  In this paper we derive the complex Hilbert space formalism of quantum theory
from four simple information theoretic axioms. It is shown that quantum theory
is the only non classical probabilistic theory satisfying the following axioms:
distinguishability, conservation, reversibility, composition. The new results
of this reconstruction compared to other reconstructions by other authors are:
(i) we get rid of axiom "subspace" in favor of axiom conservation eliminating
mathematical requirements contained in previous axiomatics; (ii) we are able to
classify all the probabilistic theories that are consistent requiring (a) only
the first two axioms (b) only the first three axioms; this could be useful in
experimental tests of quantum theory since it gives the possibility to
understand whether or not other mathematical models could be consistent with
such tests; (iii) we provide a connection between two different approaches to
quantum foundations, quantum logic and the one based on information theoretic
primitives showing that any theory satisfying the first two axioms given above
either is classical or is a theory in which physical systems are described by a
projective geometry.
