In this talk I will report on a recent work [arXiv:0908.1583], which investigates general probabilistic theories where every mixed state has a purification, unique up to reversible channels on the purifying system. The purification principle is equivalent to the existence of a reversible realization for every physical process, namely that to the fact that every physical process can be regarded as arising from the reversible interaction of the input system with an environment that is eventually discarded. From the purification principle one can also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamiolkowski isomorphism in Quantum Mechanics. Such an isomorphism allows one to prove most of the basic features of Quantum Information Processing, like e.g. no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, complementarity between correctable and deletion channels, no programming, and no bit commitment, without resorting to the mathematical framework of Hilbert spaces.