Quantum Cellular Automata, Tensor Networks, and Area Laws

TL;DR

This paper explores quantum cellular automata and quantum channels, demonstrating their relation to tensor networks and area laws for entanglement and correlations, with implications for understanding quantum information propagation.

Contribution

It establishes that quantum cellular automata can be represented as simple tensor networks with fixed bond dimension, satisfying an area law, and analyzes the properties of quantum channels regarding locality and causality.

Findings

Quantum cellular automata are equivalent to tensor networks with fixed bond dimension.

Unitary maps respecting causality obey an area law for entanglement entropy.

Non-unitary quantum channels may violate the area law for quantum correlations.

Abstract

Quantum Cellular Automata are unitary maps that preserve locality and respect causality. We identify them, in any dimension, with simple tensor networks (PEPU) 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. We define other classes of non-unitary maps, the so-called quantum channels, that either respect causality or preserve locality. We 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, the former may violate it. We also show that neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size.