  We consider a fundamental operational task, distinguishing systems in
different states, in the framework of generalized probabilistic theories and
provide a general formalism of minimum-error discrimination of states in convex
optimization. With the formalism established, we show that the
distinguishability is generally a global property assigned to the ensemble of
given states rather than other details of a given state space or pairwise
relations of given states. Then, we consider bipartite systems where ensemble
steering is possible, and show that show that with two operational tasks,
ensemble steering and the no-signaling condition, the distinguishability is
tightly determined. The result is independent to the structure of the state
space. This concludes that the distinguishability is generally determined by
the compatibility between two tasks, ensemble steering on states and the
non-signaling principle on probability distributions of outcomes.
