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.
Similar content being viewed by others
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 \(E\prec F^{\bot }.\)
References
Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics. Journal of Mathematical Physics, 27, 202–210.
Aerts, D. (1999). Foundations of quantum physics: A general realistic and operational approach. International Journal of Theoretical Physics, 38, 289–358.
Aerts, D., & Gabora, L. (2005). A state-context-property model of concepts and their combinations II: A Hilbert space representation. Kibernetes, 34, 176–204.
Aerts, D., & Sassoli de Bianchi, M. (2014). The extended Bloch representation of quantum mechanics and the hidden-measurement solution of the measurement problem. Annalen der Physik, 351, 975–1025.
Aerts, D., & Sassoli de Bianchi, M. (2017). Universal measurements how to free three birds in one move. Singapore: World Scientific.
Bell, J. S. (1964). On the Einstein–Podolski–Rosen paradox. Physics, 1, 195–200.
Bell, J. S. (1966). On the problem of hidden variables in quantum mechanics. Reviews of Modern Physics, 38, 447–452.
Beltrametti, E., & Cassinelli, G. (1981). The logic of quantum mechanics. Reading, MA: Addison-Wesley.
Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843.
Braithwaite, R. B. (1953). Scientific explanation. Cambridge: Cambridge University Press.
Busch, P., Lahti, P. J., & Mittelstaedt, P. (1996). The quantum theory of measurement. Berlin: Springer.
Carnap, R. (1966). Philosophical foundations of physics. New York: Basic Books Inc.
Dalla Chiara, M. L., Giuntini, R., & Greechie, R. (2004). Reasoning in quantum theory. Dordrecht: Kluwer.
Feyerabend, F. (1975). Against method: Outline of an anarchist theory of knowledge. London: New Left Books.
Garola, C. (1999). Semantic realism: A new philosophy for quantum physics. International Journal of Theoretical Physics, 38, 3241–3252.
Garola, C., & Persano, M. (2014). Embedding quantum mechanics into a broader noncontextual theory. Foundations of Science, 19, 217–239.
Garola, C., & Pykacz, J. (2004). Locality and measurement within the SR model for an objective interpretation of quantum mechanics. Foundations of Physics, 34, 449–475.
Garola, C., & Sozzo, S. (2010). Realistic aspects in the standard interpretation of quantum mechanics. Humana. mente Journal of Philosophical Studies, 13, 81–101.
Garola, C., Sozzo, S., & Wu, J. (2016). Outline of a generalization and a reinterpretation of quantum mechanics recovering objectivity. International Journal of Theoretical Physics, 55, 2500–2528.
Greenberger, D. M., Horne, M. A., Shimony, A., & Zeilinger, A. (1990). Bell’s theorem without inequalities. American Journal of Physics, 58, 1131–1143.
Hempel, C. C. (1965). Aspects of scientific explanation. New York: Free Press.
Kochen, S., & Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17, 59–87.
Khrennikov, A. (2009a). Contextual approach to quantum formalism. New York: Springer.
Khrennikov, A. (2009b). Interpretations of probability. Berlin: Walter de Gruyter.
Kuhn, T. S. (1962). The structure of scientific revolution. Chicago: Chicago University Press.
Ludwig, G. (1983). Foundations of quantum mechanics I. New York: Springer.
Mermin, N. D. (1993). Hidden variables and the two theorems of John Bell. Reviews of Modern Physics, 65, 803–815.
Piron, C. (1976). Foundations of quantum physics. Reading, MA: Benjamin.
Acknowledgements
We thank Dr. Karin Verelst for useful discussions at the Symposium “Worlds of Entanglement” in Brussels.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10699-018-9560-4