Conditions on the existence of maximally incompatible two-outcome measurements in general probabilistic theory

We formulate the necessary and sufficient conditions for the existence of a pair of maximally incompatible two-outcome measurements in a finite-dimensional general probabilistic theory. The conditions are on the geometry of the state space; they require the existence of two pairs of parallel exposed faces with an additional condition on their intersections. We introduce the notion of discrimination measurement and show that the conditions for a pair of two-outcome measurements to be maximally incompatible are equivalent to requiring that a (potential, yet nonexisting) joint measurement of the maximally incompatible measurements would have to discriminate affinely dependent points. We present several examples to demonstrate our results.