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PreprintarXiv licencearXiv

Multiqubit orthogonal product bases

  • Yvkai Zhao
  • Lin Chen
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Abstract

We use edge-colored complete multigraphs to study complete orthogonal product bases (OPBs) in n-qubit systems. We prove that two OPBs are equivalent if and only if their associated multigraphs are isomorphic, thereby reducing OPB classification to graph isomorphism. Within this framework, we establish the upper bound v≤ 2n−1 on the number of variables of an n-qubit OPB. We also derive binoman−1+12≤ an≤ B2n−1n for the number an of equivalence classes of n-qubit OPBs, where Bm denotes the number of partitions of an m-element set. These bounds imply the asymptotic behavior an=22n+o(n). For every OPB, the connectivity pattern of its color layers characterizes local irreducibility, which in turn implies indistinguishability by finite-round local operations and classical communication (LOCC); the existence of a complete color-splitting tree characterizes perfect distinguishability by finite-round LOCC. Finally, we give an algorithm for testing OPB equivalence and a recursive graph algorithm that constructs a finite-round LOCC protocol whenever such perfect discrimination is possible.

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Cite

BibTeXRIS
BibTeX
@misc{zhao2026multiqubit,
  title = {{Multiqubit orthogonal product bases}},
  author = {Yvkai Zhao and Lin Chen},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.18421},
  eprint = {2608.18421},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.18421},
}
RIS
TY  - UNPB
TI  - Multiqubit orthogonal product bases
AU  - Yvkai Zhao
AU  - Lin Chen
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.18421
UR  - https://arxiv.org/abs/2608.18421
AB  - We use edge-colored complete multigraphs to study complete orthogonal product bases (OPBs) in n-qubit systems. We prove that two OPBs are equivalent if and only if their associated multigraphs are isomorphic, thereby reducing OPB classification to graph isomorphism. Within this framework, we establish the upper bound v≤ 2^n−1 on the number of variables of an n-qubit OPB. We also derive binoma_n−1+12≤ a_n≤ B_2^n−1^n for the number a_n of equivalence classes of n-qubit OPBs, where B_m denotes the number of partitions of an m-element set. These bounds imply the asymptotic behavior a_n=2^2^n+o(n). For every OPB, the connectivity pattern of its color layers characterizes local irreducibility, which in turn implies indistinguishability by finite-round local operations and classical communication (LOCC); the existence of a complete color-splitting tree characterizes perfect distinguishability by finite-round LOCC. Finally, we give an algorithm for testing OPB equivalence and a recursive graph algorithm that constructs a finite-round LOCC protocol whenever such perfect discrimination is possible.
ER  -