An axiomatization of quantum computational Logic

Abstract

One of the logical proposals that arise from quantum computation is the idea to use the quantum theoretical formalism to represent parallel reasoning. To this aim a quantum computation is considered by means of convenient unitary operators assuming arguments and values in particular sets of qubit systems. Isolating some important unitary operators that have a special role in quantum computation (logical gates or quregisters) we obtain an opportunity to yield the language of Quantum Computational Logic (QCL) (cf. [Cattaneo et al., 2003; Cattaneo et al., 2004; Dalla Chiara et al., 2004]). The basic concept of the semantics of this language is the notion of quantum computational realization such that the meaning associated to any sentence is a quregister. Unlike the semantic of a standard quantum logic QCL-conjunction and QCL-disjunction do not correspond to lattice operations since they are not generally idempotent. Moreover, in QCL the weak distributivity principle breaks down and both the excluded middle and the non contradiction principles are violated. Finally, the axiomatizability of QCL is still an open problem. In the paper an axiomatization of QCL is proposed construing it as a kind of so-called Goldblatt’s binary logic. Some metalogical theorems (paraconsistency and completeness) are proved.