@article{Parthasarathy,
  author  = {Parthasarathy, K. R.},
  title   = {Extremal quantum states in coupled systems},
  journal = {Ann. Inst. H. Poincar{\'e} Probab. Statist.},
  volume  = {41},
  number  = {3},
  year    = {2005},
  pages   = {257--268}
}

@article{Rudolph,
  author  = {Rudolph, O.},
  title   = {On extremal quantum states of composite systems with fixed marginals},
  journal = {J. Math. Phys.},
  volume  = {45},
  number  = {11},
  year    = {2004},
  pages   = {4035--4041}
}

@article{ohno2010maximal,
  author  = {Ohno, H.},
  title   = {Maximal rank of extremal marginal tracial states},
  journal = {J. Math. Phys.},
  volume  = {51},
  number  = {9},
  year    = {2010},
  pages   = {092101}
}

@article{price2007extremal,
  author  = {Price, G. L. and Sakai, S.},
  title   = {Extremal marginal tracial states in coupled systems},
  journal = {Oper. Matrices},
  volume  = {1},
  number  = {2},
  year    = {2007},
  pages   = {153--163}
}

@article{Devendra2026,
  author  = {Devendra, R. and Dey, P. and Dey, S.},
  title   = {On the rank of extremal marginal states},
  journal = {Linear Algebra Appl.},
  volume  = {737},
  year    = {2026},
  pages   = {146--172}
}

@article{klyachko2004quantum,
  author  = {Klyachko, A.},
  title   = {Quantum marginal problem and representations of the symmetric group},
  journal = {arXiv preprint},
  year    = {2004},
  eprint  = {quant-ph/0409113},
  archivePrefix = {arXiv}
}

@article{christandl2014eigenvalue,
  author  = {Christandl, M. and Doran, B. and Kousidis, S. and Walter, M.},
  title   = {Eigenvalue distributions of reduced density matrices},
  journal = {Comm. Math. Phys.},
  volume  = {332},
  number  = {1},
  year    = {2014},
  pages   = {1--52}
}

@article{Choi1975,
  author  = {Choi, M.-D.},
  title   = {Completely positive linear maps on complex matrices},
  journal = {Linear Algebra Appl.},
  volume  = {10},
  number  = {3},
  year    = {1975},
  pages   = {285--290}
}

@article{LandauStreater1993,
  author  = {Landau, L. J. and Streater, R. J.},
  title   = {On Birkhoff's theorem for doubly stochastic completely positive maps on matrix algebras},
  journal = {Linear Algebra Appl.},
  volume  = {193},
  year    = {1993},
  pages   = {107--127}
}

@article{CahillMixonStrawn,
  author  = {Cahill, J. and Mixon, D. G. and Strawn, N.},
  title   = {Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames},
  journal = {SIAM J. Appl. Algebra Geom.},
  volume  = {1},
  number  = {1},
  year    = {2017},
  pages   = {38--72}
}

@article{KanmaniSatyanarayana2021,
  author  = {Kanmani, S. and Satyanarayana, S. V. M.},
  title   = {Extremal states of qubit-qutrit system with maximally mixed marginals},
  journal = {Phys. Lett. A},
  volume  = {385},
  year    = {2021},
  pages   = {126978}
}

@article{BalaRana2026,
  author  = {Bala, I. and Rana, S.},
  title   = {Extremal marginal states of maximal rank in {(d,d+m)}},
  journal = {arXiv:2605.26920},
  year    = {2026},
  eprint  = {2605.26920},
  archivePrefix = {arXiv}
}

@article{Pataki1998,
  author  = {Pataki, G.},
  title   = {On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues},
  journal = {Math. Oper. Res.},
  volume  = {23},
  number  = {2},
  year    = {1998},
  pages   = {339--358}
}

@article{Barvinok2001,
  author  = {Barvinok, A.},
  title   = {A remark on the rank of positive semidefinite matrices subject to affine constraints},
  journal = {Discrete Comput. Geom.},
  volume  = {25},
  number  = {1},
  year    = {2001},
  pages   = {23--31}
}

