Abstract
Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to . In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.
Highlights
Probability enters quantum mechanics (QM) via Born’s rule and it is usually interpreted in terms of frequencies of the outcomes that are obtained when measurements are performed
We propose a more general view, singling out a class of theories, including classical mechanics (CM), statistical mechanics (SM), and QM, in which non-Kolmogorovian probability measures are introduced as derived notions in a Kolmogorovian framework, taking into account contextuality, but making no reference to individual objects
The conditions characterizing TμMP are compatible with CM, SM, and QM, which can be maintained to belong to this class of theories
Summary
Probability enters quantum mechanics (QM) via Born’s rule and it is usually interpreted in terms of frequencies of the outcomes that are obtained when measurements are performed. In the case of QM, this means that our results hold, even if one just assumes that the values of the probability measures that are associated with quantum states are the large numbers limits of relative frequencies of measurement outcomes (the minimal interpretation of QM) They are compatible both with a statistical interpretation, according to which no reference to individual systems should be made, and with a “realistic” interpretation, according to which QM deals with individual objects and their properties (see, e.g., [1]). They are interpreted as derived notions within a Kolmogorovian framework and their non-classical character can be explained in classical terms. Approach to QM, hypothesizing a correspondence between the “localizations” that occur in their description of the measurement process and the “hidden measurements” introduced by Aerts [16,19], but they do not go into the details of this correspondence
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