Testing genuine multipartite nonlocality in phase space

Abstract

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. Quantum nonlocality is one of the most fundamental features of quantum mechanics. It refers to the correlations that cannot be explained by local hidden-variable models that satisfy a set of constraints epitomized by so-called Bell inequalities [1]. The violation of a Bell inequality reveals the existence of nonlocality in a given quantum mechanical state [2][3][4].While originally formulated for bipartite systems, Bell-like inequalities have been extended to the multipartite scenario, a noticeable example being embodied by the well-known inequality proposed by Mermin and Klyshko (MK) [5]. However, the violation of a MK-type inequality by a multipartite state does not necessarily imply the existence of genuine multipartite nonlocality, as this test can be flasified by nonlocal correlations in any reduction of the system's components. In order to demonstrate genuine tripartite nonlocality, another type of Bell inequalities should be thus considered such as the one formulated by Svetlichny [6], which rules out both local and nonlocal hidden variable models possibly imposed on any subparties. It was also noted that a stronger violation of the MK type can demonstrate genuine nonlocality for the cases with an even number of parties [7]. Experimental demonstrations of genuine multipartite nonlocality were firstly achieved by strong violations of an MK inquality with four photon polarization entanglements [8]. A violation of Svetlichnytype inequality was experimentally demonstrated recently with GHZ-type photon polarization entangled states [9]. A generalized version of Svetlichny inequality was recently proposed and studied [10].In this paper, motivated by the growing experimental capabilities of controlling and manipulating the state of tripartite quantum systems, in the optical laboratory [11] and beyond, we address the formulation of genuine three-mode continuous variable (CV) nonlocality tests in phase space. While the phase space provides a natural arena for the description of the state of multimode CV systems, it also provides us with powerful tools for the analysis of the quantum correla...

Theory of Genuine Tripartite Nonlocality of Gaussian States

“…Recently, a prescription to extend the Svetlichny inequality to the continuous variable domain using these measurements has been proposed [27]. This formalism was, however, applied in the Gaussian setting only to sparse, specialized examples [22,26,27,36].…”

“…Nonlocality tests have been studied also for continuous variable systems [19][20][21][22][23][24][25][26][27], namely systems whose canonical degrees of freedom, which can be nonclassically correlated, have a continuous spectrum [28,29]. This is the case, for instance, for quadrature modes of light, phononic momentum modes of Bose-Einstein condensates, vibrational modes of mechanical resonators, or collective spin components of cold atomic ensembles [30].…”

Genuine multipartite nonlocality of permutationally invariant Gaussian states

“…, and so on [34,36]. If ρ is a multimode Gaussian state with zero first moments and covariance matrix σ, its Wigner distribution is given by Eq.…”

“…To test multipartite nonlocality in n-mode continuous variable systems, we first choose displaced parity measurements as the operators to be measured on each mode j [27,28,34]. In the case of optical fields, the displaced parity observableP j on mode j can be measured by photon counting, preceded by a phase space displacement, the latter implemented e.g.…”

“…These states, which include squeezed and thermal states of quantized electromagnetic fields, are the theoretical pillars and the resources of choice for a number of applications in quantum information with continuous variables [20][21][22][23][24]. Their nonlocal properties have been explored in a few papers [25][26][27][28][29][30][31][32][33][34][35][36], albeit mostly limited to two or three modes. We investigate genuine multipartite nonlocality as revealed by the violation of an inequality first proposed for tripartite states by Svetlichny [12].…”