Boolean machinery for quantum logics

R. I. G. Hughes

Kenji Tokuo

Steve Faulkner

Abstract: In 2008 Tomasz Paterek et al published experiments demonstrating that quantum randomness results from logical independence. The job of this paper is to derive implications for Matrix Mechanics. Surprisingly (and apparently unwittingly), the Paterek experiments imply that faithful representation of (non-random) pure states contradicts the Quantum Postulate which imposes unitary, Hermitian and Hilbert space mathematics on all states. Paterek's Boolean formalism asserts and demands a non-unitary environment for pure states, which is freely restricted to logically independent unitary structure, wherever the creation of mixed states demands unitarity. Consequently, the Paterek experiments contradict the Quantum Postulate which imposes unitary, Hermitian and Hilbert space structures, axiomatically as blanket ontology, across the whole theory. Examination of the ‘non-unitary to unitary transition’ reveals the machinery of quantum indeterminacy. That machinery involves self-referential circularity, inaccessible history, and the geometrical ambiguity of perfect symmetry. The findings here provide answers for researchers studying Foundations of Quantum Mechanics; they make intuitive good sense of indeterminacy; they provide reason and significance for observable operators and eigenvectors; and they should be helpful for those interested in the Measurement Problem, the EPR paradox and possibly those looking for a method to quantize Gravity. Keywords: foundations of quantum theory, quantum randomness, quantum indeterminacy, logical independence, self-reference, logical circularity, mathematical undecidability.

David Finkelstein

Oľga Nánásiová, L’ubica Valášková, Viera Čerňanová

The present paper is devoted to modelling of a probability measure of logical connectives on a quantum logic via a G-map, which is a special map on it.We follow the work in which the probability of logical conjunction (AND), disjunction (OR), symmetric difference (XOR) and their negations for non-compatible propositions are studied.Now we study all remaining cases of G-maps on quantum logic, namely a probability measure of projections, of implications, and of their negations.We show that unlike classical (Boolean) logic, probability measures of projections on a quantum logic are not necessarilly pure projections.We indicate how it is possible to define a probability measure of implication using a G-map in the quantum logic, and then we study some properties of this measure which are different from a measure of implication in a Boolean algebra.Finally, we compare the properties of a G-map with the properties of a probability measure related to logical connectives on a Boolean algebra.

Mirko Navara, Pavel Pták

We consider orthomodular posets endowed with a symmetric difference. We call them ODPs. Expressed in the quantum logic language, we consider quantum logics with an XOR-type connective. We study three classes of "almost Boolean" ODPs, two of them defined by requiring rather specific behaviour of infima and the third by a Boolean-like behaviour of Frink ideals. We establish a (rather surprising) inclusion between the three classes, shadding thus light on their intrinsic properties. (More details can be found in the Introduction that follows.) Let us only note that the orthomodular posets pursued here, though close to Boolean algebras (i.e., close to standard quantum logics), still have a potential for an arbitrarily high degree of non-compatibility and hence they may enrich the studies of mathematical foundations of quantum mechanics.