After more than eighty years from its rigorous mathematical formulation, quantum theory is still mysterious. Its usual textbook presentations are merely descriptions of an abstract mathematical formalism, where``states are described by unit vectors in a Hilbert space" and ``observables are described by self-adjoint operators". However, this approach leaves aside the question about the underlying physical principles of the theory. For this reason, quantum theory is still far from providing a picture of the physical world capable to compete with the simple picture that we inherited from classical physics. In this talk I will explore the general idea that the new paradigm of physics could be that of information processing, and, specifically, I will present a new result showing that the mathematics of quantum theory can be completely reconstructed from a set of principles about information processing. The crucial feature of these principles is that they are not of mathematical nature like the usual axioms in the Hilbert space formulation. Instead, they can be formulated in a language that only refers to purely operational notions, like the notion of probabilistic mixture or the notion or reversible transformation. The key principle in our reconstruction of quantum theory is the "purification principle", stating that every mixed state of a system A can be obtained as the marginal state of some pure state of a joint system AB. In other words, the principle requires that the ignorance about a part be always compatible with the maximal knowledge of a whole. This statement reflects the original views by Schrödinger on entanglement, and introduces in our basic theory of information processing all genuine quantum features, like entanglement, the no-cloning theorem, and the possibility of state teleportation.