Concrete and abstract quantum computational logics

The theory of logical gates in quantum computation has suggested some new forms of quantum logics, called quantum computational logics. In the standard semantics of these logics, formulas denote quantum information quantities (systems of qubits, or, more generally, mixtures of systems of qubits), while the logical connectives are interpreted as logical operations defined in terms of special quantum logical gates (which have a characteristic reversible and dynamic behavior). We consider two kinds of quantum computational semantics: 1) a compositional semantics, where the meaning of any 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. Quantum computational logics can be applied to investigate different kinds of semantic phenomena where holistic and contextual patterns play an essential role (from natural languages to musical compositions). Is it sensible to look for an abstract quantum computational semantics that is not necessarily "Hilbert-space dependent''? In this perspective, we introduce a weak form of quantum computational logic (called abstract quantum computational logic), and we prove a completeness theorem for this logic.