Framework for classifying logical operators in stabilizer codes

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