Operator entanglement growth quantifies complexity of cellular automata

TL;DR

This paper introduces a quantum information-based measure called operator entanglement entropy to quantify and categorize the complexity of classical cellular automata, revealing their dynamical behavior through entropy growth.

Contribution

It develops a novel quantum-inspired framework using MPOs to measure the complexity of classical CA, bridging quantum information theory and classical dynamical systems.

Findings

Operator entanglement entropy correlates with CA complexity classes.

Entropy growth distinguishes between different dynamical behaviors.

Quantum measures provide new insights into classical system complexity.

Abstract

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.