Non-signalling Theories and Generalized Probability

Abstract

We provide mathematicaly rigorous justification of using term probability in connection to the so called non-signalling theories, known also as Popescu's and Rohrlich's box worlds. No only do we prove correctness of these models (in the sense that they describe composite system of two independent subsystems) but we obtain new properties of non-signalling boxes and expose new tools for further investigation. Moreover, it allows strightforward generalization to more complicated systems.

Tensor product of no-signaling boxes in the framework of quantum logics

“…Our previous works [21,22] showed that the following class of quantum logics is particularly important for no-signaling box models: Definition 5 (see [23], section 1.1). Let ∆ be a family of subsets of some set Ω with partial order relation given by set inclusion and A A c \ = Ω satisfying: [27]).…”

“…Let us briefly recall the structure of the logic of the arbitrary two box system; see [21,22] for detailed discussion and proof that the construction below indeed yields the logic of the arbitrary no-signaling box model.…”

“…, -box world is the quantum logic generated in the Boolean algebra 2 Γ of all subsets of Γ by all questions of the above form (see [21,22] for details). Of course, L does not have to be (and is not) a Boolean algebra anymore.…”

“…One is based on the convex set approach and is usually called a generalized or generic probability theory (GPT) (see [20] and references therein). The second approach is due to us [21,22] and describes no-signaling models in the framework of quant um logics (in the sense of [23], i.e. despite the name, it describes more general models than quantum mechanical systems).…”

Ignorance is a bliss: Mathematical structure of many-box models

“…This paper is a continuation of our program of filling this gap and trying to understand properties of the box probability theory. Previously, we characterized the mathematical structure of 2-box model with a binary input and output 17 , followed by a general characterization of arbitrary 2-box models 18 . Finally, in Ref.…”