Closed sets of nonlocal correlations

We present a fundamental concept—closed sets of correlations—for studying nonlocal correlations. We argue that sets of correlations corresponding to information-theoretic principles, or more generally to consistent physical theories, must be closed under a natural set of operations. Hence, studying the closure of sets of correlations gives insight into which information-theoretic principles are genuinely different, and which are ultimately equivalent. This concept also has implications for understanding why quantum nonlocality is limited, and for finding constraints on physical theories beyond quantum mechanics.

Figure 1

Wiring several boxes together to produce a new box.

Figure 2

(Color online) Vector fields for (a) the distillation wiring, (b) the (2-box) AND wiring, in certain two-dimensional (2D) sections of the nonsignaling polytope. Clearly, correlated nonlocal boxes [Eq. (4)] are distilled in (a). The AND wiring (b) maps isotropic boxes [Eq. (6)] above their “chord” (red line); thus restricted polytopes in case study b) (see text) are not closed. This effect can only be seen in specific projections of the polytope.

Figure 3

(Color online) Theories with limited nonlocality: (a) limiting the CHSH value by a cutoff; (b) making the extremal PR boxes noisy. Neither set is closed since—as the arrows indicate—boxes which lie inside each set can be mapped to boxes which lie outside. Note that these pictures are illustrative; the actual polytopes are eight-dimensional.

Figure 4

(Color online) Comparison of distillation protocols in the 2D slice of the polytope defined by (A1). The different lines delimit the set of boxes that can be distilled in one (or more) iterations of a given protocol. The solid blue line corresponds to the distillation protocol presented in the main text; all boxes above this line can be distilled using this distillation protocol. The dash-dotted red line is for the protocol of Ref. [13]. The dotted orange line is for the two-box AND protocol. The green solid line corresponds to the protocol of Ref. [21]. The black dashed line is an upper bound on the set of quantum correlations, derived by Navascues-Pironio-Acin (NPA) [20].

Figure 5

(Color online) Number of boxes $N$ that can be generated outside the restricted polytope ${\mathcal{R}}_b^S$. Each new box is obtained by applying the AND wiring to $N$ copies of the isotropic box [Eq. (6)], characterized by $ϵ$.

Figure 6

(Color online) 2D section of the nonsignaling polytope. The solid red line is the boundary of Uffink’s set, and the dashed black line is the upper bound on the quantum set derived by NPA [20]. The shaded region are boxes, initially inside Uffink’s set, that are mapped outside using the distillation protocol; more specifically, in one iteration of the protocol (blue region), two iterations (green region), and three iterations (red region). The lower boundary of the blue region (one iteration) is given by Eq. (B5).