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\bibitem [{\citenamefont {Kaubruegger}\ \emph {et~al.}(2023)\citenamefont
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  \BibitemOpen
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  {Vasilyev}},\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
  {Zoller}},\ }\bibfield  {title} {\bibinfo {title} {Optimal and variational
  multiparameter quantum metrology and vector-field sensing},\ }\href
  {https://doi.org/10.1103/PRXQuantum.4.020333} {\bibfield  {journal} {\bibinfo
   {journal} {PRX Quantum}\ }\textbf {\bibinfo {volume} {4}},\ \bibinfo {pages}
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  {https://arxiv.org/abs/2302.07785} {arXiv:2302.07785} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Helstrom}(1976)}]{Helstrom1976}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {C.~W.}\ \bibnamefont
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\bibitem [{\citenamefont {Holevo}(2011)}]{Holevo2011}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.~S.}\ \bibnamefont
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  della Normale},\ \bibinfo {address} {Pisa},\ \bibinfo {year}
  {2011})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ragy}\ \emph {et~al.}(2016)\citenamefont {Ragy},
  \citenamefont {Jarzyna},\ and\ \citenamefont
  {Demkowicz-Dobrza{\'n}ski}}]{Ragy2016}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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  {Demkowicz-Dobrza{\'n}ski}},\ }\bibfield  {title} {\bibinfo {title}
  {Compatibility in multiparameter quantum metrology},\ }\href
  {https://doi.org/10.1103/PhysRevA.94.052108} {\bibfield  {journal} {\bibinfo
  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo
  {pages} {052108} (\bibinfo {year} {2016})},\ \Eprint
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\bibitem [{\citenamefont {Demkowicz-Dobrza{\'n}ski}\ \emph
  {et~al.}(2020)\citenamefont {Demkowicz-Dobrza{\'n}ski}, \citenamefont
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  \BibitemOpen
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  {https://doi.org/10.1088/1751-8121/ab8ef3} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {Nagaoka}(2005)}]{Nagaoka1989}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
  {Nagaoka}},\ }\bibfield  {title} {\bibinfo {title} {A new approach to
  {Cram{\'e}r--Rao} bounds for quantum state estimation},\ }in\ \href
  {https://doi.org/10.1142/9789812563071_0009} {\emph {\bibinfo {booktitle}
  {Asymptotic Theory of Quantum Statistical Inference: Selected Papers}}},\
  \bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
  {M.}~\bibnamefont {Hayashi}}}\ (\bibinfo  {publisher} {World Scientific},\
  \bibinfo {address} {Singapore},\ \bibinfo {year} {2005})\ pp.\ \bibinfo
  {pages} {100--112},\ \bibinfo {note} {reprinted from IEICE Tech. Rep. IT89-42
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\bibitem [{\citenamefont {Nagaoka}(1991)}]{Nagaoka1991}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
  {Nagaoka}},\ }\bibfield  {title} {\bibinfo {title} {A generalization of the
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  quantum estimation theory},\ }\href {https://doi.org/10.11540/jsiamt.1.4_305}
  {\bibfield  {journal} {\bibinfo  {journal} {Trans. Jpn. Soc. Ind. Appl.
