@article{aschauer,
  author = {Aschauer, Hans and Calsamiglia, John and Hein, M. and Briegel, Hans J.},
  title = {Local invariants for multi-partite entangled states allowing for a simple entanglement criterion},
  journal = {Quantum Information and Computation},
  volume = {4},
  number = {5},
  pages = {383--395},
  year = {2004},
  doi = {10.26421/QIC4.5-4}
}
 
@article{peres,
  author = {Peres, Asher},
  title = {Separability Criterion for Density Matrices},
  journal = {Physical Review Letters},
  volume = {77},
  number = {8},
  pages = {1413--1415},
  year = {1996},
  doi = {10.1103/PhysRevLett.77.1413}
}
 
@article{horodeckiPPT,
  author = {Horodecki, Micha{\l} and Horodecki, Pawe{\l} and Horodecki, Ryszard},
  title = {Separability of mixed states: necessary and sufficient conditions},
  journal = {Physics Letters A},
  volume = {223},
  number = {1--2},
  pages = {1--8},
  year = {1996},
  doi = {10.1016/S0375-9601(96)00706-2}
}
 
@article{bennettUPB,
  author = {Bennett, Charles H. and DiVincenzo, David P. and Mor, Tal and Shor, Peter W. and Smolin, John A. and Terhal, Barbara M.},
  title = {Unextendible Product Bases and Bound Entanglement},
  journal = {Physical Review Letters},
  volume = {82},
  number = {26},
  pages = {5385--5388},
  year = {1999},
  doi = {10.1103/PhysRevLett.82.5385}
}

@article{chenwu,
  author = {Chen, Kai and Wu, Ling-An},
  title = {A matrix realignment method for recognizing entanglement},
  journal = {Quantum Information and Computation},
  volume = {3},
  number = {3},
  pages = {193--202},
  year = {2003}
}

@article{devicente,
  author = {de Vicente, Julio I.},
  title = {Separability criteria based on the Bloch representation of density matrices},
  journal = {Quantum Information and Computation},
  volume = {7},
  number = {7},
  pages = {624--638},
  year = {2007}
}

@article{hassanjoag,
  author = {Hassan, Ali S. M. and Joag, Pramod S.},
  title = {Separability criterion for multipartite quantum states based on the Bloch representation of density matrices},
  journal = {Quantum Information and Computation},
  volume = {8},
  number = {8},
  pages = {773--790},
  year = {2008}
}

@article{devicentehuber,
  author = {de Vicente, Julio I. and Huber, Marcus},
  title = {Multipartite entanglement detection from correlation tensors},
  journal = {Physical Review A},
  volume = {84},
  number = {6},
  pages = {062306},
  year = {2011},
  doi = {10.1103/PhysRevA.84.062306}
}

@article{laskowski2011,
  author = {Laskowski, Wieslaw and Markiewicz, Marcin and Paterek, Tomasz and {\v{Z}}ukowski, Marek},
  title = {Correlation-tensor criteria for genuine multiqubit entanglement},
  journal = {Physical Review A},
  volume = {84},
  number = {6},
  pages = {062305},
  year = {2011},
  doi = {10.1103/PhysRevA.84.062305}
}

@article{klocklhuber2015,
  author = {Kl{\"o}ckl, Claude and Huber, Marcus},
  title = {Characterizing multipartite entanglement without shared reference frames},
  journal = {Physical Review A},
  volume = {91},
  number = {4},
  pages = {042339},
  year = {2015},
  doi = {10.1103/PhysRevA.91.042339}
}

@article{li2014,
  author = {Li, Ming and Wang, Jing and Fei, Shao-Ming and Li-Jost, Xianqing},
  title = {Quantum separability criteria for arbitrary-dimensional multipartite states},
  journal = {Physical Review A},
  volume = {89},
  number = {2},
  pages = {022325},
  year = {2014},
  doi = {10.1103/PhysRevA.89.022325}
}

