On Quantum Event Structures Part I: The Categorical Scheme

In this paper a mathematical scheme for the analysis of quantum event structures is being proposed based on category theoretical methods. It is shown that there exists an adjunctive correspondence between Boolean presheaves of event algebras and quantum event algebras. The adjunction permits a characterization of quantum event structures as Boolean manifolds of event structures.     

Quantum Event Structures from the Perspective of Grothendieck Topoi

We develop a categorical scheme of interpretation of quantum event structures from the viewpoint of Grothendieck topoi. The construction is based on the existence of an adjunctive correspondence between Boolean presheaves of event algebras and Quantum event algebras, which we construct explicitly. We show that the established adjunction can be transformed to a categorical equivalence if the base category of Boolean event algebras, defining variation, is endowed with a suitable Grothendieck topology of covering systems. The scheme leads to a sheaf theoretical representation of Quantum structure in terms of variation taking place over epimorphic families of Boolean reference frames.     

On Quantum Event Structures. Part III: Object of Truth Values

In this work we expand the foundational perspective of category theory on quantum event structures by showing the existence of an object of truth values in the category of quantum event algebras, characterized as subobject classifier. This object plays the corresponking role that the two-valued Boolean truth values object plays in a classical event structure. 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.     

A Categorial Semantic Representation of Quantum Event Structures

The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved syntactic language to tackle these problems from the set-theoretic to the category-theoretic mode, together with a study of the consequent semantic transition in the logical interpretation of quantum event structures. In the present work, this is realized by representing categorically the global structure of a quantum algebra of events (or propositions) in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of topos theory with reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a revised realist account of truth in the quantum domain of discourse. We also include an Appendix, where we compare our topos-theoretic representation scheme of quantum event algebras with other categorial and topos-theoretic approaches.     

Interpreting Observables in a Quantum World from the Categorial Standpoint

We develop a relativistic perspective on structures of quantum observables, in terms of localization systems of Boolean coordinatizing charts. This perspective implies that the quantum world is comprehended via Boolean reference frames for measurement of observables, pasted together along their overlaps. The scheme is formalized categorically, as an instance of the adjunction concept. The latter is used as a framework for the specification of a categorical equivalence signifying an invariance in the translational code of communication between Boolean localizing contexts and quantum systems. Aspects of the scheme semantics are discussed in relation to logic. The interpretation of coordinatizing localization systems, as structure sheaves, provides the basis for the development of an algebraic differential geometric machinery suited to the quantum regime.     

Tensor Products of Quantum Structures and Their Applications in Quantum Measurements

Tensor products of quantum logics and effect algebras with some known problems are reviewed. It is noticed that although tensor products of effect algebras having at least one state exist, in the category of complex Hilbert space effect algebras similar problems as with tensor products of projection lattices occur. Nevertheless, if one of the two coupled physical systems is classical, tensor product exists and can be considered as a Boolean power. Some applications of tensor products (in the form of Boolean powers) to quantum measurements are reviewed.     

Kochen-Specker theorem in the modal interpretation of quantum mechanics

According to the modal interpretation of quantum mechanics, subsystems of a quantum mechanical system have definite properties, the set of definite properties forming a partial Boolean algebra. It is shown that these partial Boolean algebras have no common extension (as a partial Boolean subalgebra of the properties of the total system) that is embeddable in a Boolean algebra. One has thus either to restrict the rules to preferred subsystems (Healey), or to advocate a shift in metaphysics (Dieks).     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Quantum logical solution to the measurement problem of quantum mechanics

In this paper I propose a reformulation and solution of the measurement problem of quantum mechanics. The reformulation depends on a quantum logical interpretation of quantum mechanics, broadly construed. The solution depends on a theorem about partial Boolean algebras which is proved here.     

Coreflections in Algebraic Quantum Logic

Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.     

