  The lack of methods to experimentally detect and quantify entanglement in
quantum matter impedes our ability to identify materials hosting highly
entangled phases, such as quantum spin liquids. We thus investigate the
feasibility of using inelastic neutron scattering (INS) to implement a
model-independent measurement protocol for entanglement based on three
entanglement witnesses: one-tangle, two-tangle, and quantum Fisher information
(QFI). We perform high-resolution INS measurements on Cs$_2$CoCl$_4$, a close
realization of the $S=1/2$ transverse-field XXZ spin chain, where we can
control entanglement using the magnetic field, and compare with density-matrix
renormalization group calculations for validation. The three witnesses allow us
to infer entanglement properties and make deductions about the quantum state in
the material. We find QFI to be a particularly robust experimental probe of
entanglement, whereas the one- and two-tangles require more careful analysis.
Our results lay the foundation for a general entanglement detection protocol
for quantum spin systems.
