Deriving Born’s Rule from an Inference to the Best Explanation

Abstract

In previous articles we presented a simple set of axioms named "Contexts, Systems and Modalities" (CSM), where the structure of quantum mechanics appears as a result of the interplay between the quantized number of modalities accessible to a quantum system, and the continuum of contexts that are required to define these modalities. In the present article we discuss further how to obtain (or rather infer) Born's rule within this framework. Our approach is compared with other former and recent derivations, and its strong links with Gleason's theorem are particularly emphasized.

Contextual Inferences, Nonlocality, and the Incompleteness of Quantum Mechanics

“…It does, because it indicates a set of contexts, corresponding to all the observables, including as an eigenvector, where the associated measurement result (eigenvalue) is predictable with certainty. In recent papers [ 23 , 24 , 25 , 26 , 27 ], we introduced a framework that makes a careful distinction between the usual without a context and the physical state within a context—called a modality (see also Appendix A ).…”

“…In order to admit the idea of contextual inference, an intellectual quantum jump is required to accept that, in quantum mechanics, one has to take into consideration both the systems and the contexts in which they evolve. A simple way not to forget this requirement is to postulate that the “object” carrying well-defined properties is a composite: a (quantum) system within a (classical) context [ 17 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 31 ].…”

“…In other words, refers to a property of Bob’s particle, not in and by itself, but within a context defined by Alice’s result. In the CSM language [ 22 , 23 , 24 , 25 , 26 , 27 , 28 ], defines a modality for Bob’s particle, in Alice’s context. The correspondance between the respective modalities and can only be probabilistic, with probability 1 if [ 24 ].…”

“…Arguments based on the CSM formalism [ 31 ]. QM is a fundamentally probabilistic theory: this is a consequence of contextual quantization [ 23 , 27 ]. Nevertheless, QM allows measurements results to be predicted with certainty, either by repeating them in the same measurement context (this defines a modality), or by observing fully connected results betwen different contexts (this defines an equivalence classe of modalities, called an extravalence class).…”

Quantum Violation of the Suppes-Zanotti Inequalities and “Contextuality”

“…Let me add a cautionary remark on the widely held opinion that violations of classical Boolean criteria such as the Suppes-Zanotti-Brodi inequalities suggest or even imply "contextuality". Presently the term "contextual" [42][43][44][45][46][47][48][49][50][51][52] is often heuristically used as "violation of some inequality that is derived by assuming classical probability distributions" [53,54]. There are a variety of notions [55] and accompanying measures [45,[56][57][58] for the term "contextuality".…”