Characterization of quantum logics

Enrico G. Beltrametti, Maciej Mączyński

M. K. Mukherjee

Daniel J. Lehmann

When a physicist performs a quantic measurement, new information about the system at hand is gathered. This article studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the and then operation that combines old and new information. The and then connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the and then connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch in [3] satisfies all the properties studied in this article. All properties satisfied by Finch's; operation in modular lattices are valid in Quantum Logic. It is not known whether all properties of Quantum Logic are satisfied by Finch's; operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.

Han Zhao-wei

The notion of quantum Bchi automaton(LVBA for short)is introduced,by means of quantum state construction, the equivalence of an LVBA and an LVSBA with crisp transition function is proved,based on this,the algebraic and level characterizations and also the Bchi characterization of quantum infinite regular languages are investigated,and also the closed properties of those quantum infinite regular languages under some regular operations are dealt with.By providing the concept of monadic secondorder quantum logic(LVMSO in short),the monadic second-order logic characterizations of infinite regular languages recognized by quantum Biichi automata are presented,which deepen and generalize the fundamental Bchi theorem to quantum setting.

Maciej Mączyński

María Luisa Dalla Chiara, Roberto Giuntini

Quantum logic is a particular example of a fuzzy quantum logic. QL is semantically characterized by the class of all quantum MV algebras. The standard quantum MV algebra is based on the set of all effects in a Hilbert space. From the physical point of view, effects represent physical properties that may be noisy and ambiguous.