Leaks: quantum, classical, intermediate, and more

We introduce the notion of a leak for general process theories, and identify quantum theory as a theory with minimal leakage, while classical theory has maximal leakage. We provide a construction that adjoins leaks to theories, an instance of which describes the emergence of classical theory by adjoining decoherence-leaks to quantum theory. Finally, we show that defining a notion of purity for processes in general process theories has to make reference to the leaks of that theory ---a feature missing in standard definitions--- hence, we propose a refined definition and study the resulting notion of purity for quantum, classical and intermediate theories.