@article{HaagerupMusatRuskai2021,
  author  = {Haagerup, U. and Musat, M. and Ruskai, M. B.},
  title   = {Extreme points and factorizability for new classes of unital quantum channels},
  journal = {Ann. Henri Poincar{\'e}},
  volume  = {22},
  year    = {2021},
  pages   = {3455--3496}
}

@article{DykemaStrawn,
  author  = {Dykema, K. and Strawn, N.},
  title   = {Manifold structure of spaces of spherical tight frames},
  journal = {Int. J. Pure Appl. Math.},
  volume  = {28},
  number  = {2},
  year    = {2006},
  pages   = {217--256}
}

@article{BenedettoFickus,
  author  = {Benedetto, J. J. and Fickus, M.},
  title   = {Finite normalized tight frames},
  journal = {Adv. Comput. Math.},
  volume  = {18},
  number  = {2--4},
  year    = {2003},
  pages   = {357--385}
}

@book{BochnakCosteRoy,
  author    = {Bochnak, J. and Coste, M. and Roy, M.-F.},
  title     = {Real Algebraic Geometry},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {36},
  publisher = {Springer},
  address   = {Berlin},
  year      = {1998}
}

@misc{Coste,
  author       = {Coste, M.},
  title        = {An Introduction to Semialgebraic Geometry},
  howpublished = {RAAG Notes, Institut de Recherche Math{\'e}matique de Rennes},
  year         = {2002}
}



@article{Parthasarathy98,
 author = {Parthasarathy, K. R.},
 title = {Extreme points of the convex set of stochastic maps on a {{\(C^*\)}}-algebra},
 fjournal = {Infinite Dimensional Analysis, Quantum Probability and Related Topics},
 journal = {Infin. Dimens. Anal. Quantum Probab. Relat. Top.},
 issn = {0219-0257},
 volume = {1},
 number = {4},
 pages = {599--609},
 year = {1998},
 language = {English},
 doi = {10.1142/S0219025798000326},
 keywords = {46L05,81S25,46N30,46L65},
 zbMATH = {1270036},
 Zbl = {0940.46041}
}


@book{Watrous_2018, 
place={Cambridge},
title={The Theory of Quantum Information}, publisher={Cambridge University Press}, 
author={Watrous, John},
year={2018}}




  @article{jamiolkowski1972linear,
  title={Linear transformations which preserve trace and positive semidefiniteness of operators},
  author={Jamio{\l}kowski, Andrzej},
  journal={Reports on mathematical physics},
  volume={3},
  number={4},
  pages={275--278},
  year={1972},
  publisher={Elsevier}
}



@article{Horodecki2009,
  title     = {Quantum entanglement},
  author    = {Horodecki, Ryszard and Horodecki, Pawe{\l} and Horodecki, Micha{\l} and Horodecki, Karol},
  journal   = {Rev. Mod. Phys.},
  volume    = {81},
  number    = {2},
  pages     = {865--942},
  year      = {2009},
  month     = jun,
  publisher = {American Physical Society},
  doi       = {10.1103/RevModPhys.81.865}
}

@article{Coleman1963,
  title   = {Structure of Fermion Density Matrices},
  author  = {Coleman, A. J.},
  journal = {Rev. Mod. Phys.},
  volume  = {35},
  pages   = {668--686},
  year    = {1963},
  doi     = {10.1103/RevModPhys.35.668}
}

@article{Parthasarathy2004CES,
  title   = {On the maximal dimension of a completely entangled subspace for finite level quantum systems},
  author  = {Parthasarathy, K. R.},
  journal = {Proc. Indian Acad. Sci. Math. Sci.},
  volume  = {114},
  number  = {4},
  pages   = {365--374},
  year    = {2004},
  doi     = {10.1007/BF02829441}
}

@article{Bhat2006CES,
  title   = {A completely entangled subspace of maximal dimension},
  author  = {Bhat, B. V. Rajarama},
  journal = {Int. J. Quantum Inf.},
  volume  = {4},
  number  = {2},
  pages   = {325--330},
  year    = {2006},
  doi     = {10.1142/S0219749906001797}
}


