  Cellular automata (CA) exemplify systems where simple local interaction rules
can lead to intricate and complex emergent phenomena at large scales. The
various types of dynamical behavior of CA are usually categorized empirically
into Wolfram's complexity classes. Here, we propose a quantitative measure,
rooted in quantum information theory, to categorize the complexity of classical
deterministic cellular automata. Specifically, we construct a Matrix Product
Operator (MPO) of the transition matrix on the space of all possible CA
configurations. We find that the growth of entropy of the singular value
spectrum of the MPO reveals the complexity of the CA and can be used to
characterize its dynamical behavior. This measure defines the concept of
operator entanglement entropy for CA, demonstrating that quantum information
measures can be meaningfully applied to classical deterministic systems.
