Theory of Genuine Tripartite Nonlocality of Gaussian States

Abstract

We investigate the genuine multipartite nonlocality of three-mode Gaussian states of continuous variable systems. For pure states, we present a simplified procedure to obtain the maximum violation of the Svetlichny inequality based on displaced parity measurements, and we analyze its interplay with genuine tripartite entanglement measured via Rényi-2 entropy. The maximum Svetlichny violation admits tight upper and lower bounds at fixed tripartite entanglement. For mixed states, no violation is possible when the purity falls below 0.86. We also explore a set of recently derived weaker inequalities for three-way nonlocality, finding violations for all tested pure states. Our results provide a strong signature for the nonclassical and nonlocal nature of Gaussian states despite their positive Wigner function, and lead to precise recipes for its experimental verification.

Genuine multipartite nonlocality of permutationally invariant Gaussian states

“…As clear from the inset of the Figure, one finds that there exists, for any odd n, a threshold valueã n of a such thatp n (a) = 0 and S opt n (a) = 1 for 1 ≤ a ≤ã n , meaning that no genuine n-partite nonlocality can be detected below the threshold using displaced parity measurements, despite the fact that pure permutationally invariant Gaussian states are fully inseparable for any n as soon as a > 1 [37,40,44,45]. This was already noted in [36] in the case n = 3. The threshold valueã n to violate the Svetlichny inequality, as well as the optimal settingp n (a) to reach the largest violation provided a >ã n , can be determined analytically in principle by solving Eqs.…”

“…2(a). As the inset of the Figure shows, and as numerical calculations confirm, in the case of even n there is no threshold for the violation of the 1 Note that there was a typo in the expression corresponding top 3 (a) in [36], while Eq. (21) gives the correct formula.…”

“…, 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.…”

“…For displaced parity measurements, we extend the results of [36] from three to an arbitrary number of modes, providing a prescription to identify the phase space settings leading to the largest violations. However, we show that (as in the case of two and arXiv:1610.01928v2 [quant-ph] 18 Jan 2017 three modes) these violations do not reach the absolute maximum allowed by quantum mechanics, that is the multipartite analogue of the Tsirelson bound [1].…”

“…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].…”