Reasoning About Quantum Systems

A new logic is proposed for reasoning about quantum systems. The logic embodies the postulates of quantum physics and it was designed from the semantics upwards by identifying quantum models with superpositions of classical models. This novel approach to quantum logic is completely different from the traditional approach of Birkhoff and von Neumann. It has the advantage of making quantum logic an extension of classical logic. The key new ingredient of the language of the proposed logic is a rather general modal operator. The logic incorporates probabilistic reasoning (in the style of Nilsson) in order to deal with uncertainty on the outcome of measurements. The logic also incorporates dynamic reasoning (in the style of Hoare) in order to cope with the evolution of quantum systems. A Hilbert calculus for the logic is sketched. A quantum key distribution protocol is specified and analyzed. 1 Motivation and Related Work A new logic is proposed for modeling and reasoning about quantum systems, embodying all that is stated in the postulates of quantum physics (as presented, for instance, in [1]). The logic was designed from the semantics upwards starting with the key idea of adopting superpositions of classical models as the models of the quantum logic. This novel approach to quantum logic semantics is completely different from the traditional approach [2, 3] to the problem, as initially proposed by Birkhoff and von Neumann [4] focusing on the lattice of closed subspaces of a Hilbert space. Our semantics has the advantage of closely guiding the design of the language around the underlying concepts of quantum physics while keeping the classical connectives and was inspired by the possible worlds approach originally proposed by Kripke [5] for modal logic. It is also akin to the society semantics introduced in [6] for many-valued logic and to the possible translations semantics proposed in [7] for paraconsistent logic. The possible worlds approach was also used in [8–12] for probabilistic logic. Our semantics to quantum logic, although inspired by modal logic, is also completely different from the alternative Kripke semantics given to traditional quantum logics (as first proposed in [13]) still closely related to the lattice-oriented operations. Contrarily to traditional quantum logics that replace the classical connectives by new connectives inspired by the lattice-oriented operations, by adopting superpositions of classical models as the models of the quantum logic we are led