Preparation Uncertainty Implies Measurement Uncertainty in a Class of Generalized Probabilistic Theories

Abstract

In quantum theory, it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values, and that there is no joint measurement of them. They are called preparation uncertainty and measurement uncertainty respectively, and research has unveiled that they are not independent from but related with each other in a quantitative way. This study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories (GPTs). In particular, a certain class of GPTs is considered which can be characterized by transitivity and self-duality and regarded as extensions of quantum theory. It is proved that there are close connections expressed quantitatively between two types of uncertainty on a pair observables also in those theories: if preparation uncertainty exists, then measurement uncertainty also exists, and they are described by similar inequalities. Our results manifest that their correspondences are not specific to quantum theory but more universal ones.

44 pages (12 pages for appendices), 1 figure

References in corpus (6)

• A generalized no-broadcasting theorem
• Higher-order interference and single-system postulates characterizing quantum theory
• All measurements in a probabilistic theory are compatible if and only if the state space is a simplex
• Heisenberg's uncertainty principle for simultaneous measurement of positive-operator-valued measures
• Universal joint-measurement uncertainty relation for error bars
• Generalized Landau-Pollak Uncertainty Relation