Probing the Meaning of Quantum Mechanics

The following sections are included:

• EDITORIAL PREFACE
• ACKNOWLEDGMENTS
• CONTENTS

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen-Specker's theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event algebras. We show explicitly that the latter category is equipped with an object of truth values, or classifying object, which constitutes the appropriate tool for assigning truth values to propositions describing the behavior of quantum systems. Effectively, this category-theoretic representation scheme circumvents consistently the semantic ambiguity with respect to truth valuation that is inherent in conventional quantum mechanics by inducing an objective contextual account of truth in the quantum domain of discourse. The philosophical implications of the resulting account are analyzed. We argue that it subscribes neither to a pragmatic instrumental nor to a relative notion of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature. In this light, the transcendence condition of the usual conception of correspondence truth is superseded by a reflective-like transcendental reasoning of the proposed account of truth that is suitable to the quantum domain of discourse.

In this paper we consider the phenomenon of superpositions in quantum mechanics and suggest a way to deal with the idea in a logical setting from a syntactical point of view, that is, as subsumed in the language of the formalism, and not semantically. We restrict the discussion to the propositional level only. Then, after presenting the motivations and a possible world semantics, the formalism is outlined and we also consider within this scheme the claim that superpositions may involve contradictions, as in the case of the Schrödinger's cat, which (it is usually said) is both alive and dead. We argue that this claim is a misreading of the quantum case. Finally, we sketch a new form of quantum logic that involves three kinds of negations and present the relationships among them. The paper is a first approach to the subject, introducing some main guidelines to be developed by a ‘syntactical’ logical approach to quantum superpositions.

The study of the foundations of Quantum Mechanics, especially after the advent of Quantum Computation and Information, has benefited from the application of category-theoretic tools and modal logics to the analysis of Quantum processes: we witness a wealth of theoretical frameworks casted in either of the two languages. This paper explores the interplay of the two formalisms in the peculiar context of Quantum Theory.

After a review of some influential abstract frameworks, we show how different modal logic frames can be extracted from the category of finite dimensional Hilbert spaces, connecting the Categorical Quantum Mechanics approach to some modal logics that have been proposed for Quantum Computing. We then apply a general version of the same technique to two other categorical frameworks, the ‘topos approach’ of Doering and Isham and the sheaf-theoretic work on contextuality by Abramsky and Brandenburger, suggesting how some key features can be expressed with modal languages.

The standard theory of quantum computation relies on the idea that the qubit — the basic information quantity — is represented by a superposition of elements of the standard quantum computational basis B⁽²⁾ = {|0〉, |1〉}. In the present paper we focus on the case of qutrits where the standard quantum computational bases is replaced by a three-valued computational basis . Recently in [13], unary gates on the Hilbert space ℂ³ were considered. In this work we propose an extensive method that allows to extend binary gates to the framework of qutrits.

We present a very general geometrico-dynamical description of physical or more abstract entities, called the general tension-reduction (GTR) model, where not only states, but also measurement-interactions can be represented, and the associated outcome probabilities calculated. Underlying the model is the hypothesis that indeterminism manifests as a consequence of unavoidable uctuations in the experimental context, in accordance with the hidden-measurements interpretation of quantum mechanics. When the structure of the state space is Hilbertian, and measurements are of the universal kind, i.e., are the result of an average over all possible ways of selecting an outcome, the GTR-model provides the same predictions of the Born rule, and therefore provides a natural completed version of quantum mechanics. However, when the structure of the state space is non-Hilbertian and/or not all possible ways of selecting an outcome are available to be actualized, the predictions of the model generally differ from the quantum ones, especially when sequential measurements are considered. Some paradigmatic examples will be discussed, taken from physics and human cognition. Particular attention will be given to some known psychological effects, like question order effects and response replicability, which we show are able to generate non-Hilbertian statistics. We also suggest a realistic interpretation of the GTR-model, when applied to human cognition and decision, which we think could become the generally adopted interpretative framework in quantum cognition research.

In classical physics, probabilistic or statistical knowledge has been always related to ignorance or inaccurate subjective knowledge about an actual state of affairs. This idea has been extended to quantum mechanics through a completely incoherent interpretation of the Fermi-Dirac and Bose-Einstein statistics in terms of “strange” quantum particles. This interpretation, naturalized through a widespread “way of speaking” in the physics community, contradicts Born's physical account of Ψ as a “probability wave” which provides statistical information about outcomes that, in fact, cannot be interpreted in terms of ‘ignorance about an actual state of affairs’. In the present paper we discuss how the metaphysics of actuality has played an essential role in limiting the possibilities of understating things differently. We propose instead a metaphysical scheme in terms of immanent powers with definite potentia which allows us to consider quantum probability in a new light, namely, as providing objective knowledge about a potential state of affairs.

