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Computing quantum entanglement with machine learning

Andrea Bulgarelli, Elia Cellini, Karl Jansen, Stefan Kühn, Alessandro Nada, Shinichi Nakajima, Kim A. Nicoli, Marco Panero

TL;DR

The paper tackles the challenge of computing entanglement measures in lattice field theories, where Renyi entropies $S_n$ are expressed via the replica trick as $S_n = \frac{1}{1-n} \log \frac{Z_n}{Z^n}$ and universal finite contributions are captured by the entropic c-function $C_n = \frac{l^{D-1}}{|\partial A|} \frac{\partial S_n}{\partial l}$. It introduces a defect-focused Normalizing Flow framework that acts on a localized region near the replica-cut endpoint to transform a simple prior into the replicated ensemble, enabling direct estimation of partition-function ratios. Across a $(2+1)$-dimensional $\phi^4$ theory at criticality, the approach yields significant efficiency gains and high-precision estimates of $C_n$ on large lattices, outperforming traditional Monte Carlo methods. The work presents a transferable paradigm for entanglement-related observables in lattice field theories, compatible with equivariant-flow ideas and extendable to gauge theories and other observables.

Abstract

Entanglement calculations in quantum field theories are extremely challenging and typically rely on the replica trick, where the problem is rephrased in a study of defects. We demonstrate that the use of deep generative models drastically outperforms standard Monte Carlo algorithms. Remarkably, such a machine-learning method enables high-precision estimates of Rényi entropies in three dimensions for very large lattices. Moreover, we propose a new paradigm for studying lattice defects with flow-based sampling.

Computing quantum entanglement with machine learning

TL;DR

The paper tackles the challenge of computing entanglement measures in lattice field theories, where Renyi entropies are expressed via the replica trick as and universal finite contributions are captured by the entropic c-function . It introduces a defect-focused Normalizing Flow framework that acts on a localized region near the replica-cut endpoint to transform a simple prior into the replicated ensemble, enabling direct estimation of partition-function ratios. Across a -dimensional theory at criticality, the approach yields significant efficiency gains and high-precision estimates of on large lattices, outperforming traditional Monte Carlo methods. The work presents a transferable paradigm for entanglement-related observables in lattice field theories, compatible with equivariant-flow ideas and extendable to gauge theories and other observables.

Abstract

Entanglement calculations in quantum field theories are extremely challenging and typically rely on the replica trick, where the problem is rephrased in a study of defects. We demonstrate that the use of deep generative models drastically outperforms standard Monte Carlo algorithms. Remarkably, such a machine-learning method enables high-precision estimates of Rényi entropies in three dimensions for very large lattices. Moreover, we propose a new paradigm for studying lattice defects with flow-based sampling.

Paper Structure

This paper contains 6 sections, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The replica trick
  3. Flow-based sampling
  4. Defect coupling layers
  5. Numerical results
  6. Conclusions

Figures (3)

  • Figure 1: Left panel: system of two independent lattices. Right panel: the two replicas are coupled through a set of links along the subsystem $A$ at a fixed Euclidean time.
  • Figure 2: $(1 + 1)$-dimensional replicated lattice ($\tau$ is the Euclidean-time direction). Purple links connect different replicas. The lattice is divided in three parts in the NF: the environment (black sites), which does not enter the coupling layer; frozen sites (empty cyan circles), that are inputs of the neural network; active sites (orange diamonds), which are transformed by the NF.
  • Figure 3: Defect-coupling-layer-based NFs compared with other flow-based samplers. Left panel: total simulation time to reach a given accuracy. Right panel: entropic c-function computed with different samplers at fixed statistics.