Testing genuine multipartite nonlocality in phase space

We demonstrate genuine three-mode nonlocality based on phase-space formalism. A Svetlichny-type Bell inequality is formulated in terms of the $s$-parametrized quasiprobability function. We test such a tool using exemplary forms of three-mode entangled states, identifying the ideal measurement settings required for each state. We thus verify the presence of genuine three-mode nonlocality that cannot be reproduced by local or nonlocal hidden variable models between any two out of three modes. In our results, GHZ- and W-type nonlocality can be fully discriminated. We also study the behavior of genuine tripartite nonlocality under the effects of detection inefficiency and dissipation induced by local thermal environments. Our formalism can be useful to test the sharing of genuine multipartite quantum correlations among the elements of some interesting physical settings, including arrays of trapped ions and intracavity ultracold atoms.

Figure 1

Behavior of the $s$-parametrized MK and Svetlichny parameter $\left|\mathcal{M}\right|$ and $\left|\mathcal{S}\right|$ for (a) the single-photon W state $|\varphi \rangle$ (with variable $p$), (b) the three-mode squeezed vacuum state ($r$ being the degree of squeezing), and (c) the GHZ-type ECS ($\zeta$ is the amplitude of each coherent-state component).

Figure 2

Genuine tripartite nonlocality phase diagram for a three-mode ECS. We plot the Svetlichny parameter $\left|\mathcal{S}\right|$ against the amplitude of the coherent-state components $\zeta$. The W-type and GHZ-type nonlocal characters, best revealed by tests built on the Husimi ($s=-1$, dashed line) and Wigner function ($s=0$, solid line), respectively, cross for $\zeta \simeq 0.455$.

Figure 3

Tripartite nonlocality phase diagram for a three-mode ECS against the detection efficiency $\eta$. As in Fig. 2, dashed (solid) traits are for $s=-1$ ($s=0$).

Figure 4

Behavior of the MK parameter (a) and the Svetlichny one (b) against the section efficiency $\eta$ for a three-mode ECS ($\zeta =1.0,0.1$) and a three-mode squeezed state ($r=0.5$).

Figure 5

Dynamics of genuine GHZ-type ($\zeta =1,2$) and W-type ($\zeta =0.1$) nonlocality of a three-mode ECS interacting with individual thermal baths at equilibrium (mean occupation number $\overline{n}=0$).

Figure 6

Exemplary physical systems where our phase-space formalism could be applied to investigate genuine multipartite nonlocality: (a) an ultracold atomic system loaded into an intracavity double-well potential [39]; (b) a Coulomb crystal of ions confined on a planar on-chip trap (the dashed triangle identified a subsystem of three ions whose state can be studied with our proposed tools. Red (blue) arrows show nearest-neighbor (next-to-nearest-neighbor) interactions [36]; (c) a mechanical membrane in the middle of a cavity [40]; (d) a doped microtoroid coupled to a fiber [41]; (e) a hybrid optomechanical device including a mechanical mode and a trapped particle [44] or Bose-Einstein condensate [42].