  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.
