Quantum Cellular Automata, Tensor Networks, and Area Laws

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Quantum Cellular Automata are unitary maps that preserve locality and respect causality. I will show that in one spatial dimension they correspond to matrix product unitary operators, and that one can classify them in the presence of symmetries, giving rise to phenomenon analogous to symmetry protection. I will then show that in higher dimensions, they correspond to other tensor networks that fulfill an extra condition and whose bond dimension does not grow with the system size. As a result, they satisfy an area law for the entanglement entropy they can create. I will also define other classes of non-unitary maps, the so-called quantum channels, that either respect causality or preserve locality and show that, whereas the latter obey an area law for the amount of quantum correlations they can create, as measured by the quantum mutual information, theformer may violate it. Additionally, neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size.