@article{shen2016,
  author = {Shen, Shu-Qian and Yu, Juan and Li, Ming and Fei, Shao-Ming},
  title = {Improved Separability Criteria Based on Bloch Representation of Density Matrices},
  journal = {Scientific Reports},
  volume = {6},
  pages = {28850},
  year = {2016},
  doi = {10.1038/srep28850}
}

@article{sarbicki2020,
  author = {Sarbicki, Gniewomir and Scala, Giovanni and Chru{\'s}ci{\'n}ski, Dariusz},
  title = {Family of multipartite separability criteria based on a correlation tensor},
  journal = {Physical Review A},
  volume = {101},
  number = {1},
  pages = {012341},
  year = {2020},
  doi = {10.1103/PhysRevA.101.012341}
}

@article{zhao2020,
  author = {Zhao, Hui and Zhang, Mei-Ming and Jing, NaiHuan and Wang, Zhi-Xi},
  title = {Separability criteria based on Bloch representation of density matrices},
  journal = {Quantum Information Processing},
  volume = {19},
  number = {1},
  year = {2020},
  doi = {10.1007/s11128-019-2504-2}
}

@misc{jingzhang2023,
  author = {Jing, Naihuan and Zhang, Mingming},
  title = {Criteria of genuine multipartite entanglement based on correlation tensors},
  year = {2023},
  eprint = {2301.06463},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/2301.06463}
}

@misc{huang2024moments,
  author = {Huang, Xin and Jing, Naihuan},
  title = {Separability criteria based on the correlation tensor moments for arbitrary dimensional states},
  year = {2024},
  eprint = {2402.13162},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/2402.13162}
}

@misc{huang2024extended,
  author = {Huang, Xin and Zhang, Tianyi and Jing, Naihuan},
  title = {A unifying separability criterion based on extended correlation tensor},
  year = {2024},
  eprint = {2406.17230},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/2406.17230}
}

@misc{liyaoyangfei2025,
  author = {Li, Lei and Yao, Hongyang and Yang, Chao and Fei, Shao-Ming},
  title = {Separability criteria of quantum states based on generalized Bloch representation},
  year = {2025},
  eprint = {2510.24110},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/2510.24110}
}

@article{vollbrechtwerner2001,
  author = {Vollbrecht, Karl Gerd H. and Werner, Reinhard F.},
  title = {Entanglement measures under symmetry},
  journal = {Physical Review A},
  volume = {64},
  number = {6},
  pages = {062307},
  year = {2001},
  doi = {10.1103/PhysRevA.64.062307}
}

@misc{fattal2004stabilizer,
  author = {Fattal, David and Cubitt, Toby S. and Yamamoto, Yoshihisa and Bravyi, Sergey and Chuang, Isaac L.},
  title = {Entanglement in the stabilizer formalism},
  year = {2004},
  eprint = {quant-ph/0406168},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/quant-ph/0406168}
}

@article{tothguhne2005stabilizer,
  author = {T{\'o}th, G{\'e}za and G{\"u}hne, Otfried},
  title = {Entanglement detection in the stabilizer formalism},
  journal = {Physical Review A},
  volume = {72},
  number = {2},
  pages = {022340},
  year = {2005},
  doi = {10.1103/PhysRevA.72.022340},
  note = {arXiv:quant-ph/0501020}
}

@article{sharmachan2012spinadapted,
  author = {Sharma, Sandeep and Chan, Garnet Kin-Lic},
  title = {Spin-adapted density matrix renormalization group algorithms for quantum chemistry},
  journal = {The Journal of Chemical Physics},
  volume = {136},
  number = {12},
  pages = {124121},
  year = {2012},
  doi = {10.1063/1.3695642}
}

@article{menczer2024quarterpetaflops,
  author = {Menczer, Andor and van Damme, Maarten and Rask, Alan and Huntington, Lee and Hammond, Jeff and Xantheas, Sotiris S. and Ganahl, Martin and Legeza, {\"O}rs},
  title = {Parallel Implementation of the Density Matrix Renormalization Group Method Achieving a Quarter petaFLOPS Performance on a Single DGX-H100 GPU Node},
  journal = {Journal of Chemical Theory and Computation},
  volume = {20},
  number = {19},
  pages = {8397--8404},
  year = {2024},
  doi = {10.1021/acs.jctc.4c00903},
  note = {arXiv:2407.07411}
}

