Testing axioms for quantum theory on probabilistic toy-theories

Abstract

In D’Ariano in Philosophy of Quantum Information and Entanglement, Cambridge University Press, Cambridge, UK (2010), one of the authors proposed a set of operational postulates to be considered for axiomatizing Quantum Theory. The underlying idea is to derive Quantum Theory as the mathematical representation of a fair operational framework, i.e. a set of rules which allows the experimenter to make predictions on future events on the basis of suitable tests, e.g. without interference from uncontrollable sources and having local control and low experimental complexity. In addition to causality, two main postulates have been considered: PFAITH (existence of a pure preparationally faithful state), and FAITHE (existence of a faithful effect). These postulates have exhibited an unexpected theoretical power, excluding all known nonquantum probabilistic theories. In the same paper also postulate PURIFY-1 (purifiability of all states) has been introduced, which later has been reconsidered in the stronger version PURIFY-2 (purifiability of all states unique up to reversible channels on the purifying system) in Chiribella et al. (Reversible realization of physical processes in probabilistic theories, arXiv:0908.1583). There, it has been shown that Postulate PURIFY-2, along with causality and local discriminability, narrow the probabilistic theory to something very close to the quantum one. In the present paper we test the above postulates on some nonquantum probabilistic models. The first model—the two-box world—is an extension of the Popescu–Rohrlich model (Found Phys, 24:379, 1994), which achieves the greatest violation of the CHSH inequality compatible with the no-signaling principle. The second model—the two-clock world— is actually a full class of models, all having a disk as convex set of states for the local system. One of them corresponds to—the two-rebit world— namely qubits with real Hilbert space. The third model—the spin-factor—is a sort of n-dimensional generalization of the clock. Finally the last model is the classical probabilistic theory. We see how each model violates some of the proposed postulates, when and how teleportation can be achieved, and we analyze other interesting connections between these postulate violations, along with deep relations between the local and the non-local structures of the probabilistic theory.