Concrete Quantum Logics

Concrete quantum logics are quantum logics which allow for a set representation.They seem to be of significant conceptual value within quantum axiomatics andthey play an important role in the theory of orthomodular structures asset-representable orthomodular posets or lattices and they also sometimes constitutea domain for investigations in noncommutative. measure theory. This paperpresents a survey of recent results on this class of logics. Stress is put on thealgebraic and measure-theoretic aspects. Several open questions relevant tothe logicoalgebraic foundation of quantum theories are posed.     

Quantum Logics and Instruments

In a combined quantum logic and convexityapproach, an abstract notion of an instrument (statetransformer) is introduced to describe quantummeasurements. Some important classes of instruments(first kind, repeatable, ideal, Luders) and relations amongthem are investigated.     

Towards Quantum Computational Logics

Quantum computational logics have recently stirred increasing attention (Cattaneo et al. in Math. Slovaca 54:87–108, 2004; Ledda et al. in Stud. Log. 82(2):245–270, 2006; Giuntini et al. in Stud. Log. 87(1):99–128, 2007). In this paper we outline their motivations and report on the state of the art of the approach to the logic of quantum computation that has been recently taken up and developed by our research group.     

Order Topology Orthosummability in Quantum Logics

By using the Antosik–Mikusinski infinite matrix convergence theorem in quantum logics, we prove a theorem on orthosummability with respect to order topology in quantum logics.     

Fock Theories and Quantum Logics

The notion of Fock theory is introduced in the framework of quantum logics, which are here orthomodular atomic lattices satisfying the covering property. It is shown that there are some fundamental facts concerning particles, which may be successfully discussed in this general context. One of these facts is to establish the theoretical conditions for considering particles as sharply defined entities. The other refers to the theoretical circumstances, which almost impose to consider that some particles have a structure, meaning they are composed from other particles. This last problem is strongly related with the conservative time evolutions.     

Quantum Logics and Convex Spaces

An orthomodular -lattice with rich set ofstates satisfying the property that every affinefunctional from the set of states into the unit intervalof the reals corresponds to an expectational functional of exactly one real observable (so-calledu-spectral logic) is compared with the noncommutativespectral theory of Alfsen and Shultz. Necessary andsufficient conditions are found under which these two approaches are in correspondence.     

Quantum Logics that are Symmetric-difference-closed

International Journal of Theoretical Physics - In this note we contribute to the recently developing study of “almost Boolean” quantum logics (i.e. to the study of orthomodular...     

Antosik-Mikusinski Matrix Convergence Theorem in Quantum Logics

In this paper we establish the order topology type Antosik-Mikusinski infinite matrix convergence theorem in quantum logics. As application, we prove the Hahn-Schur summation theorem in quantum logics, too.     

Quantum Conservative Gates for Finite-Valued Logics

We introduce some conservative gates for finite-valued logics which are able to realize all the main connectives of the many-valued logics of ?ukasiewicz, the MV-algebras of Chang and Brower–Zadeh algebras. After a brief exposition of the motivations for this work, the gates are defined and their properties are explored. Finally, a possible quantum realization of them is proposed, using three techniques: a “brute force” method--an extension of the Conditional Quantum Control argument, and a new technique which we call the Constants Method. For all these techniques, the unitary operator which describes the gate is a sum of local operators.     

Quantum Logics and Relative Lindenbaum Property

The relative Lindenbaum property is considered in orthomodular quantum logic and in partial class ical logic. The properties of these models are connected with the possibility of hidden-variable reconstruction of particular (1/2-spin particle e.g.) quantum physical systems.     

Quantum Computational Logics and Possible Applications

In quantum computational logics meanings of formulas are identified with quantum information quantities: systems of qubits or, more generally, mixtures of systems of qubits. We consider two kinds of quantum computational semantics: (1) a compositional semantics, where the meaning of a compound formula is determined by the meanings of its parts; (2) a holistic semantics, which makes essential use of the characteristic “holistic” features of the quantum-theoretic formalism. The compositional and the holistic semantics turn out to characterize the same logic. In this framework, one can introduce the notion of quantum-classical truth table, which corresponds to the most natural way for a quantum computer to calculate classical tautologies. Quantum computational logics can be applied to investigate different kinds of semantic phenomena where holistic, contextual and gestaltic patterns play an essential role (from natural languages to musical compositions).     

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.     

Local Automorphisms of Some Quantum Mechanical Structures

Let H be a separable infinite-dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented.     

Fuzzy Quantum Logics as a Basis for Quantum Probability Theory

Representation of an abstract quantum logic withan ordering set of states S in the form of a family L(S) of fuzzy subsets of S which fulfils conditionsanalogous to Kolmogorovian conditions imposed on -algebra of random events allows us toconstruct quantum probability calculus in a waycompletely parallel to the classical Kolmogorovianprobability calculus. It is shown that the quantumprobability calculus so constructed is a propergeneralization of the classical Kolmogorovian one. Someindications for building a phase-space representation ofquantum mechanics free of the problem of negativeprobabilities are given.     

On Twisting Real Spectral Triples by Algebra Automorphisms

We systematically investigate ways to twist a real spectral triple via an algebra automorphism and in particular, we naturally define a twisted partner for any real graded spectral triple. Among other things, we investigate consequences of the twisting on the fluctuations of the metric and possible applications to the spectral approach to the Standard Model of particle physics.     

Differential Calculi on Quantum Spaces Determined by Automorphisms

If the bimodule of 1-forms of a differential calculus over an associative algebra A is the direct sum of 1-dimensional bimodules, a relation with automorphisms of A shows up. This happens for some familiar quantum-space calculi.     

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.