  We investigate general probabilistic theories in which every mixed state has
a purification, unique up to reversible channels on the purifying system. We
show that the purification principle is equivalent to the existence of a
reversible realization of every physical process, namely that every physical
process can be regarded as arising from a reversible interaction of the system
with an environment, which is eventually discarded. From the purification
principle we 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 mechanics, like e.g.
existence of pure bipartite states giving perfect correlations in independent
experiments, no information without disturbance, no joint discrimination of all
pure states, no cloning, teleportation, no programming, no bit commitment,
complementarity between correctable channels and deletion channels,
characterization of entanglement-breaking channels as measure-and-prepare
channels, and others, without resorting to the mathematical framework of
Hilbert spaces.
