Nonsignaling

Nonsignaling, in the context of quantum information theory and related disciplines, refers to the no-signaling principle. It states that the outcome probabilities for one party's measurements cannot be used to infer or influence the choices of measurements made by a distant party. In probabilistic terms, for a joint distribution P(a,b|x,y) describing outputs a and b given inputs x and y, the marginal distribution of a is independent of y, and the marginal distribution of b is independent of x.

Formally, a bipartite nonsignaling distribution satisfies: sum_b P(a,b|x,y) = P(a|x) for all a,x,y, and sum_a P(a,b|x,y) = P(b|y) for all b,x,y. This ensures that no communication or signaling is possible through measurement choices, regardless of the underlying physical theory. The concept generalizes to multiple parties and various outcome sets.

Nonsignaling encompasses all correlations that respect the prohibition on instantaneous signaling. Quantum mechanics yields correlations that are nonsignaling, but not all nonsignaling correlations are quantum. A famous example is the Popescu–Rohrlich (PR) box, a hypothetical nonsignaling resource that achieves maximal Bell inequality violation beyond quantum limits, yet cannot be realized within quantum theory.

Historically, the idea arose to explore why quantum nonlocality is limited despite no-signaling allowing stronger correlations. It has become a tool for studying the geometry of possible correlations, the separation between classical, quantum, and nonsignaling theories, and for device-independent protocols in quantum information, such as randomness certification and cryptography.

In practice, the set of nonsignaling correlations forms a convex polytope whose extremal points include both deterministic strategies and PR-type boxes, illustrating a clear distinction from the quantum set.