Sampling reduced density matrix to extract fine levels of entanglement spectrum and restore entanglement Hamiltonian

TL;DR

This paper introduces a quantum Monte Carlo method to accurately extract the reduced density matrix, enabling detailed analysis of entanglement spectra and the reconstruction of entanglement Hamiltonians in large quantum systems.

Contribution

The authors develop a low-barrier quantum Monte Carlo scheme for precise RDM extraction and demonstrate its effectiveness in revealing fine entanglement spectra and restoring entanglement Hamiltonians.

Findings

Successful extraction of entanglement spectrum for 1D and 2D models

Reconstruction of entanglement Hamiltonian from RDM data

Application to large system sizes with high accuracy

Abstract

The reduced density matrix (RDM) plays a key role in quantum entanglement and measurement, as it allows the extraction of almost all physical quantities related to the reduced degrees of freedom. However, restricted by the degrees of freedom in the environment, the total system size is often limited, let alone the subsystem. To address this challenge, we propose a quantum Monte Carlo scheme with a low technical barrier, enabling precise extraction of the RDM. To demonstrate the power of the method, we present the fine levels of the entanglement spectrum (ES), which is the logarithmic eigenvalues of the RDM. We clearly show the ES for a $1$D ladder with a long entangled boundary, and that for the $2$D Heisenberg model with a tower of states. Furthermore, we put forward an efficient way to restore the entanglement Hamiltonian in operator-form from the sampled RDM data. Our simulation results, utilizing unprecedentedly large system sizes, establish a practical computational framework for determining entanglement quantities based on the RDM, such as the ES, particularly in scenarios where the environment has a huge number of degrees of freedom.