On Locally Finite Orthomodular Lattices

This paper is a sequel to arXiv:0902.2355 and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance.

This paper describes a computational approach to studying certain aspects of small finite orthomodular lattices (OML’s). The program has two main parts: one, to construct a representation of an OML L, and to perform some simple analyses on L; and two, to construct at least a portion of S 0 (L), the Foulis semigroup (or Baer *-semigroup) of the Sasaki projection functions on L. The representation is constructed from an input which represents the Greechie diagram of L. Such a diagram depicts an orthoposet as a collection of maximal Boolean subalgebras, and if an input does not represent a lattice, the program indicates why not. While the operations of the program are not asserted to be optimal, extensive efforts have been made to use information about the problem area to find efficient methods of performing the various tasks of the program. The methods used and some results are described, and some remarks about possible future work are also given.

It is shown that an orthomodular lattice can be axiomatized as an ortholattice with a unique operation of identity (bi-implication) instead of the operation of implication, and a corresponding algebraic unified quantum logic is formulated. A statistical YES-NO physical interpretation of the quantum logical propositions is then provided to establish a support for a novel YES-NO representation of quantum logic which prompts a conjecture about a possible completion of quantum logic by means of probabilistic forcing. In this paper we shall anwser two basic questions of quantum' logics, present a novel probabilistic representation of the logic and of quantum measurements, and propose a probabilistic forcing as a possible tool for completing quantum logic. The questions are, first, whether there is a unique object language operation which can take over the role of the unique classical operation of implication (conditional, set-theoretic inclusion), and, second, whether there is a relation which is more appropriate for set-theoretic representation of quantum-theoretic measurements than the usual irreflexive and symmetric orthogonality relation. We answer the first question in the positive by substituting the biimplication for the implication and the equality for the ordering relation, thus at the same time making the ordering within quantum sets irrelevant. This renders the usual techniques of quantum logic as a deductive inferential theory inappropriate and ascribes quantum deductive logic a particular equational meaning. The result is obtained in Section 2.

In the orthodox quantum logic approach to foundations of quantum mechanics originated by Birkhoff and von Neumann [1] the object called quantum logic is not a logic in the very sense of this word but an algebraic structure: orthomodular lattice or poset. These orthomodular structures, abstracted from structures of Hilbert spaces used to describe quantum phenomena are weakenings of Boolean algebras. They can be thought of as representing Lindenbaum algebras of systems of propositions about quantum objects, the propositions belonging to the quantum logic proper. Since Lindenbaum algebras of propositional systems which obey the rules of classical logic are Boolean algebras it is inferred that the quantum logic proper, whatever it means, is non-classical.

Quantum logic has been studied with orthomodular lattices. The semantics required for quantum logic can also be provided by OM-models, whose nature is almost equivalent to the notion of orthomodular lattices. However, the development of OM-models is in its infancy, and important notions of orthomodular lattices such as OM laws, atomicity and covering laws cannot yet be fully described. Thus, in this paper, we develop OM-models in an attempt to solve these problems.

We show that an orthomodular lattice is embeddable in a Hilbert space if and only if states of a certain kind exist. A physical motivation for the existence of such states is given and a connection is provided between the quantum logic, algebraic, and operational approaches to quantum mechanics.

Resumen

With each orthomodular lattice L we associate a spectral presheaf \(\underline{\varSigma }^{L}\), generalising the Stone space of a Boolean algebra, and show that (a) the assignment \(L\mapsto \underline{\varSigma }^{L}\) is contravariantly functorial, (b) \(\underline{\varSigma }^{L}\) is a complete invariant of L, and (c) for complete orthomodular lattices there is a generalisation of Stone representation in the sense that L is mapped into the clopen subobjects of the spectral presheaf \(\underline{\varSigma }^{L}\). The clopen subobjects form a complete bi-Heyting algebra, and by taking suitable equivalence classes of clopen subobjects, one can regain a complete orthomodular lattice isomorphic to L. We interpret our results in the light of quantum logic and in the light of the topos approach to quantum theory.

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

Fuzzy set representations of some quantum structures

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.

This paper investigates mathematical properties of a finite-dimensional real algebra of linear operators which are generated by an orthomodular lattice of filters in the sense of Mielnik [4]. Properties of filter decomposability and a representation theorem for the vector space underlying the algebra mentioned are derived.

By combining the advantages of quantum computing and soft computing, the paper shows that rough sets can be used with quantum logic for classification and recognition systems. We suggest the new definition of rough set theory as quantum logic theory. Rough approximations are essential elements in rough set theory, the quantum rough set model for set-valued data directly construct set approximation based on a kind of quantum similarity relation which is presented here. Theoretical analyses demonstrate that the new model for quantum rough sets has new type of decision rule with less redundancy which can be used to give accurate classification using principles of quantum superposition and non-linear quantum relations. To our knowledge, this is the first attempt aiming to define rough sets in representation of a quantum rather than logic or sets. The experiments on data-sets have demonstrated that the proposed model is more accuracy than the traditional rough sets in terms of finding optimal classifications.

Structural semantics for polymorphic data types (preliminary report)

Turing machines based on quantum logic can solve undecidableproblems. In this paper we will give recursion-theoreticalcharacterization of the computational power of this kind of quantumTuring machines. In detail, for the unsharp case, it is proved that&#931<sup>0</sup><sub>1</sub>&#8746&#928<sup>0</sup><sub>1</sub>&#8838L<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))&#8838&#928<sup>0</sup><sub>2</sub>when the truth value lattice is locally finite and the operation &#8743is computable, whereL<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))denotes theclass of quantum language accepted by these Turing machine indepth-first model (respectively, width-first model);for the sharp case, we can obtain similar results for usual orthomodular lattices.