Quantization of cellular automata

Take a cellular automaton, consider that each configuration is a basis vector in some vector space, and linearize the global evolution function. If lucky, the result could actually make sense physically, as a valid quantum evolution; but does it make sense as a quantum cellular automaton? That is the main question we address in this paper. In every model with discrete time and space, two things are required in order to qualify as a cellular automaton: invariance by translation and locality. We prove that this locality condition is so restrictive in the quantum case that every quantum cellular automaton constructed in this way — i.e., by linearization of a classical one — must be reversible. We also discuss some subtleties about the extent of nonlocality that can be encountered in the one-dimensional case; we show that, even when the quantized version is non local, still, under some conditions, we may be unable to use this nonlocality to transmit information nonlocally.