  Although entanglement is both a central ingredient in our understanding of
quantum many-body systems and an essential resource for quantum technologies,
we only have a limited ability to quantify entanglement in real quantum
materials. Thus far, entanglement metrology in quantum materials has been
limited to measurements involving Hermitian operators, such as the detection of
spin entanglement using inelastic neutron scattering. Here, we devise a method
to extract the quantum Fisher information (QFI) from non-Hermitian operators
and formulate an entanglement witness for resonant inelastic x-ray scattering
(RIXS). Our approach is then applied to the model iridate dimer system
Ba$_3$CeIr$_2$O$_9$ and used to directly test for entanglement of the
electronic orbitals between neighboring Ir sites. We find that entanglement is
challenging to detect under standard conditions, but that it could be achieved
by analyzing the outgoing x-ray polarization or via specific choices of
momentum and energy. Our protocol provides a new handle for entanglement
detection, which offers routes to related types of entanglement witness (such
as orbitally-resolved measurements) and to the generalization to
out-of-equilibrium settings accessed in ultrafast settings.
