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arXiv:1511.01570 · Electronic Proceedings in Theoretical Computer Science

Quotient-Comprehension Chains

Kenta Cho, Bart Jacobs, Bas Westerbaan, Bram Westerbaan

Published Nov 4, 2015• 15 citations
arXivHTMLPDF

Abstract

Quotients and comprehension are fundamental mathematical constructions that can be described via adjunctions in categorical logic. This paper reveals that quotients and comprehension are related to measurement, not only in quantum logic, but also in probabilistic and classical logic. This relation is presented by a long series of examples, some of them easy, and some also highly non-trivial (esp. for von Neumann algebras). We have not yet identified a unifying theory. Nevertheless, the paper contributes towards such a theory by introducing the new quotient-and-comprehension perspective on measurement instruments, and by describing the examples on which such a theory should be built.

DOI
10.4204/eptcs.195.10

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Quotients and comprehension are fundamental mathematical constructions that can be described via adjunctions in categorical logic. This paper reveals that quotients and comprehension are related to measurement, not only in quantum logic, but also in probabilistic and classical logic. This relation is presented by a long series of examples, some of them easy, and some also highly non-trivial (esp. for von Neumann algebras). We have not yet identified a unifying theory. Nevertheless, the paper contributes towards such a theory by introducing the new quotient-and-comprehension perspective on measurement instruments, and by describing the examples on which such a theory should be built.
This paper proposes a comprehensive categorical framework for unifying classical and quantum probabilistic logics. We leverage the inherent compositional and structural advantages of category theory to construct a common mathematical language capable of expressing both Kolmogorovian probability and the non-Boolean structures characteristic of quantum mechanics. By treating systems and processes as objects and morphisms within suitable monoidal categories, we demonstrate how core concepts like states, observables, measurements, and conditioning can be abstractly represented and seamlessly transitioned between classical and quantum domains. The framework utilizes probability monads for classical uncertainty and dagger compact closed categories for quantum processes, showing how these distinct structures can be seen as instances of a more general categorical paradigm. This unification not only provides a deeper foundational understanding of uncertainty across different physical theories but also opens new avenues for developing hybrid probabilistic-logical systems in fields such as artificial intelligence and quantum computation.
We introduce a foundational sheaf theoretical scheme for the comprehension of quantum event structures, in terms of localization systems consisting of Boolean coordinatization coverings induced by measurement. The scheme is based on the existence of a categorical adjunction between presheaves of Boolean event algebras and Quantum event algebras. On the basis of this adjoint correspondence we prove the existence of an object of truth values in the category of quantum logics, characterized as subobject classifier. This classifying object plays the equivalent role that the two-valued Boolean truth values object plays in classical event structures. We construct the object of quantum truth values explicitly and argue that it constitutes the appropriate choice for the valuation of propositions describing the behavior of quantum systems.
Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of Boole-Bell type consistency constraints on joint probabilities, as well as the Kochen-Specker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of two-valued states interpretable as classical truth assignments. A more complete introduction of the author can be found in the book "Quantum Logic" (Springer, 1998)
Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (non-empty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality–which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are ‘lifted’ to complex vector spaces, then the mathematical framework of quantum mechanics (QM) is obtained. Partition logic models the indefiniteness of QM while subset logic models the definiteness of classical physics. Hence partition logic may provide the backstory so the old idea of ‘objective indefiniteness’ in QM can be fleshed out to a full interpretation of quantum mechanics. In that case, QM will be the ‘killer application’ of partition logic.
Summary Foundational studies in quantum theory have consisted in large part of attempts to avoid paradoxical situations in describing properties of a physical system. Quantum logic addresses some of the perplexing problems that plague interpretations of quantum theory by invoking a non-classical propositional calculus for the set of propositions of a quantum system. Historically, quantum logic derives from von Neumann’s observation that the set of projection operators on a Hilbert space constitute a ‘logic’ of experimental propositions. More than two decades passed from the pioneering work of Birkho and von Neumann before interest in quantum logic was re-ignited by Mackey’s probabilistic analysis of quantum theory. Piron’s axiomatization provided a signicant
The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors [E,Sets]\mathsf{[E,Sets]}[E,Sets] where E\mathsf{E}E is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called quantosquantosquantos – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative C∗C^*C∗-algebras, which in turn are studied in their quantum habitat of noncommutative C∗C^*C∗-algebras – more fundamental structures than commutative C∗C^*C∗-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors [E,QSets]\mathsf{[E,QSets]}[E,QSets] is constructed where QSets\mathsf{QSets}QSets is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.