All measurements in a probabilistic theory are compatible if and only if the state space is a simplex Martin PlávalaMathematical Institute, Slovak Academy of Sciences,Štefánikova 49, Bratislava, SlovakiaWe study the compatibility of measurements on finite-dimensional compact convex state space in the framework of general probabilistic theory. Our main emphasis is on formulation of necessary and sufficient conditions for two-outcome measurements to be compatible and we use these conditions to show that there exist incompatible measurements whenever the state space is not a simplex. We also formulate the linear programming problem for the compatibility of two-outcome measurements.

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 existence of two pairs of parallel exposed faces with 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 non-existing) joint measurement of the maximally incompatible measurements would have to discriminate affinely dependent points. We present several examples to demonstrate our results.

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