Connections among quantum logics. Part 1. Quantum propositional logics

In this paper, we propose a theory of quantum logics which is general enough to enable us to reexamine previous work on quantum logics in the context of this theory. It is then easy to assess the differences between the different systems studied. The quantum logical systems which we incorporate are divided into two groups which we call quantum propositional logics and quantum event logics. We include the work of Kochen and Specker (partial Boolean algebras), Greechie and Gudder (orthomodular partially ordered sets), Domotar (quantum mechanical systems), and Foulis and Randall (operational logics) in quantum propositional logics; and Abbott (semi-Boolean algebras) and Foulis and Randall (manuals) in quantum event logics. In this part of the paper, we develop an axiom system for quantum propositional logics and examine the above structures in the context of this system.     

Complementarity in Categorical Quantum Mechanics

We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.     

Robustness of interferometric complementarity under decoherence

Interferometric complementarity is known to be one of the most nonclassical manifestations of the quantum formalism. It is commonly known as wave-particle duality and has been studied presently from the perspective of quantum information theory where wave and particle nature of a quantum system, called quanton, are characterised by coherence and path distinguishability respectively. We here consider the effect of noisy detectors on the complementarity relation. We report that by suitably choosing the initial quanton and the detector states along with the proper interactions between the quanton and the detectors, one can reduce the influence of noisy environment on complementarity, thereby pushing it towards saturation. To demonstrate this, three kinds of noise on detectors and their roles on the saturation of the complementarity relation are extensively studied. We also observe that for fixed values of parameters involved in the process, asymmetric quanton state posses low value of coherence while it can have a higher amount of distinguishability, and hence it has the potential to enhance the duality relation.     

On the energy-time uncertainty relation. Part I: Dynamical time and time indeterminacy

The problem of the validity and interpretation of the energy-time uncertainty relation is briefly reviewed and reformulated in a systematic way. The Bohr-Einsteinphoton-box gedanken experiment is seen to illustrate the complementarity of energy andevent time. A more recent experiment with amplitude-modulated Mößbauer quanta yields evidence for the genuine quantum indeterminacy of event time. In this way, event time arises as a quantum observable.Dedicated to my parents on the occasion of their 60th birthdays.     

On Monoids in the Category of Sets and Relations

The category R e l is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, R e l is a monoidal category. Moreover, R e l is a locally posetal 2-category, since every homset R e l(A,B) is a poset with respect to inclusion. We examine the 2-category of monoids R e l M o n in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.     

From Quantum Quasi-Shuffle Algebras to Braided Rota–Baxter Algebras

In this letter, we use quantum quasi-shuffle algebras to construct Rota–Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota–Baxter algebras, the relevant object of Rota–Baxter algebras in a braided tensor category. Examples of such new algebras are provided using quantum multi-brace algebras in a category of Yetter–Drinfeld modules.     

Lattice Properties of Ring-like Quantum Logics

Generalized Boolean quasirings (GBQRs) are extensions of partial algebras thatare in one-to-one correspondence to bounded lattices with an involutoryantiautomorphism. This correspondence generalizes the bijection betweenBoolean rings and Boolean algebras and provides for a large variety of presumptivequantum logics (including logics which can be defined by means of Mackey'sprobability function). It is shown how properties of the corresponding latticesare reflected in GBQRs and what the implications are of the associativity of the+-operation of GBQRs, which can be interpreted as some kind of an exclusiveor-operation. We prove that under very weak conditions, which, however, seemto be essential for experimental verifications, the associativity of + implies theclassicality of the considered quantum mechanical system.     

Measures Defined on Quantum Logics of Sets

We study families formed with subsets of any set X which are quantum logics but which are not Boolean algebras. We consider sequences of measures defined on a sets quantum logics and valued on an effect algebra and obtain a sufficient condition for a sequences of such measures to be uniformly strongly additive with respect to order topology of effect algebras.     

Generalized EMV-Effect Algebras

Recently in Dvure?enskij and Zahiri (2017), new algebraic structures, called EMV-algebras which generalize both MV-algebras and generalized Boolean algebras, were introduced. We present equivalent conditions for EMV-algebras. In addition, we define a partial algebraic structure, called a generalized EMV-effect algebra, which is close to generalized MV-effect algebras. Finally, we show that every generalized EMV-effect algebra is either an MV-effect algebra or can be embedded into an MV-effect algebra as a maximal ideal.