Quantum logics with bounded additive operators

The theory of gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics, where the meaning of a sentence is identi ed with a system of qubits in a pure or, more generally, mixed state. In this framework, any formula of the language gives rise to a quantum circuit that transforms the state associated to the atomic subformulas into the state associated to the formula and vice versa. On this bases, some holistic semantic situations can be described, where the meaning of whole determine the meaning of the parts, by non-linear and anti-unitary operators. We prove that the semantics with such operators and the semantics with unitary operators turn out to characterize the same logic.