Non-signaling boxes and quantum logics

We analyze the structure of the so called non-signaling theories respecting relativistic causality but allowing correlations violating bounds imposed by quantum mechanics such as CHSH inequality. We discuss relations among such theories, quantum mechanics, and classical physics. In particular we reconstruct the probability theory adequate for the simplest instance of a non-signaling theory, the two non-signaling boxes world, and exhibit its differences in comparison with classical and quantum probabilities. We show that the question whether such a theory can be treated as a kind of "generalization" of the quantum theory of the two-qubit system cannot be answered positively. Some of its features put it closer to the quantum world, for example measurements must be destructive, on the other hand the Heisenberg uncertainty relations are not satisfied. Another interesting property contrasting it from quantum mechanics is that the subset of "classically correlated states," i.e.\ the states with only classical correlations, does not reproduce the classical world of two two-state systems. Our results establish a new link between quantum information theory and the well-developed theory of quantum logics and can shed new light on the problem why quantum mechanics is distinguished among non-signaling theories.