Multipartite Cellular Automata and the Superposition Principle

Cellular automata can show well known features of quantum mechanics, such as a linear updating rule that resembles a discretized form of the Schr\"odinger equation together with its conservation laws. Surprisingly, a whole class of "natural" Hamiltonian cellular automata, which are based entirely on integer-valued variables and couplings and derived from an Action Principle, can be mapped reversibly to continuum models with the help of Sampling Theory. This results in "deformed" quantum mechanical models with a finite discreteness scale $l$, which for $l\rightarrow 0$ reproduce the familiar continuum limit. Presently, we show, in particular, how such automata can form "multipartite" systems consistently with the tensor product structures of nonrelativistic many-body quantum mechanics, while maintaining the linearity of dynamics. Consequently, the Superposition Principle is fully operative already on the level of these primordial discrete deterministic automata, including the essential quantum effects of interference and entanglement.