Sequent Calculus and Quantum Parallelism

In the last years, quantum computational logics have been developed in order to describe the logical setting of quantum computation [1]. An important challenge is the development of proof theoretical tools for quantum computation, so that the physical process of computation can correspond to a logical process of computation. In particular, the challenge is that quantum computational speed up, due to superposition and entanglement, can find an explanation in terms of logical proofs. The object of our research is a sequent calculus developed in the framework of basic logic, that is a logical platform to study extensional logics, including quantum logics [2, 3]. In basic logic, logical connectives and their rules are the result of importing suitable links at the metalevel into the object level. We suppose that a particular kind of dynamics between the object level and the metalevel is created when considering a quantum system rather than a classical one. This is due to a specific treatment of variables, that, in our view, can create the logical setting for a holistic, rather than compositional, treatment of information, proper of entangled states [4]. Then we develop a predicative sequent calculus. In it, superposition is built by quantifying a propo-sition, associated with a quantum system, on a first order domain determined by the experiments on such system. Then a convenient treatment of variables can create dynamically entangled states in logica