Entropic Uncertainty Relations in a Class of Generalized Probabilistic Theories

Abstract: Entropic uncertainty relations play an important role in both fundamentals and applications of quantum theory. Although they have been well-investigated in quantum theory, little is known about entropic uncertainty in generalized probabilistic theories (GPTs). The current study explores two types of entropic uncertainty relations, preparation and measurement uncertainty relations, in a class of GPTs which can be considered generalizations of quantum theory. Not only a method for obtaining entropic preparation uncertainty relations but also an entropic measurement uncertainty relation similar to the quantum one by Buscemi et al. [Phys. Rev. Lett., 112, 050401] are proved in those theories. It manifests that the entropic structure of uncertainty relations in quantum theory is more universal. Concrete calculations of our relations in GPTs called the regular polygon theories are also demonstrated.