Information in Many-body Eigenstates: A Question of Learnability

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.