Testing axioms for Quantum Mechanics on Probabilistic toy-theories

TL;DR

This paper evaluates several proposed axioms for quantum mechanics against nonquantum probabilistic models, revealing which models satisfy or violate these axioms and exploring their implications for nonlocality and teleportation.

Contribution

It systematically tests key quantum axioms on diverse probabilistic models, highlighting their limitations and connections to nonlocal phenomena.

Findings

The two-box world model violates some postulates, indicating limits of axioms.

The two-clock world models show varied compliance, illustrating the role of state space geometry.

Classical theory violates certain quantum postulates, emphasizing differences between classical and quantum structures.

Abstract

In Ref. [1] one of the authors proposed postulates for axiomatizing Quantum Mechanics as a "fair operational framework", namely regarding the theory as a set of rules that allow the experimenter to predict future events on the basis of suitable tests, having local control and low experimental complexity. In addition to causality, the following 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. Later in Ref. [2] in addition to causality and PFAITH, postulate LDISCR (local discriminability) and PURIFY (purifiability of all states) have been considered, narrowing the probabilistic theory to something very close to Quantum Mechanics. 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, 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 "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.