Deriving Three-Outcome Permutationally Invariant Bell Inequalities

Abstract

We present strategies to derive Bell inequalities valid for systems composed of many three-level parties. This scenario is formalized by a Bell experiment with N observers, each of which performs one out of two possible three-outcome measurements on their share of the system. As the complexity of the set of classical correlations prohibits its full characterization in this multipartite scenario, we consider its projection to a lower-dimensional subspace spanned by permutationally invariant one- and two-body observables. This simplification allows us to formulate two complementary methods for detecting nonlocality in multipartite three-level systems, both having a complexity independent of N. Our work can have interesting applications in the detection of Bell correlations in paradigmatic spin-1 models, as well as in experiments with solid-state systems or atomic ensembles.

Keywords: Bell correlations; Bell inequalities; Bell nonlocality; multipartite quantum correlations.

Figure 1

Schematic illustration of the $(N,2,3)$ Bell scenario. Each party ${\mathcal{A}}_i$ performs measurement $x_i$ and observes outcome $a_i$. After many repetitions, one estimates the conditional probability $p(a_1\dots a_N|x_1\dots x_N)$ to check if it is compatible with a local hidden variable description. If this is not the case, Bell nonlocality is revealed.

Figure 2

Illustration of the local polytope $\mathbb{P}$ and of its projection ${\mathbb{P}}^S$ to a lower-dimensional subspace. The red dashed line indicates a resulting Bell inequality in the projected space.

Figure 3

Slices of the local polytope ${\mathbb{P}}^S$ (red circles and black dashed line), compared to a slice of its outer approximation provided by the SdP (26) (blue asterisks and blue dotted line), for different number of parties N. These slices are taken on a plane defined by two orthonormal directions ${\mathit{v}}_1,{\mathit{v}}_2$ randomly chosen in the 14-dimensional space spanned by Equation (11). One observes that the outer approximation becomes relatively tighter as N increases [63].