@article{huberklepmagronvolcic2022werner,
  author = {Huber, Felix and Klep, Igor and Magron, Victor and Vol{\v{c}}i{\v{c}}, Jurij},
  title = {Dimension-Free Entanglement Detection in Multipartite Werner States},
  journal = {Communications in Mathematical Physics},
  volume = {396},
  pages = {1051--1070},
  year = {2022},
  doi = {10.1007/s00220-022-04485-9},
  note = {arXiv:2108.08720}
}

@article{heinEisertBriegel2004,
  author = {Hein, M. and Eisert, J. and Briegel, H. J.},
  title = {Multiparty entanglement in graph states},
  journal = {Physical Review A},
  volume = {69},
  number = {6},
  pages = {062311},
  year = {2004},
  doi = {10.1103/PhysRevA.69.062311}
}

@article{huangkuengpreskill2020,
  author = {Huang, Hsin-Yuan and Kueng, Richard and Preskill, John},
  title = {Predicting many properties of a quantum system from very few measurements},
  journal = {Nature Physics},
  volume = {16},
  number = {10},
  pages = {1050--1057},
  year = {2020},
  doi = {10.1038/s41567-020-0932-7}
}

@article{delathauwer,
  author = {De Lathauwer, Lieven and De Moor, Bart and Vandewalle, Joos},
  title = {A multilinear singular value decomposition},
  journal = {SIAM Journal on Matrix Analysis and Applications},
  volume = {21},
  number = {4},
  pages = {1253--1278},
  year = {2000},
  doi = {10.1137/S0895479896305696}
}

@misc{appleby2009,
  author = {Appleby, D. M.},
  title = {Properties of the extended Clifford group with applications to SIC-POVMs and MUBs},
  year = {2009},
  eprint = {0909.5233},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  doi = {10.48550/arXiv.0909.5233},
  url = {https://arxiv.org/abs/0909.5233}
}

@misc{applebyBengtssonChaturvedi2007,
  author = {Appleby, D. M. and Bengtsson, Ingemar and Chaturvedi, Subramanyan},
  title = {Spectra of phase point operators in odd prime dimensions and the extended Clifford group},
  year = {2007},
  eprint = {0710.3013},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/0710.3013}
}

@article{vanDamHoward2011,
  author = {van Dam, Wim and Howard, Mark},
  title = {Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs},
  journal = {Physical Review A},
  volume = {84},
  number = {1},
  pages = {012117},
  year = {2011},
  doi = {10.1103/PhysRevA.84.012117}
}

@article{poppHiesmayr2024,
  author = {Popp, C. and Hiesmayr, B. C.},
  title = {Special features of the Weyl-Heisenberg Bell basis imply unusual entanglement structure of Bell-diagonal states},
  journal = {New Journal of Physics},
  volume = {26},
  number = {1},
  pages = {013039},
  year = {2024},
  doi = {10.1088/1367-2630/ad1f10}
}

@article{sutterPoppHiesmayr2026,
  author = {Sutter, T. C. and Popp, C. and Hiesmayr, B. C.},
  title = {Group-Theoretic Perspective on the {PPT} and Realignment Criteria in the Magic Simplex for Bipartite Qutrits},
  journal = {Physical Review A},
  volume = {113},
  number = {3},
  pages = {032440},
  year = {2026},
  doi = {10.1103/PhysRevA.113.032440}
}


@book{kato1995perturbation,
  author    = {Kato, Tosio},
  title     = {Perturbation Theory for Linear Operators},
  publisher = {Springer-Verlag},
  series    = {Classics in Mathematics},
  year      = {1995},
  note      = {Reprint of the 1980 second edition}
}


@article{lewissendov2005,
  author  = {Lewis, A. S. and Sendov, H. S.},
  title   = {Nonsmooth analysis of singular values. {P}art {I}: Theory},
  journal = {Set-Valued Analysis},
  volume  = {13},
  number  = {3},
  pages   = {213--241},
  year    = {2005},
  doi     = {10.1007/s11228-004-7197-7}
}
 
