# Entanglement Area Law for Shallow and Deep Quantum Neural Network States

## TL;DR

This paper investigates how shallow and deep quantum neural network states exhibit entanglement area laws, revealing that locality constraints influence entanglement properties and impact the computational power in image classification tasks.

## Contribution

It introduces the concept of local quasi-product states and demonstrates that various neural network states obey the entanglement area law, linking entanglement properties to neural network efficiency.

## Key findings

- Shallow and deep neural network states obey the entanglement area law.
- Locality constraints enforce the area law in neural network states.
- Efficient image classification tasks must obey the entanglement area law.

## Abstract

A study of the artificial neural network representation of quantum many-body states is presented. The locality and entanglement properties of states for shallow and deep quantum neural networks are investigated in detail. By introducing the notion of local quasi-product states, for which the locally connected shallow feed-forward neural network states and restricted Boltzmann machine states are special cases, we show that R\'{e}nyi entanglement entropies of all these states obey the entanglement area law. Besides, we also investigate the entanglement features of deep Boltzmann machine states and show that locality constraints imposed on the neural networks make the states obey the entanglement area law. Finally, as an application, we apply the notion of R\'{e}nyi entanglement entropy to understanding the power of neural networks and show that image classification problems which can be efficiently solved must obey the area law.