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The Logos Categorical Approach to Quantum Mechanics: II. Quantum Superpositions and Intensive Values

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Abstract

In this paper we attempt to consider quantum superpositions from the perspective of the logos categorical approach presented in de Ronde and Massri (30) . We will argue that our approach allows us not only to better visualize the structural features of quantum superpositions providing an anschaulich content to all terms, but also to restore —through the intensive valuation of graphs and the notion of immanent power— an objective representation of what QM is really talking about. In particular, we will discuss how superpositions relate to some of the main features of the theory of quanta, namely, contextuality, paraconsistency, probability and measurement.

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Notes

  1. As we discussed in detail in [30], the classical representation amounts to an understanding of physical reality in terms of an actual state of affairs; more specifically, in terms of ‘systems’ constituted by definite valued ‘properties’.

  2. Given a quantum system represented by a superposition of more than one term, \( \sum {c}_i\mid {\alpha}_i\Big\rangle \), when in contact with an apparatus ready to measure, |R0〉, QM predicts that system and apparatus will become “entangled” in such a way that the final ‘system + apparatus’ will be described by \( \sum {c}_i\mid {\alpha}_i\left\rangle \mid {R}_i\right\rangle \). Thus, as a consequence of the quantum evolution, the pointers have also become —like the original quantum system— a superposition of pointers \( \sum {c}_i\mid {R}_i\Big\rangle \). This is why the measurement problem can be stated as a problem only in the case the original quantum state is described by a superposition of more than one term.

  3. For an insightful critique of the idea that quantum computation implies the existence of many worlds, see [51].

  4. According to Wallace [54], there are only three possible choices when attempting to address the interpretational problems of quantum theory. The first possibility is to give up on philosophy and accept instrumentalism, the second is to give up on physics and provide a new explanation of why superpositions are not being observed, the third and last possibility is to believe in the existence of many worlds.

  5. In fact, there are many different possibilities to consider. See for a related discussion and analysis [4].

  6. In fact, there are several well known interpretations which have attempted to produce a non-classical picture of what is going on according to QM. See, e.g., [3, 44, 52].

  7. Recalling the definition provided in [24], a Meaningful Operational Statement can be defined in the following manner: Every operational statement within a theory capable of predicting the outcomes of possible measurements must be considered as meaningful with respect to the representation of physical reality provided by that theory.

  8. We might remark that in the case of entangled states what is measured and compared are the outcomes pertaining to two different systems. While each one of these systems is, in general, in a superposition the measurement of entangled states does not try to account for each superposition. Instead, what is measured is the correlation between the superpositions.

  9. The semantic interpretation used in order to interpret the syntactical level of the quantum formalism presupposes implicitly the principles of existence, non-contradiction and identity. This “common sense” classical interpretation has been uncritically accepted without considering the required necessary coherency between the addressed semantical and syntactical levels of the theory.

  10. For a detailed exposition of how the measurement process is explained in terms of the notion of immanent cause see [23].

  11. In QM the commuting relation for incompatible observables, such as e.g. position and momentum, is given by [Xi,Xj] = 0, [Pi,Pj] = 0, \( \left[{X}_i,{P}_j\right]=\mathrm{i}\mathrm{\hslash }{\delta}_{ij}\mathbf{1} \).

  12. In general the dimension might be arbitrarily large. We choose a small dimension to make the picture more manageable.

  13. In the orthodox interpretation the choice of the basis has been interpreted as “the act of observing the quantum system”. This relation is not a direct one. Obviously, I can always choose a basis (on paper) without the need of performing any experiment (in the lab). Later on, the imposition of a binary valuation to the chosen basis is interpreted as something that “actually takes place in reality”. The inference derived by Bohr in his reply to EPR [12] is that the choice of the context determines the object under study. Today, in the words of Butterfiled [13], the widespread conclusion is that: “the properties of a system are different whether you look at them or not.” For a detailed analysis we refer to [27, 30].

  14. Of course, another possibility considered by many approaches to QM is to consider the notions of ‘system’, ‘state’ and ‘property’ as contextual ones (e.g. [2, 39, 42, 43]). As discussed in [29], this re-interpretation of the notions of ‘system’, ‘state’ and ‘property’ implies the abandonment of the invariant character of the notions themselves. Such contextual approaches abandon the attempt to provide a global non-contextual account of projection operators (interpreted as properties) and introduce —following Bohr’s famous reply to EPR [12]— the measurement set-up as a necessary condition in order to assign definite values to quantum properties in a contextual manner. See for a detailed analysis and discussion: [27].

  15. For an interesting analysis of the fruitfulness of diagrams with respect to the representation of mathematical abstract relations, see [15].

  16. This obviously does not imply the naive realist claim that —consequently— we are finally representing reality as it is. See for a discussion of this important point: [22].

  17. Wittgenstein’s assertion that “the limits of my language mean the limits of my world”, captures the fact we are always speaking from within a particular representation.

  18. This is of course in case we do not want to overpopulate reality with unobservable worlds in order to explain a ‘pointer reading’ we did not observe, or shift completely the focus to observability itself and end up discussing the problem of human consciousness.

  19. This is completely analogous to the interpretation of quantum possibility discussed in [31, 35].

  20. Let us remark that when considering the problem of representing quantum physical reality, the application of the Bayesian interpretation of subjective probability misses completely the point. An ontological question cannot be addressed from an epistemological perspective. QBism does use the Bayesian subjectivist interpretation of probability, but at the cost of denying the reference of QM to physical reality itself [37, 38].

  21. This result is just an expression of the conclusion derived by Schrödinger in [50, p. 153] regarding the notion of state in QM: “The classical notion of state becomes lost [in QM], in that at most half of a complete set of variables can be assigned definite numerical values”.

  22. The discussion about what exactly is this relation exceeds the scope of the present paper. We leave this discussion for a future work.

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Acknowledgements

We would like to thank an anonymous referee for her/his careful reading, comments and corrections. This work was partially supported by the following grants: FWO project G.0405.08 and FWO-research community W0.030.06. CONICET RES. 4541-12 and the Project PIO-CONICET-UNAJ (15520150100008CO) “Quantum Superpositions in Quantum Information Processing”.

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Authors and Affiliations

  1. Philosophy Institute Dr. A. Korn, University of Buenos Aires - CONICET, Buenos Aires, Argentina

    C. de Ronde

  2. Center Leo Apostel for Interdisciplinary Studies, Foundations of the Exact Sciences - Vrije Universiteit Brussel, Brussel, Belgium

    C. de Ronde

  3. Institute of Engineering, National University Arturo Jauretche, Buenos Aires, Argentina

    C. de Ronde

  4. Institute of Mathematical Investigations Luis A. Santaló, UBA - CONICET, Buenos Aires, Argentina

    C. Massri

  5. University CAECE, Buenos Aires, Argentina

    C. Massri

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de Ronde, C., Massri, C. The Logos Categorical Approach to Quantum Mechanics: II. Quantum Superpositions and Intensive Values. Int J Theor Phys 58, 1968–1988 (2019). https://doi.org/10.1007/s10773-019-04091-x

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