Measuring resources in quantum spin liquids with valence bond shadows

The question of whether a given candidate material really is a quantum spin liquid has plagued the field of frustrated magnetism for decades. Thanks to modern resource theories, we can now formalize these questions, turning them into a measurement. Yet, current measurement techniques, such as shadow tomography and classical shadows, are ill-equipped to measure resources. What is needed is a subtle kind of shadow tomography method, one that is neither easily invertible nor non-invertible, one that is ill-conditioned yet for which resources are efficiently estimated. We formalize the coherence resource theory of valence bond singlets and present valence bond shadows, an iterative random walk procedure that collapses a pure state in the total spin 0 sector onto a single valence bond singlet state (labeled by perfect matchings) with each shot. We construct two coherence witnesses using the maximum weight perfect matching algorithm and, since each shot contains finite entanglement, a lower bound on entanglement. Our work simultaneously lays the foundation to establish whether a quantum spin liquid is coherent and presents a novel shadow tomography algorithm capable of efficiently measuring resources.