Low-Rank Approximability and Entropy Area Laws for Ground States of Unbounded Hamiltonians

Abstract: We show how local bounded interactions in an unbounded Hamiltonian lead to eigenfunctions with favorable low-rank properties. To this end, we utilize ideas from quantum entanglement of multi-particle spin systems. We begin by analyzing the connection between entropy area laws and low-rank approximability. The characterization for 1D chains such as Matrix Product States (MPS) / Tensor Trains (TT) is rather extensive though incomplete. We then show that a Nearest Neighbor Interaction (NNI) Hamiltonian has eigenfunctions that are approximately separable in a certain sense. Under a further assumption on the approximand, we show that this implies a constant entropy bound. To the best of our knowledge, this work is the first analysis of low-rank approximability for unbounded Hamiltonians. Moreover, it extends previous results on entanglement entropy area laws to unbounded operators. The assumptions include a variety of self-adjoint operators and have a physical interpretation. The weak points are the aforementioned assumption on the approximand and that the validity is limited to MPS/TT formats.