $n=6$, (g) $n=16$, and (h) $n=24$. The vertical axis in each panel ranges from 1 to the maximum quantum bound $S_n^Q$, so that only values of $S_n$ violating the Svetlichny inequality (14) are shown. All the plotted quantities are dimensionless.

Figure 2

Optimal Svetlichny parameter $S_n^{\text{opt}}$ (solid curves) for pure permutationally invariant $n$-mode Gaussian states using displaced parity measurements, plotted versus the covariance parameter $a$ for (from bottom to top): (a) $n=2$ (red online), $n=4$ (green online), $n=6$ (blue online), and (b) $n=3$ (red online), $n=5$ (green online), $n=7$ (blue online). The insets detail the regime of small $a$, showing that a threshold for violations of the Svetlichny inequality exists in the odd $n$ case (b), but not in the even $n$ case (a). In both panels, the dashed horizontal lines indicate the maximum value $S_n^Q$ of the Svetlichny parameter allowed by quantum mechanics, given by (from bottom to top) $S_2^Q=S_3^Q=\sqrt{2}$ (dashed red online), $S_4^Q=S_5^Q=2\sqrt{2}$ (dashed green online), and $S_6^Q=S_7^Q=4\sqrt{2}$ (dashed blue online), respectively. All the plotted quantities are dimensionless.

Figure 3

Optimal Svetlichny parameter $S_n^{\text{opt}}$ for the three-mode Gaussian states of Eq. (27), using pseudospin observables (blue points) and displaced parity measurements (solid green line), plotted versus the squeezing parameter $r$. As in Fig. 2, the dashed horizontal (red) line indicates the maximum value $S_3^Q=\sqrt{2}$ of the Svetlichny parameter allowed by quantum mechanics. All the plotted quantities are dimensionless.

Figure 4

Numerical study of the sequence $f\left(n\right)$ as defined in Eq. (38). Red dots indicate the numerically calculated value of $f\left(n\right)$. Notice how the asymptotic behavior for large $n$ appears well approximated by $\simeq 0.282n^{-3/2}$ (black solid line), which has been obtained via a power-law fit. The sequence $n^{-3/2}$ is also shown for comparison (blue dashed line). Logarithmic scale is used on both axes. All the plotted quantities are dimensionless.