Estimation of the Reduced Density Matrix and Entanglement Entropies Using Autoregressive Networks

Matrix elements of the density matrix of small quantum systems have recently become experimentally accessible within the cold atoms simulations. Numerical estimation of these quantities is known to be computationally challenging. Tracing out a part of the studied system and hence accessing the reduced density matrices offers similar difficulties, but it is more interesting, as the reduced density matrix is the basis of entanglement entropy measures. In this work, we present a method of estimating the individual elements of the reduced density matrices that uses autoregressive neural networks and the path integral quantization approach. We use a hierarchy of neural networks capable of estimating conditional probabilities of consecutive spins to evaluate elements of reduced density matrices directly. Using the Ising chain as an example, we calculate the continuum limit of the ground state’s von Neumann and Rényi bipartite entanglement entropies of an interval built of up to 5 spins. We demonstrate that our architecture is able to estimate all the needed matrix elements with just a single training for a fixed temperature discretization and lattice volume. Our method can be applied to other types of spin chains, possibly with defects, as well as to estimating entanglement entropies of thermal states at non-zero temperature. Hence, it may offer an attractive computational approach as a counterpart to the experimental effort. Abstract Published by the Jagiellonian University 2025 authors