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\bibitem [{\citenamefont {Hayashi}(1999)}]{Hayashi1999}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
  {Hayashi}},\ }\bibfield  {title} {\bibinfo {title} {On simultaneous
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  {\bibinfo {booktitle} {Development of Infinite-Dimensional Noncommutative
  Analysis}}},\ \bibinfo {series} {RIMS K{\^o}ky{\^u}roku}, Vol.\ \bibinfo
  {volume} {1099},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
  {\bibfnamefont {A.}~\bibnamefont {Hora}}}\ (\bibinfo  {publisher} {Research
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  {Kyoto},\ \bibinfo {year} {1999})\ pp.\ \bibinfo {pages} {96--118},\ \bibinfo
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\bibitem [{\citenamefont {Conlon}\ \emph {et~al.}(2021)\citenamefont {Conlon},
  \citenamefont {Suzuki}, \citenamefont {Lam},\ and\ \citenamefont
  {Assad}}]{Conlon2021}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {L.~O.}\ \bibnamefont
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  {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Assad}},\ }\bibfield  {title}
  {\bibinfo {title} {Efficient computation of the {Nagaoka}--{Hayashi} bound
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  {https://doi.org/10.1038/s41534-021-00414-1} {\bibfield  {journal} {\bibinfo
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\bibitem [{\citenamefont {Suzuki}(2024)}]{Suzuki2024}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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  problem},\ }\href {https://doi.org/10.1587/transfun.2023TAP0014} {\bibfield
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\bibitem [{\citenamefont {Conlon}\ \emph {et~al.}(2025)\citenamefont {Conlon},
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {L.~O.}\ \bibnamefont
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  \bibinfo {author} {\bibfnamefont {P.~K.}\ \bibnamefont {Lam}},\ and\ \bibinfo
  {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Assad}},\ }\bibfield  {title}
  {\bibinfo {title} {Role of the extended {Hilbert} space in the attainability
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  }\href {https://doi.org/10.1016/j.physleta.2025.130445} {\bibfield  {journal}
  {\bibinfo  {journal} {Phys. Lett. A}\ }\textbf {\bibinfo {volume} {542}},\
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\bibitem [{\citenamefont {Personick}(1971)}]{Personick1971}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
  {Personick}},\ }\bibfield  {title} {\bibinfo {title} {Application of quantum
  estimation theory to analog communication over quantum channels},\ }\href
  {https://doi.org/10.1109/TIT.1971.1054643} {\bibfield  {journal} {\bibinfo
  {journal} {IEEE Trans. Inf. Theory}\ }\textbf {\bibinfo {volume} {17}},\
  \bibinfo {pages} {240} (\bibinfo {year} {1971})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Rubio}\ and\ \citenamefont
  {Dunningham}(2020)}]{Rubio2020}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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  multiparameter quantum metrology with limited data},\ }\href
  {https://doi.org/10.1103/PhysRevA.101.032114} {\bibfield  {journal} {\bibinfo
   {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo
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  {https://arxiv.org/abs/1906.04123} {arXiv:1906.04123} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Albarelli}\ \emph {et~al.}(2026)\citenamefont
  {Albarelli}, \citenamefont {Branford},\ and\ \citenamefont
  {Rubio}}]{Albarelli2026}%
  \BibitemOpen
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  {Rubio}},\ }\bibfield  {title} {\bibinfo {title} {Measurement incompatibility
  in {Bayesian} multiparameter quantum estimation},\ }\href
  {https://doi.org/10.22331/q-2026-08-13-2192} {\bibfield  {journal} {\bibinfo
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  {2192} (\bibinfo {year} {2026})},\ \Eprint {https://arxiv.org/abs/2511.16645}
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\bibitem [{\citenamefont {Zhao}\ \emph {et~al.}(2026)\citenamefont {Zhao},
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {K.-H.}\ \bibnamefont
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  and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Suzuki}},\
  }\href@noop {} {\bibinfo {title} {{Bayesian} {Gill--Massar} bound: An
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  {2026}),\ \Eprint {https://arxiv.org/abs/2607.07031} {arXiv:2607.07031}
  \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Bia}\ and\ \citenamefont {Gagatsos}(2026)}]{Bia2026}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
  {Bia}}\ and\ \bibinfo {author} {\bibfnamefont {C.~N.}\ \bibnamefont
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  {Nagaoka}--{Hayashi} bound within numerical precision for the depolarization
  {SU}(2) rotation channel}} (\bibinfo {year} {2026}),\ \bibinfo {note} {phys.
