Quantum Cellular Automata, Tensor Networks, and Area Laws.

Area of Science:

• Quantum Information Theory
• Condensed Matter Physics
• Quantum Computing

Background:

• Quantum cellular automata (QCA) are fundamental models in quantum information processing.
• Understanding the properties of QCA, such as locality and causality preservation, is crucial.
• Tensor networks provide a powerful framework for studying quantum many-body systems.

Purpose of the Study:

• To classify unitary and nonunitary quantum maps using tensor network representations.
• To investigate the entanglement properties and correlation structures of these maps.
• To determine the relationship between quantum maps and the area law for entanglement entropy.

Main Methods:

• Identifying quantum cellular automata with projected entangled pair unitary (PEPU) tensor networks.
• Analyzing the bond dimension of tensor networks to understand entanglement scaling.
• Defining and studying nonunitary quantum channels that preserve locality or causality.
• Quantifying quantum correlations using quantum mutual information.

Main Results:

• Quantum cellular automata correspond to PEPU tensor networks with a constant bond dimension, satisfying an area law for entanglement entropy.
• Nonunitary quantum channels preserving locality obey an area law for quantum mutual information.
• Nonunitary quantum channels preserving causality may violate the area law for quantum mutual information.
• Neither unitary nor nonunitary quantum channels can be represented by tensor networks with a system-size-independent bond dimension.

Conclusions:

• The tensor network framework provides a unified view of quantum cellular automata and quantum channels.
• The area law for entanglement entropy is a key characteristic of unitary quantum maps like QCA.
• Nonunitary quantum channels exhibit more diverse entanglement and correlation behaviors, with potential violations of the area law.