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Analog of Formula of Total Probability for Quantum Observables Represented by Positive Operator Valued Measures

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Abstract

We represent Born’s rule as an analog of the formula of total probability (FTP): the classical formula is perturbed by an additive interference term. In this note we consider practically the most general case: generalized quantum observables given by positive operator valued measures and measurement feedback on states described by atomic instruments. This representation of Born’s rule clarifies the probabilistic structure of quantum mechanics (QM). The probabilistic counterpart of QM can be treated as the probability update machinery based on the special generalization of classical FTP. This is the essence of the Växjö interpretation of QM: statistical realist contextual and local interpretation. We analyze the origin of the additional interference term in quantum FTP by considering the contextual structure of the two slit experiment which was emphasized by R. Feynman.

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Notes

  1. 1 Nowadays realism is not fashionable in QM, especially local realism. Besides the Växjö interpretation, we can mention the interpretation due to Garola [14–16] (completed in cooperation with Sozzo [17, 18]) and known as the extended semantic realism (ESR). We stress that the latter is non-contextual.

  2. 2 See Bohr [4, 5]: “This crucial point, which was to become a main theme of the discussions reported in the following, implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled. Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.”

  3. 3 They are also both local and it seems that a consistent subjectivist has to handle probability in the contextual framework.

  4. 4 However, the problem “quantum versus classical probability” is even more complicated, cf. Nanasiova [35, 36] and Nanasiova and Valaskova [37]. If additional randomness, namely, of selection of experimental contexts, is taken into account then it is possible to construct the Kolmogorov representation for data collected for incompatible contexts, e.g., contexts C i ,i=0,1,C 01 in the two slit experiment [32], and even for incompatible contexts involved in Bell’s test - corresponding to selection of different pairs of orientations of polarization beam splitters [33]. Thus the terminology “non-Kolmogorovian probability has to be used with caution, cf., e.g., with its wide exploring in [1, 2, 28, 29].

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  1. International Center for Mathematical Modelling in Physics and Cognitive Sciences, Linnaeus University, Växjö, SE-351 95, Sweden

    Andrei Khrennikov

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Appendix: Växjö Viewpoint on Quantum Realism

Appendix: Växjö Viewpoint on Quantum Realism

Now we plan to discuss in more detail the Växjö viewpoint on quantum realism. We recall that the Växjö interpretation of QM, claiming that it is statistical, realist, contextual and local. At the same time this interpretation does not assume the existence of individual objects possessing individual properties, but only deals with statistical ensembles. One can ask: What kind of realism is the Växjö interpretation of QM the about?

As was pointed out in Section 5, Växjö realism is represented in the claim that it is in principle possible to construct a mathematical model describing causally subquantum processes and generating (via some correspondence rules) the observational quantum model described by the mathematical formalism of QM. This approach matches well with philosophy of the ontic-epistemic description of physical reality (Atmanspacher and Primas, see, e.g., [3]). The ontic description is about “reality as it is”, i.e., when nobody performs measurements. The epistemic description is about outputs of measurements, i.e., this is the observational description. The epistemic quantities cannot be treated as the objective properties. In this approach QM is considered as the epistemic (observational) model. In particular, this model is contextual, because its quantities are determined not only by features of physical systems, but also by the corresponding experimental contexts. Thus by the Växjö interpretation the QM-formalism is epistemic, but understanding of its observational and contextual nature does not exclude the possibility to create a causal ontic model. As was emphasized, this possibility is Växjö’s quantum realism.

Finally, as one of possible candidates for aforementioned ontic model, we can present the subquantum model of the random field type, so-called prequantum classical statistical field theory (PCSFT). Another candidate is the Bohmian model. However, in contrast to PCSFT, the Bohmian mechanics is nonlocal. We have already mentioned the Garola-Sozzo interpretation [14–18] of QM based on the extended semantic realism (ERS) model. This model is not only local, but even non-contextual.

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Khrennikov, A. Analog of Formula of Total Probability for Quantum Observables Represented by Positive Operator Valued Measures. Int J Theor Phys 55, 3859–3874 (2016). https://doi.org/10.1007/s10773-016-3015-x

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