Quantum Computational Logics. A Survey

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 quregister, a system of qubits, representing a possible pure state of a compound quantum system. The generalization to mixed states, which might be useful to analyse entanglement-phenomena, is due to Gudder. Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subformulas of the sentence into the quregister associated to the sentence.