Bounding Entanglement Entropy Using Zeros of Local Correlation Matrices

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 the subsystem EE by local measurements. This protocol utilizes the local correlation matrix and focuses on its (approximate) zero eigenvalues. Given a quantum state, each (approximate) zero eigenvalue can be used to define a local projection operator. 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 of small 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 a quantum simulator.