Information in Many-body Eigenstates: A Question of Learnability

TL;DR

This paper introduces a machine learning-based measure called learnability to quantify how much information eigenstates encode about their Hamiltonian, revealing differences based on spectral position.

Contribution

It proposes a novel neural network approach to assess the information content of eigenstates, providing a new perspective on eigenstate properties in many-body quantum systems.

Findings

Spectral-edge eigenstates are more learnable than mid-spectrum eigenstates.

Fewer spectral-edge eigenstates are needed to accurately reconstruct the Hamiltonian.

Learnability correlates with the entanglement and structure of eigenstates.

Abstract

To what extent do individual eigenstates encode information of their underlying Hamiltonian, and how does this depend on their spectral position? For many-body quantum systems, this issue is widely understood in terms of the differing nature of the eigenstates near the spectral edges (low-entanglement, highly-structured eigenstates) and those far from the spectral edges (high-entanglement, near-random eigenstates). Utilizing the availability of machine learning tools, we introduce a new way to quantify the information contained in eigenstates: for a particular learning architecture, how precisely can the Hamiltonian be reconstructed from a single eigenstate? We refer to this property as learnability; it serves as a new, alternative measure of the information content of eigenstates, made possible by machine learning. Using an encoder-decoder neural network and a physics-inspired loss function, we demonstrate how the distinction between two types of eigenstates is manifested as a difference in learnability. For spectral-edge eigenstates, the prediction accuracy is much better, and fewer eigenstates are required to learn the Hamiltonian, compared to mid-spectrum eigenstates.