Abstract
According to a standard view, quantum mechanics (QM) is a contextual theory and quantum probability does not satisfy Kolmogorov’s axioms. We show, by considering the macroscopic contexts associated with measurement procedures and the microscopic contexts (μ-contexts) underlying them, that one can interpret quantum probability as epistemic, despite its non-Kolmogorovian structure. To attain this result we introduce a predicate language L(x), a classical probability measure on it and a family of classical probability measures on sets of μ-contexts, each element of the family corresponding to a (macroscopic) measurement procedure. By using only Kolmogorovian probability measures we can thus define mean conditional probabilities on the set of properties of any quantum system that admit an epistemic interpretation but are not bound to satisfy Kolmogorov’s axioms. The generalized probability measures associated with states in QM can then be seen as special cases of these mean probabilities, which explains how they can be non-classical and provides them with an epistemic interpretation. Moreover, the distinction between compatible and incompatible properties is explained in a natural way, and purely theoretical classical conditional probabilities coexist with empirically testable quantum conditional probabilities.
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Notes
We stress that our general framework does not constitute a hidden variables theory for QM in a standard sense. Indeed, μ-contexts are associated (generally many-to-one) with measurement procedures, not with properties or states of the measured entity. Our perspective complies instead with Aerts’ (1986) hidden measurements approach.
More generally, according to the standard epistemological conception, or received view (see, e.g., Braithwaite 1953; Hempel 1965; Carnap 1966), a fully-developed physical theory T, as QM, is in principle expressible by means of a metalanguage in which a theoretical language\(L_{T}\), an observational language \(L_{O}\) and correspondence rules\(R_{C}\) connecting \(L_{T}\) and \(L_{O}\) can be distinguished. The theoretical apparatus of T, expressed by means of \(L_{T}\), includes a mathematical structure and, usually, an intended interpretation that is a direct and complete physical model of the mathematical structure. The observational language \(L_{O}\) describes instead an empirical domain, hence it has a semantic interpretation, so that the correspondence rules \(R_{C}\) provide an empirical interpretation of the mathematical structure that is indirect and incomplete in the sense specified above.
The received view has been criticized by several authors (see, e.g. Kuhn 1962; Feyerabend 1975) and is nowadays maintained to be outdated by some scholars. However, we retain here some of its basic ideas that we consider epistemologically relevant.
We have emphasized in some previous papers (see, e.g., Garola 1999; Garola and Pykacz 2004; Garola and Sozzo 2010; Garola and Persano 2014) that the epistemological clause “the laws of QM have to be preserved in every conceivable physical situation” is essential in the proofs of Bell’s and Kochen–Specker’s theorems. Nevertheless, this clause generally is not explicitly noticed or stated, possibly because it seems to be unquestionably justified by the outstanding success of QM. Yet it must be observed that all the proofs mentioned above proceed ab absurdo, hypothesizing physical situations in which noncompatible physical properties are assumed to be simultaneously possessed by an individual object. In such situations the quantum laws that are applied cannot be simultaneously tested, hence the assumption that they hold anyway seems more consistent with a classical than with a quantum view. One can therefore try to give up the aforesaid clause, but then the proofs of Bell’s an Kochen–Specker’s theorems cannot be completed. This conclusion opens the way to the attempt at recovering noncontextual interpretations of QM (Garola et al. 2016). The arguments in this paper, however, apply to every theory in which contexts can be defined, irrespective of whether the results of measurements are context-depending (locally, or also at a distance) or not.
Following a standard terminology we call classical probability space here any triple \((\Omega ,\Sigma ,\mu )\), where \(\Omega\) is a set, \(\Sigma\) is a Boolean \(\sigma\)-subalgebra of \({\mathbb {P}} (\Omega )\), and \(\mu :\Delta \in \Sigma \longrightarrow \mu (\Delta )\in [0,1]\) is a mapping satisfying the following conditions: (i) \(\mu (\Omega )=1\); (ii) if \(\left\{ \Delta _{i}\right\} _{i\in {{N}}}\) is a family of pairwise disjoint elements of \(\Sigma\), then \(\mu (\cup _{i}\Delta _{i})=\Sigma _{i}\mu (\Delta _{i})\).
We recall that the Aerts and Sassoli de Bianchi proposal finds its roots in the hidden measurement approach (see, e.g., Aerts 1986). This approach led the author to introduce state property systems (see, e.g., Aerts 1999), that successively evolved in the state-context-property (SCoP) formalism (see, e.g., Aerts and Gabora 2005; this formalism was mainly used for working out a theory of concepts, in particular in the field of quantum cognition). It is then possible to show that the SCoP formalism can be (partially) translated into the formalism developed in the present paper, and conversely, which explains the conceptual similarities pointed out above. For the sake of brevity we do not deal with this issue in detail here.
We recall that \(^{\bot }\mathcal {\ }\)is a unary operation on \(({\mathcal {E}} ,\prec )\) such that, for every \(E,F\in \mathcal {E}\), \(E^{\bot \bot }=E\), \(E\prec F\) implies \(F^{\bot }\prec E^{\bot }\), \(E\Cap E^{\bot }=\mathsf {O}\) and \(E\Cup E^{\bot }=\mathsf {U}\). Then \(\bot\) is the non-reflexive and symmetric binary relation on \(\mathcal {E}\) defined by setting, for every \(E,F\in \mathcal {E}\), \(E\bot F\) iff \(E\prec F^{\bot }.\)
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Acknowledgements
We thank Dr. Karin Verelst for useful discussions at the Symposium “Worlds of Entanglement” in Brussels.
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Garola, C. An Epistemic Interpretation of Quantum Probability via Contextuality. Found Sci 25, 105–120 (2020). https://doi.org/10.1007/s10699-018-9560-4
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DOI: https://doi.org/10.1007/s10699-018-9560-4