The preparation procedure, an undefined notion in quantum theory, has not had the relevance that it deserves in the interpretation of quantum mechanical formalism. Here we utilize the concepts of identical and similar preparation procedures to show the conceptual differences and mutual interconnections between the statistical and the conventional interpretation of quantum formalism. Although the statistical understanding, and its final logical consequence, hidden variables theories (this connexion being explained in the text), have a great intuitive appeal due to its fewer ontological difficulties, both recent experimental results and some theoretical developments seem to support an epistemic alternative closer to the conventional one. Nevertheless, we must not rule out the possibility that new theorems or new explanatory principles may modify the reigning supremacy of this interpretation.
The preparation procedure, an undefined notion in quantum theory, has not had the relevance that it deserves in the interpretation of quantum mechanical formalism. Here we utilize the concepts of identical and similar preparation procedures to show the conceptual differences and mutual interconnections between the statistical and the conventional interpretation of quantum formalism. Although the statistical understanding, and its final logical consequence, hidden variables theories (this connexion being explained in the text), have a great intuitive appeal due to its fewer ontological difficulties, both recent experimental results and some theoretical developments seem to support an epistemic alternative closer to the conventional one. Nevertheless, we must not rule out the possibility that new theorems or new explanatory principles may modify the reigning supremacy of this interpretation.
The preparation procedure, an undefined notion in quantum theory, has not had the relevance that it deserves in the interpretation of quantum mechanical formalism.Here we utilize the concepts of identical and similar preparation procedures to show the conceptual differences between the statistical and the conventional interpretation of quantum formalism.Although the statistical understanding, and its final logical consequence, hidden variables theories (this connexion being explained in the text), have a great intuitive appeal due to its fewer ontological difficulties, both recent experimental results and theoretical developments seem to support an epistemic alternative closer to the conventional one.Nevertheless, we must not rule out the possibility that new theorems or new explanatory principles may modify the reigning supremacy of this interpretation. 1.-Introduction.Although the conventional interpretation of the quantum mechanical formalism (CIQM), very close to the Copenhagen interpretation, is still dominant (is used in almost all textbooks), there are physical situations that ask for an interpretation of the quantum mechanical formalism (QMF) in terms of the Statistical Interpretation of Quantum Mechanics (SIQM) or even for hidden variables theories (HVT).The SIQM and HVT are different research programs, but they are closely related, as we will see further below.To introduce properly the distinctions between the three previous acronyms we will ask an old and apparently simple question.Is the wave function associated to an individual system or to a statistical ensemble?A first tentative answer could be formulated as follows.QMF introduces probabilities, and probabilities are related to statistical ensembles.Otherwise, the same concept of probability would lose its meaning.In QM, if it is said that you would get some concrete result with 0,63 probability it means that if the experiment is carried out 100 times, that concrete result would appear approximately 63 times.Hence, the obvious answer will be that the wave function should be related with statistical ensembles.However, despite its intuitiveness this idea does not capture completely what we want to express.The real problem to analyze could be best introduced with the following example.Imagine that we prepare identically two radioactive atoms with half-life of 30 minutes.By preparation we understand all those theoretical methods and physical procedures necessary to fully characterize the state of the system.It is, thus, a macroscopic reproducible act that lefts the system, without destroying it, in a concrete state after the procedure.In quantum theory, this preparation is represented by a vector state (or a density matrix).Identically in this context will mean identity of procedures, that is that the experimenter has repeated all previous operations, so that there is no difference whatsoever between them.Note, however, that identical preparations are not necessarily followed by identical results, so there is
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.
Previously suggested hidden time interpretation of quantum mechanics allows to reproduce the same predictions as standard quantum mechanics provides, since it is based on Feynman many - paths formulation of QM. While new experimental consequences of this interpretation are under investigation, some advantages can be enumerated. (1) The interpretation is much field theoretic - like in classical sense, so it is local in mathematical sense, though quantum (physical) non-locality is preserved. (2) The interpretation is based on one type of mathematical objects, rather than two different (Hilbert space vectors and operators). (3) The interpretation, as it was argued, overcomes the problem of hidden variables in a radically new way, with no conflict to Bell's theorem. Recently an important argument against hidden variables - like formulations of quantum theory was risen - "no protocol" argument. It is argued in the paper, that hidden time interpretation successfully overcomes this argument.
The main question under investigation in this thesis was the reality of the quantum state. We approached this problem by studying a recent no-go result by Pusey, Barrett and Rudolph, namely the PBR theorem. In Section 1 we provided a short introduction to quantum theory. A few relevant examples were covered, namely decompositions of quantum states, impossibility of the ignorance interpretation and the distinguishability of pure quantum states. Section 2 developed measure theoretic tools needed in the framework of ontological models. The Radon-Nikodym theorem was proven in order to justify the use of an alternative expression of the variational distance of probability measures. In Section 3 we developed a general framework for ontological models. In short, the ontological models framework can be considered the most general framework currently available for hidden variable theories and no-go results. We used the measure theoretic tools of Section 2 to make the formalism rigorous. Original research included in Section 3 was the characterization of a property called antidistinguishability for qubit states. A necessary and sufficient condition for antidistinguishability was given. It was then proven that any set of pure qubit states that is not antidistinguishable can be made antidistinguishable by adding a single carefully chosen state into the set. In the end of Section 3, some original observations were presented. Namely, a specific connection with contextuality was proven for sometimes ψ-ontic and never ψ-ontic ontological models. However, it was then noted that these observations were not enough to exclude never ψ-ontic ontological models, a feat that would have provided a new no-go theorem on its own. Contextuality in the ontological models framework was studied in Section 4. The main results were establishing preparation contextuality from the failure of maximal ψ-epistemicity and a proof of Bell's theorem based on preparation contextuality established this way. We concluded that ψ-ontic models cannot be Bell local based on these results. Finally in Section 5 the implications of ψ-ontology were discussed. By careful analysis of the results, it was noted that accepting the reality of the quantum state exposes explanatory gaps in the formalism. Possible solutions to the challenges introduced by these explanatory gaps were discussed through interpretations of quantum mechanics and the so-called measurement problem.
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.
The discussion of the foundations of quantum mechanics is complicated by the fact that a number of different issues are closely entangled. Three of these issues are i) the interpretation of probability, ii) the choice between realist and empiricist interpretations of the mathematical formalism of quantum mechanics, iii) the distinction between measurement and preparation. It will be demonstrated that an interpretation of violation of Bell's inequality by quantum mechanics as evidence of non-locality of the quantum world is a consequence of a particular choice between these alternatives. Also a distinction must be drawn between two forms of realism, viz. a) realist interpretations of quantum mechanics, b) the possibility of hidden-variables (sub-quantum) theories.