  Rev. A (to be published), doi:10.1103/p7vp-vyxf},\ \Eprint
  {https://arxiv.org/abs/2607.15398} {arXiv:2607.15398} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Yamagata}(2026)}]{Yamagata2026}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
  {Yamagata}},\ }\href@noop {} {\bibinfo {title} {Sufficient support size of
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  {https://arxiv.org/abs/2604.21323} {arXiv:2604.21323} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Holevo}(1973)}]{Holevo1973}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.~S.}\ \bibnamefont
  {Holevo}},\ }\bibfield  {title} {\bibinfo {title} {Statistical decision
  theory for quantum systems},\ }\href
  {https://doi.org/10.1016/0047-259X(73)90028-6} {\bibfield  {journal}
  {\bibinfo  {journal} {J. Multivariate Anal.}\ }\textbf {\bibinfo {volume}
  {3}},\ \bibinfo {pages} {337} (\bibinfo {year} {1973})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Hayashi}\ and\ \citenamefont
  {Ouyang}(2023)}]{Hayashi2023}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
  {Hayashi}}\ and\ \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
  {Ouyang}},\ }\bibfield  {title} {\bibinfo {title} {Tight {Cram{\'e}r--Rao}
  type bounds for multiparameter quantum metrology through conic programming},\
  }\href {https://doi.org/10.22331/q-2023-08-29-1094} {\bibfield  {journal}
  {\bibinfo  {journal} {Quantum}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo
  {pages} {1094} (\bibinfo {year} {2023})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Terhal}(2000)}]{Terhal2000}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {B.~M.}\ \bibnamefont
  {Terhal}},\ }\bibfield  {title} {\bibinfo {title} {{Bell} inequalities and
  the separability criterion},\ }\href
  {https://doi.org/10.1016/S0375-9601(00)00401-1} {\bibfield  {journal}
  {\bibinfo  {journal} {Phys. Lett. A}\ }\textbf {\bibinfo {volume} {271}},\
  \bibinfo {pages} {319} (\bibinfo {year} {2000})},\ \Eprint
  {https://arxiv.org/abs/quant-ph/9911057} {arXiv:quant-ph/9911057}
  \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Eldar}\ and\ \citenamefont
  {Forney}(2002)}]{Eldar2002}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Y.~C.}\ \bibnamefont
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  {Optimal tight frames and quantum measurement},\ }\href
  {https://doi.org/10.1109/18.985949} {\bibfield  {journal} {\bibinfo
  {journal} {IEEE Trans. Inf. Theory}\ }\textbf {\bibinfo {volume} {48}},\
  \bibinfo {pages} {599} (\bibinfo {year} {2002})},\ \Eprint
  {https://arxiv.org/abs/quant-ph/0106070} {arXiv:quant-ph/0106070}
  \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Chiribella}\ and\ \citenamefont
  {D'Ariano}(2006)}]{Chiribella2006}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
  {Chiribella}}\ and\ \bibinfo {author} {\bibfnamefont {G.~M.}\ \bibnamefont
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  measurements},\ }\href {https://doi.org/10.1063/1.2349481} {\bibfield
  {journal} {\bibinfo  {journal} {J. Math. Phys.}\ }\textbf {\bibinfo {volume}
  {47}},\ \bibinfo {pages} {092107} (\bibinfo {year} {2006})},\ \Eprint
  {https://arxiv.org/abs/quant-ph/0603168} {arXiv:quant-ph/0603168}
  \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Calsamiglia}\ \emph {et~al.}(2009)\citenamefont
  {Calsamiglia}, \citenamefont {Aspachs}, \citenamefont {Mu{\~n}oz-Tapia},\
  and\ \citenamefont {Bagan}}]{Calsamiglia2009}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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  {Mu{\~n}oz-Tapia}},\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
  {Bagan}},\ }\bibfield  {title} {\bibinfo {title} {Phase-covariant quantum
  benchmarks},\ }\href {https://doi.org/10.1103/PhysRevA.79.050301} {\bibfield
  {journal} {\bibinfo  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
  {79}},\ \bibinfo {pages} {050301(R)} (\bibinfo {year} {2009})},\ \Eprint
  {https://arxiv.org/abs/0807.5126} {arXiv:0807.5126} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Kaliuzhnyi-Verbovetskyi}\ \emph
  {et~al.}(2009)\citenamefont {Kaliuzhnyi-Verbovetskyi}, \citenamefont
  {Spitkovsky},\ and\ \citenamefont {Woerdeman}}]{Kaliuzhnyi2009}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {D.~S.}\ \bibnamefont
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  \bibnamefont {Spitkovsky}},\ and\ \bibinfo {author} {\bibfnamefont {H.~J.}\
  \bibnamefont {Woerdeman}},\ }\bibfield  {title} {\bibinfo {title} {Matrices
  with normal defect one},\ }\href {https://doi.org/10.7153/oam-03-24}
