In a recent paper [1], the mathematical structure of finite dimensional Hilbert spaces and the description of states, evolutions and measurements through density matrices, completely positive maps and POVMs has been derived from principles ruling information processing tasks. This result provides an information-theoretic basis for the abstract formalism of the theory of quantum systems.

In the present work we review subsequent efforts [2] aimed at an extension of the information-theoretical foundations to encompass also the strictly mechanical part of Quantum Theory. In particular, we will review the derivation of Weyl's, Dirac's and Maxwell's equations based on Quantum Cellular Automata, and from simple principles regarding their main computational features. These principles are unitarity, linearity, locality, homogeneity, isotropy.

Quantum systems constituting the Quantum Cellular automaton can be organised in a graph--where edges connect directly interacting systems--which is more precisely the Cayley graph of some finitely presented group. Under the further assumption that the group is (quasi-isometrically) embeddable in a Euclidean manifold, we will show that the only two possible unitary automata satisfying all our requirements give rise, in an appropriate limit, to Weyl's equation, and can be combined to obtain both Dirac [2] or Maxwell's [3] equations for free quantum fields.

[1] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. A *84**, 012311 (2011).

[2] G. M. D'Ariano and P. Perinotti, arXiv:1306.1934.

[3] A. Bisio, G. M. D'Ariano, and P. Perinotti, in preparation.