Bounding Entanglement Entropy Using Zeros of Local Correlation Matrices

TL;DR

This paper introduces a protocol to estimate bounds on entanglement entropy in quantum many-body states using local measurements of correlation matrices, linking zero eigenvalues to subsystem entanglement properties.

Contribution

The authors propose a novel method to bound entanglement entropy through local correlation matrices and eigenvalue analysis, enabling experimental investigation of quantum states.

Findings

EE bounds are related to ground-state degeneracy of an auxiliary Hamiltonian.

Including nonzero eigenvalues allows bounding EE by thermal entropy.

Protocol is applicable to quantum simulators for exotic states.

Abstract

Correlation functions and entanglement are two different aspects to characterize quantum many-body states. While many correlation functions are experimentally accessible, entanglement entropy (EE), the simplest characterization of quantum entanglement, is usually difficult to measure. In this Letter, we propose a protocol to bound EE by local measurements. This protocol utilizes local correlation matrices and focuses on their (approximate) zero eigenvalues. Given a quantum state, each (approximate) zero eigenvalue can be used to define a set of local projection operators. An auxiliary Hamiltonian can then be constructed by summing these projectors. When the construction only involves projectors of zero eigenvalues, we prove the EE of a subsystem is bounded by the ground-state degeneracy of the auxiliary Hamiltonian on this subsystem. When projectors from nonzero eigenvalues are included, we show the EE can be bounded by a thermal entropy of the subsystem. Our protocol can be applied experimentally to investigate exotic quantum many-body states prepared in quantum simulators.