Genuine Multi-Entropy in the Toric Code

@misc{akella2026genuine,
  title = {{Genuine Multi-Entropy in the Toric Code}},
  author = {Sriram Akella and Norihiro Iizuka and Akihiro Miyata},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.06050},
  eprint = {2607.06050},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.06050},
}

TY  - UNPB
TI  - Genuine Multi-Entropy in the Toric Code
AU  - Sriram Akella
AU  - Norihiro Iizuka
AU  - Akihiro Miyata
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2607.06050
UR  - https://arxiv.org/abs/2607.06050
AB  - 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. We conjecture that, for stabilizer states and q≥4, the q-partite genuine multi-entropy at replica index n<q collapses to a linear combination of multi-entropies involving at most q−2 parties. We establish this pattern explicitly for q=4 in the toric code stabilizer sector: for n=2,3, the genuine multi-entropy is proportional to the tripartite information I_3 and, for the Kitaev-Preskill partition, contains no independent genuine four-partite information beyond that captured by the topological entanglement entropy. At n=4, however, this reduction breaks down: the genuine multi-entropy is no longer proportional to I_3, but remains a topological invariant of the toric-code stabilizer ground states under Kitaev-Preskill-preserving deformations that do not change whether the numbers of vertex- and plaquette-type junctions coincide. For generic non-stabilizer superpositions within the ground-state manifold and for coherent superpositions of local excitations, the low-n reduction 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.
ER  -