Product Lukasiewicz quantum logic

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a density operator (called qumix). In this framework, any sentence α of the language gives rise to a quantum circuit that transforms the qumix associated to the atomic subformulas of α into the qumix associated to α. In this paper we enrich the language by adding a new connective which expresses truncated sum.