@article{zanardirasetti1997,
  author  = {Zanardi, Paolo and Rasetti, Mario},
  title   = {Noiseless Quantum Codes},
  journal = {Physical Review Letters},
  volume  = {79},
  number  = {17},
  pages   = {3306--3309},
  year    = {1997},
  doi     = {10.1103/PhysRevLett.79.3306}
}
 
@article{kribslaflammepoulin2005,
  author  = {Kribs, David and Laflamme, Raymond and Poulin, David},
  title   = {Unified and Generalized Approach to Quantum Error Correction},
  journal = {Physical Review Letters},
  volume  = {94},
  number  = {18},
  pages   = {180501},
  year    = {2005},
  doi     = {10.1103/PhysRevLett.94.180501}
}

@article{jungnitschmorodergühne2011,
  author = {Jungnitsch, Bastian and Moroder, Tobias and G{\"u}hne, Otfried},
  title = {Taming Multiparticle Entanglement},
  journal = {Physical Review Letters},
  volume = {106},
  number = {19},
  pages = {190502},
  year = {2011},
  doi = {10.1103/PhysRevLett.106.190502}
}

@article{guhneseevinck2010,
  author = {G{\"u}hne, Otfried and Seevinck, Michael},
  title = {Separability criteria for genuine multiparticle entanglement},
  journal = {New Journal of Physics},
  volume = {12},
  pages = {053002},
  year = {2010},
  doi = {10.1088/1367-2630/12/5/053002}
}

@article{novomorodergühne2013,
  author = {Novo, Leonardo and Moroder, Tobias and G{\"u}hne, Otfried},
  title = {Genuine multiparticle entanglement of permutationally invariant states},
  journal = {Physical Review A},
  volume = {88},
  pages = {012305},
  year = {2013},
  doi = {10.1103/PhysRevA.88.012305}
}

@article{tothetal2010pitomography,
  author = {T{\'o}th, G{\'e}za and Wieczorek, Witlef and Gross, David and Krischek, Roland and Schwemmer, Christian and Weinfurter, Harald},
  title = {Permutationally Invariant Quantum Tomography},
  journal = {Physical Review Letters},
  volume = {105},
  number = {25},
  pages = {250403},
  year = {2010},
  doi = {10.1103/PhysRevLett.105.250403}
}

@article{moroderhyllustoth2012,
  author = {Moroder, Tobias and Hyllus, Philipp and T{\'o}th, G{\'e}za and Schwemmer, Christian and Niggebaum, Alexander and Gaile, Sarah and G{\"u}hne, Otfried and Weinfurter, Harald},
  title = {Permutationally invariant state reconstruction},
  journal = {New Journal of Physics},
  volume = {14},
  pages = {105001},
  year = {2012},
  doi = {10.1088/1367-2630/14/10/105001}
}

@article{bergmanngühne2013,
  author = {Bergmann, Marcus and G{\"u}hne, Otfried},
  title = {Entanglement criteria for Dicke states},
  journal = {Journal of Physics A: Mathematical and Theoretical},
  volume = {46},
  pages = {385304},
  year = {2013},
  doi = {10.1088/1751-8113/46/38/385304}
}

@article{nevenetal2021,
  author = {Neven, Antoine and Carrasco, Jose and Vitale, Vittorio and Kokail, Christian and Elben, Andreas and Dalmonte, Marcello and Calabrese, Pasquale and Zoller, Peter and Vermersch, Benoit and Kueng, Richard and Kraus, Barbara},
  title = {Symmetry-resolved entanglement detection using partial transpose moments},
  journal = {npj Quantum Information},
  volume = {7},
  pages = {152},
  year = {2021},
  doi = {10.1038/s41534-021-00487-y}
}

