Complex Hadamard Matrices - Quantum Symmetries, Equivalence and Non-Local Games

@misc{brannan2026complex,
  title = {{Complex Hadamard Matrices - Quantum Symmetries, Equivalence and Non-Local Games}},
  author = {Michael Brannan and Daniel Gromada and Roberto Hernández Palomares and Nicholas Priebe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.14858},
  eprint = {2608.14858},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14858},
}

TY  - UNPB
TI  - Complex Hadamard Matrices - Quantum Symmetries, Equivalence and Non-Local Games
AU  - Michael Brannan
AU  - Daniel Gromada
AU  - Roberto Hernández Palomares
AU  - Nicholas Priebe
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2608.14858
UR  - https://arxiv.org/abs/2608.14858
AB  - We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of s-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not s-quantum equivalent for any choice of s. We also show that s-quantum equivalence is operationally modeled by a synchronous non-local ``Hadamard equivalence'' game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak--Sołtan--Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.
ER  -