On the Modal Logic of the Non-orthogonality Relation Between Quantum States

It is well known that the non-orthogonality relation between the (pure) states of a quantum system is reflexive and symmetric, and the modal logic KTB is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that satisfies each one of these two properties and both, respectively. We also show that the modal logics involved are decidable.