# Estimation of the reduced density matrix and entanglement entropies using autoregressive networks

arXiv: https://arxiv.org/abs/2506.04170 · HTML: https://mlsanity.com/paper/2506.04170/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 31 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 5 of 5)

- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Flow-Based Sampling for Entanglement Entropy and the Machine Learning of Defects](https://mlsanity.com/paper/2410.14466/) (2024)
- [Hierarchical autoregressive neural networks for statistical systems](https://mlsanity.com/paper/2203.10989/) (2022)
- [Hierarchical autoregressive neural networks in three-dimensional statistical system](https://mlsanity.com/paper/2503.08610/) (2025)

## Built on by (most cited first, 2 of 2)

- [Sampling two-dimensional spin systems with transformers](https://mlsanity.com/paper/2604.27738/) (2026)
- [Variational Autoregressive Networks with probability priors](https://mlsanity.com/paper/2605.16020/) (2026)

## Abstract (arXiv)

We present an application of autoregressive neural networks to Monte Carlo simulations of quantum spin chains using the correspondence with classical two-dimensional spin systems. 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 R'enyi 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 time 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.