@book{IrelandRosen1990,
  title     = {A Classical Introduction to Modern Number Theory},
  author    = {Ireland, Kenneth and Rosen, Michael},
  edition   = {2nd},
  series    = {Graduate Texts in Mathematics},
  volume    = {84},
  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1990}
}


@book{dummit2003abstract,
  title    ={Abstract Algebra},
  author   ={Dummit, David S. and Foote, Richard M.},
  edition  ={3rd},
  year     ={2003},
  publisher={John Wiley \& Sons},
  address  ={Hoboken, NJ},
  isbn     ={978-0471433347}
}

@book{han2007frames,
  title     = {Frames for Undergraduates},
  author    = {Han, Deguang and Kornelson, Keri and Larson, David and Weber, Eric},
  year      = {2007},
  publisher = {American Mathematical Society},
  series    = {Student Mathematical Library},
  volume    = {40},
  address   = {Providence, Rhode Island}
}

@article{zbMATH00537274,
 author = {Bhat, Rajarama and Pati, V. and Sunder, V. S.},
 title = {On some convex sets and their extreme points},
 fjournal = {Mathematische Annalen},
 journal = {Math. Ann.},
 issn = {0025-5831},
 volume = {296},
 number = {4},
 pages = {637--648},
 year = {1993},
 language = {English},
 doi = {10.1007/BF01445126},
 keywords = {46A55,46L37},
 url = {https://eudml.org/doc/165106},
 zbMATH = {537274},
 Zbl = {0791.46006}
}

@article{tregub1986bistochastic,
  title={Bistochastic operators on finite-dimensional von Neumann algebras},
  author={Tregub, SL},
  journal={Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika},
  number={3},
  pages={75--77},
  year={1986},
  publisher={Kazan (Volga region) Federal University}
}

@article{zbMATH04020083,
 author = {K{\"u}mmerer, Burkhard and Maassen, Hans},
 title = {The essentially commutative dilations of dynamical semigroups on {{\(M_ n\)}}},
 fjournal = {Communications in Mathematical Physics},
 journal = {Commun. Math. Phys.},
 issn = {0010-3616},
 volume = {109},
 pages = {1--22},
 year = {1987},
 language = {English},
 doi = {10.1007/BF01205670},
 keywords = {60B15,81P20,47D03,60J35},
 zbMATH = {4020083},
 Zbl = {0627.60015}
}





@article{BhatPatiSunder1993,
  author  = {Bhat, B. V. Rajarama and Pati, V. and Sunder, V. S.},
  title   = {On some convex sets and their extreme points},
  journal = {Mathematische Annalen},
  volume  = {296},
  number  = {4},
  pages   = {637--648},
  year    = {1993}
}

@article{Tregub1986,
  author  = {Tregub, S. L.},
  title   = {Bistochastic operators on finite-dimensional von {N}eumann algebras},
  journal = {Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika},
  volume  = {3},
  pages   = {75--77, 88},
  year    = {1986}
}

@article{KummererMaassen1987,
  author  = {K{\"u}mmerer, Burkhard and Maassen, Hans},
  title   = {The essentially commutative dilations of dynamical semigroups on {$M_n$}},
  journal = {Communications in Mathematical Physics},
  volume  = {109},
  number  = {1},
  pages   = {1--22},
  year    = {1987}
}

@book{zbMATH00967875,
 author = {Washington, Lawrence C.},
 title = {Introduction to cyclotomic fields.},
 edition = {2nd ed.},
 fseries = {Graduate Texts in Mathematics},
 series = {Grad. Texts Math.},
 issn = {0072-5285},
 volume = {83},
 isbn = {0-387-94762-0},
 year = {1997},
 publisher = {New York, NY: Springer},
 language = {English},
 keywords = {11R18,11-01,11-02,11R23},
 zbMATH = {967875},
 Zbl = {0966.11047}
}