摘要
We study genuine multi-entropy as a diagnostic of multipartite entanglement in the toric code, which provides a controlled setting for probing multipartite structures in topologically ordered states. Our main question is whether genuine multi-entropy captures information that is not reducible to conventional lower-party entropic data, such as topological entanglement entropy. We first analyze toric-code ground states that admit a stabilizer-state description, where the relevant quantities can be evaluated exactly. In this sector, genuine multi-entropy reflects the topological structure and symmetries of the toric code, while exhibiting highly constrained relations to lower-party multi-entropies. We conjecture that, for stabilizer states andq≥4 , theq -partite genuine multi-entropy at replica indexn<qcollapses to a linear combination of multi-entropies involving at mostq-2parties. We establish this pattern explicitly forq=4in the toric code stabilizer sector: forn=2,3 , the genuine multi-entropy is proportional to the tripartite informationI₃and, for the Kitaev–Preskill partition, contains no independent genuine four-partite information beyond that captured by the topological entanglement entropy. Atn=4 , however, this reduction breaks down: the genuine multi-entropy is no longer proportional toI₃ , but remains a topological invariant of the toric-code stabilizer ground states. For generic non-stabilizer superpositions within the ground-state manifold and for coherent superpositions of local excitations, the low- nreduction also fails. These results show that genuine multi-entropy probes multipartite entanglement structure beyond the tripartite information, and hence beyond the topological entanglement entropy in the Kitaev–Preskill partition, whereas for stabilizer states at low replica index it reduces to lower-partite entropic data.