Topological correlation: anyonic states cannot be determined by local operations and classical communication

Anyonic system not only has potential applications in the construction of topological quantum computer, but also presents a unique property known as topological entanglement entropy in quantum many-body systems. How to understand topological entanglement entropy is one of the most concerned problems for physicists. For an anyonic bipartite system, we define an operational measure of topological correlation based on the principle of maximal entropy, where the topological correlation is the information that cannot be accessed by local operations constrained by anyonic superselection rules and classical communication. This measure can be extended to measure non-local resources of other compound quantum systems in the presence of superselection rules. For a given anyonic bipartite state with maximal rank, we prove that its topological correlation is equal to its entropy of anyonic charge entanglement that has been shown in the literature to be able to derive topological entanglement entropy. This measure provides a more refined classification of correlations in a multipartite system with superselection rules and an illuminating approach to topological phase classification.