It has been taken for granted for a long time that orthomodular lattices are the “algebraic counterpart” of orthomodular quantum logic. Pavičic and Megill have questioned this claim by pointing out that orthomodular quantum logic is sound and complete with respect to a proper supervariety of the variety OML of orthomodular lattices (the so-called weakly orthomodular lattices). The same authors conclude that “in the syntactical structure of quantum logic there is nothing orthomodular”. After reviewing in a certain detail some concepts from Abstract Algebraic Logic, especially Blok and Pigozzi's theory of algebraisable logics, we argue that the weakly orthomodular semantics introduced by Pavičic and Megill is not a good semantics, and that the role of OML as an algebraic counterpart of quantum logic is unaffected by their allegations.

The following sections are included:

• Introduction
• Partial Structures and Quasi-Truth
• Partial Isomorphism and Partial Homomorphism
• Von Neumann, Logic and the Application of Mathematics
• Conclusion
• Acknowledgements
• Bibliography

In the libertarian “agent causation” view of free will, free choices are attributable only to the choosing agent, as opposed to a specific cause or causes outside the agent. An often-repeated claim in the philosophical literature on free will is that agent causation necessarily implies lawlessness, and is therefore “antiscientific.” That claim is critiqued and it is argued, on the contrary, that the volitional powers of a free agent need not be viewed as anomic, specifically with regard to the quantum statistical law (the Born Rule). Assumptions about the role and nature of causation, taken as bearing on volitional agency, are examined and found inadequate to the task. Finally, it is suggested that quantum theory may constitute precisely the sort of theory required for a nomic grounding of libertarian free will.

In the framework of quantum computation with mixed states, a fuzzy representation of CNOT gate is introduced. In this representation, the incidence of non-factorizability is specially investigated.

This paper is a continuation of the authors' attempts to deal with the notion of indistinguishability (or indiscernibility) from a logical point of view. Now we introduce a two-sorted first-order modal logic to enable us to deal with objects of two different species. The intended interpretation is that objects of one of the species obey the rules of standard S5, while the objects of the other species obey only the rules of a weaker notion of indiscernibility. Quantum mechanics motivates the development. The basic idea is that in the ‘actual’ world things may be indiscernible but in another accessible world they may be distinguished in some way. That is, indistinguishability needs not be seen as a necessary relation. Contrariwise, things might be distinguished in the ‘actual’ world, but they may be indiscernible in another world. So, while two quantum systems may be entangled in the actual world, in some accessible world, due to a measurement, they can be discerned, and on the other hand, two initially separated quantum systems may enter in a state of superposition, losing their individualities. Two semantics are sketched for our system. The first is constructed within a standard set theory (the ZFC system is assumed at the metamathematics). The second one is constructed within the theory of quasi-sets, which we believe suits better the purposes of our logic and the mathematical treatment of certain situations in quantum mechanics. Some further philosophically related topics are considered.

According to de Ronde it was Bohr's interpretation of Quantum Mechanics (QM) which closed the possibility of understanding physical reality beyond the realm of the actual, so establishing the Orthodox Line of Research. In this sense, it is not the task of any physical theory to look beyond the language and metaphysics supposed by classical physics, in order to account for what QM describes. If one wishes to maintain a realist position (though not nave) regarding physical theories, one seems then to be trapped by an array of concepts that do not allow to understand the main principles involved in the most successful physical theory thus far, mainly: the quantum postulate, the principle of indetermination and the superposition principle. If de Ronde is right in proposing QM can only be completed as a physical theory by the introduction of `new concepts' that admit as real a domain beyond actuality, then a new ontology that goes beyond Aristotelian and Newtonian actualism is needed. It was already in the early 20th century that misunderstood philosopher Alexius von Meinong proposed a Theory of Objects that admits a domain of being beyond existence-actuality. Member of the so called`School of Brentano', Meinong's concerns were oriented to provide an ontology of everything that can be thought of, and at the same time an intentionality theory of how objects are thought of. I wish to argue that in Meinong's theory of objects we find the rudiments of the ontology and the intentionality theory we need to account for QM's basic principles: mainly the possibility of predicating properties of non-entities, or in other words, the possibility of objectively describing a domain of what is, that is different from the domain of actual existence.

To what extent is a logical characterization of entanglement possible? We investigate some correlations that hold between the concept of entanglement of formation for Werner states and for isotropic states and the probabilistic behavior of some quantum logical gates.

I propose an understanding of Everett and Wheeler's relative-state interpretation of quantum mechanics, which restores the feature of indeterminism to the theory. This incorporates a theory of probability as truth values in a many-valued logic for future statements, and a contextual theory of truth which gives objective and subjective perspectives equal validity.

In this paper, we show how a particular form of paraconsistent logic arises naturally from a topos-theoretic formulation of quantum theory. We argue that paraconsistent approaches have the potential to shed new light on some conceptual problems in the interpretation of quantum theory, and suggest some potential directions for related mathematical research.

Entangled states are in conflict with a general physical principle which expresses that a composite entity exists if and only if its components also exist, and the hypothesis that pure states represent the actuality of a physical entity, i.e., its ‘existence’. A possible way to solve this paradox consists in completing the standard formulation of quantum mechanics, by adding more pure states. We show that this can be done, in a consistent way, by using the extended Bloch representation of quantum mechanics, recently introduced to provide a possible solution to the measurement problem. Hence, with the solution proposed by the extended Bloch representation of quantum mechanics, the situation of entangled states regains full intelligibility.