Entanglement, as studied in quantum information science, and nonlocal quantum correlations, as studiedin condensed matter physics, are fundamentally akin to each other. However, their relationship is often hard toquantify due to the lack of a general approach to study both on the same footing. In particular, while entanglementand nonlocal correlations are properties of states, both arise from symmetries of global operators that commutewith the system Hamiltonian. Here, we introduce a framework for completely classifying the local and nonlocalproperties of all such global operators, given the Hamiltonian and a bipartitioning of the system. This frameworkis limited to descriptions based on stabilizer quantum codes, but may be generalized. We illustrate the use of thisframework to study entanglement and nonlocal correlations by analyzing global symmetries in topological order,distribution of entanglement, and entanglement entropy.
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