Observable-enriched entanglement

We introduce methods of characterizing entanglement on the example of the quantum skyrmion Hall effect, 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 entanglement spectra, unveiling 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 the study of quantum systems and the fundamental generalization of the interpretation and treatment of spin within the framework of the quantum skyrmion Hall effect, concepts of observable-enriched entanglement introduced here are broadly applicable to myriad problems of quantum systems.