  {\bibfield  {journal} {\bibinfo  {journal} {Oper. Matrices}\ }\textbf
  {\bibinfo {volume} {3}},\ \bibinfo {pages} {401} (\bibinfo {year} {2009})},\
  \Eprint {https://arxiv.org/abs/0903.0090} {arXiv:0903.0090} \BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {Wasson}\ and\ \citenamefont
  {Woerdeman}(2013)}]{Wasson2013}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {R.~D.}\ \bibnamefont
  {Wasson}}\ and\ \bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont
  {Woerdeman}},\ }\bibfield  {title} {\bibinfo {title} {The normal defect of
  some classes of matrices},\ }\href
  {https://doi.org/10.1016/j.laa.2012.12.046} {\bibfield  {journal} {\bibinfo
  {journal} {Linear Algebra Appl.}\ }\textbf {\bibinfo {volume} {438}},\
  \bibinfo {pages} {3530} (\bibinfo {year} {2013})},\ \Eprint
  {https://arxiv.org/abs/1211.4145} {arXiv:1211.4145} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Bierly}(2026)}]{Bierly2026}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
  {Bierly}},\ }\href@noop {} {\bibinfo {title} {The normal {Procrustes}
  problem: A {Riemannian} optimization approach}} (\bibinfo {year} {2026}),\
  \Eprint {https://arxiv.org/abs/2608.19513} {arXiv:2608.19513} \BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {Fabre}\ \emph {et~al.}(2023)\citenamefont {Fabre},
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  \BibitemOpen
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  {author} {\bibfnamefont {L.~L.}\ \bibnamefont {S{\'a}nchez-Soto}},\
  }\bibfield  {title} {\bibinfo {title} {Local sampling of the {SU(1,1)}
  {Wigner} function},\ }\href {https://doi.org/10.1116/5.0134784} {\bibfield
  {journal} {\bibinfo  {journal} {AVS Quantum Sci.}\ }\textbf {\bibinfo
  {volume} {5}},\ \bibinfo {pages} {014404} (\bibinfo {year}
  {2023})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Wang}(2026)}]{Wang2026preliminary}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Q.}~\bibnamefont
  {Wang}},\ }\href {https://doi.org/10.5281/zenodo.22083724} {\bibinfo {title}
  {Nonorthogonal {Nagaoka} equality supports under {Bayesian} weak
  commutativity}} (\bibinfo {year} {2026}),\ \bibinfo {note} {{Zenodo}
  preprint, doi:10.5281/zenodo.22083724}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Law}\ and\ \citenamefont
  {Bern{\'a}d}(2026)}]{Law2026}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {C.~K.~D.}\
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  \bibnamefont {Bern{\'a}d}},\ }\href@noop {} {\bibinfo {title} {{Bayesian}
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  (\bibinfo {year} {2026}),\ \Eprint {https://arxiv.org/abs/2608.10230}
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  {Mart{\'i}nez}, \citenamefont {Pereira}, \citenamefont {Sawada},
  \citenamefont {Gonz{\'a}lez}, \citenamefont {Cari{\~n}e}, \citenamefont
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  {G.}~\bibnamefont {Lima}},\ }\bibfield  {title} {\bibinfo {title}
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  {https://doi.org/10.1103/gpfl-wdgg} {\bibfield  {journal} {\bibinfo
  {journal} {Phys. Rev. Res.}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo
  {pages} {033119} (\bibinfo {year} {2025})},\ \Eprint
  {https://arxiv.org/abs/2412.14915} {arXiv:2412.14915} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Glashoff}\ and\ \citenamefont
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  their asymptotic metric properties and relation to approximate joint
  diagonalization},\ }\href {https://doi.org/10.1016/j.laa.2013.07.013}
  {\bibfield  {journal} {\bibinfo  {journal} {Linear Algebra Appl.}\ }\textbf
  {\bibinfo {volume} {439}},\ \bibinfo {pages} {2503} (\bibinfo {year}
  {2013})},\ \Eprint {https://arxiv.org/abs/1305.2135} {arXiv:1305.2135}
  \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Noferini}\ and\ \citenamefont
  {Zhukov}(2026)}]{Noferini2026}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont
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  again using {Riemannian} optimization}} (\bibinfo {year} {2026}),\ \Eprint
  {https://arxiv.org/abs/2608.28545} {arXiv:2608.28545} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Lasserre}(2001)}]{Lasserre2001}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont
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  {https://doi.org/10.1137/S1052623400366802} {\bibfield  {journal} {\bibinfo
  {journal} {SIAM J. Optim.}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo
  {pages} {796} (\bibinfo {year} {2001})}\BibitemShut {NoStop}%
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