@article{rathetal2023,
  author = {Rath, Aniket and Vitale, Vittorio and Murciano, Sara and Votto, Matteo and Dubail, Jerome and Kueng, Richard and Branciard, Cyril and Calabrese, Pasquale and Vermersch, Benoit},
  title = {Entanglement Barrier and its Symmetry Resolution: Theory and Experiment},
  journal = {PRX Quantum},
  volume = {4},
  pages = {010318},
  year = {2023},
  doi = {10.1103/PRXQuantum.4.010318}
}

@misc{tarabungahaug2025,
  author = {Tarabunga, Poetri Sonya and Haug, Tobias},
  title = {Quantifying mixed-state entanglement via partial transpose and realignment moments},
  year = {2025},
  eprint = {2507.13840},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/2507.13840}
}

@article{wyderkaketterer2023,
  author = {Wyderka, Nikolai and Ketterer, Andreas},
  title = {Probing the Geometry of Correlation Matrices with Randomized Measurements},
  journal = {PRX Quantum},
  volume = {4},
  pages = {020325},
  year = {2023},
  doi = {10.1103/PRXQuantum.4.020325}
}

@article{jungnitschmorodergühne2011graphstates,
  author = {Jungnitsch, Bastian and Moroder, Tobias and G{\"u}hne, Otfried},
  title = {Entanglement Witnesses for Graph States: General Theory and Examples},
  journal = {Physical Review A},
  volume = {84},
  pages = {032310},
  year = {2011},
  doi = {10.1103/PhysRevA.84.032310}
}

@article{kampermannguhnewilmottbruss2012,
  author = {Kampermann, Hermann and G{\"u}hne, Otfried and Wilmott, Colin and Bru{\ss}, Dagmar},
  title = {Algorithm for characterizing stochastic local operations and classical communication classes of multiparticle entanglement},
  journal = {Physical Review A},
  volume = {86},
  pages = {032307},
  year = {2012},
  doi = {10.1103/PhysRevA.86.032307}
}

@article{higuchisudbery2000,
  author = {Higuchi, A. and Sudbery, A.},
  title = {How entangled can two couples get?},
  journal = {Physics Letters A},
  volume = {273},
  number = {4},
  pages = {213--217},
  year = {2000},
  doi = {10.1016/S0375-9601(00)00500-8}
}

@article{paraschivmiklinmorodergühne2018,
  author = {Paraschiv, Marius and Miklin, Nikolai and Moroder, Tobias and G{\"u}hne, Otfried},
  title = {Proving genuine multiparticle entanglement from separable nearest-neighbor marginals},
  journal = {Physical Review A},
  volume = {98},
  pages = {062102},
  year = {2018},
  doi = {10.1103/PhysRevA.98.062102}
}

@article{clivazhuberlamimurta2017,
  author = {Clivaz, Fabien and Huber, Marcus and Lami, Ludovico and Murta, Gl{\'a}ucia},
  title = {Genuine-multipartite entanglement criteria based on positive maps},
  journal = {Journal of Mathematical Physics},
  volume = {58},
  pages = {082201},
  year = {2017},
  doi = {10.1063/1.4998433},
  note = {arXiv:1609.08126}
}

@misc{weinbrennerricogoodenoughyugühne2026,
  author = {Weinbrenner, Lisa T. and Rico, Albert and Goodenough, Kenneth and Yu, Xiao-Dong and G{\"u}hne, Otfried},
  title = {Complete Hierarchies for the Geometric Measure of Entanglement},
  year = {2026},
  eprint = {2601.23243},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  note = {Very recent (Jan.\ 2026); addresses the geometric measure / overlap-with-product-states problem via multi-copy hierarchies, a related but distinct question from the correlation-tensor witness of Section~\ref{sec:gme-cluster-witness}. Check for overlap before submission.}
}

@misc{aschauer2026a,
  author = {Aschauer, Hans},
  title = {Symmetric Shadow Maps and Multipartite Correlation Criteria},
  year = {2026},
  note = {Unpublished}
}

@misc{aschauer2026b,
  author = {Aschauer, Hans},
  title = {Symmetry-adapted block decomposition of the shadow map},
  year = {2026},
  note = {Unpublished}
}