  We introduce methods of characterizing entanglement, in which entanglement
measures are enriched by the matrix representations of operators for
observables. These observable operator matrix representations can enrich the
partial trace over subsets of a system's degrees of freedom, yielding reduced
density matrices useful in computing various measures of entanglement, which
also preserve the observable expectation value. We focus here on applying these
methods to compute observable-enriched entanglement spectra, unveiling new
bulk-boundary correspondences of canonical four-band models for topological
skyrmion phases and their connection to simpler forms of bulk-boundary
correspondence. Given the fundamental roles entanglement signatures and
observables play in study of quantum many body systems, observable-enriched
entanglement is broadly applicable to myriad problems of quantum mechanics.
