  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.
