# Conditions on the existence of maximally incompatible two-outcome measurements in General Probabilistic Theory

## TL;DR

This paper establishes geometric conditions in finite-dimensional General Probabilistic Theories for the existence of maximally incompatible two-outcome measurements, linking incompatibility to state space geometry and discrimination capabilities.

## Contribution

It provides necessary and sufficient geometric conditions for maximal incompatibility, introducing the notion of discrimination measurement in GPTs.

## Key findings

- Conditions involve pairs of parallel exposed faces with specific intersection properties.
- Maximal incompatibility relates to the ability of hypothetical joint measurements to discriminate affinely dependent points.
- Several examples illustrate the geometric criteria and their implications.

## Abstract

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.