Information in Many-body Eigenstates: A Question of Learnability

Abstract

To what extent do individual eigenstates encode information about their parent Hamiltonian, and how does this encoding vary across the spectrum? We introduce \emph{learnability} as a new framework to quantify this information, measured by the precision with which a machine learning model can reconstruct a Hamiltonian from a limited set of eigenstates. For many-body quantum systems, there is a contrast between the eigenstates near the spectral edges (low-entanglement, highly-structured states) and those far from the spectral edges (high-entanglement, near-random states). Using an encoder-decoder neural network for a non-integrable spin chain, we show that this dichotomy results in a stark difference in learnability: spectral-edge eigenstates allow for higher-accuracy Hamiltonian reconstruction using significantly fewer eigenstates, compared to mid-spectrum eigenstates. Our results provide a new lens through which to view the spectral structure of many-body systems.