  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.
