Zespolone proste równokątne i hipoteza Zaunera

• Autorzy: Słomczyński Wojciech , Szymusiak Anna

Warianty tytułu

EN

Complex equiangular lines and Zauner conjecture

• Języki publikacji: PL
• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Wiadomości Matematyczne
• Rocznik: 2020
• Tom: T. 56, Nr 2
• Strony: 197--213
• Opis fizyczny: Bibliogr. 44 poz.

Twórcy

autor

Słomczyński Wojciech

• Wydział Matematyki i Informatyki Uniwersytet Jagielloński

• https://orcid.org/0000-0003-2388-8930

autor

Szymusiak Anna

• Wydział Matematyki i Informatyki Uniwersytet Jagielloński

• https://orcid.org/0000-0002-8512-4947

Bibliografia

• [1] O. Andersson, I. Dumitru, Aligned SICs and embedded tight frames in even dimensions, J. Phys. A 52 (2019), 425302.
• [2] D. M. Appleby, SIC-POVMs and the extended Clifford Group, J. Math. Phys. 46 (2005), 052107.
• [3] M.Appleby, C.A. Fuchs, H. Zhu,Group theoretic, Lie algebraic and Jordan algebraic formulations of the SIC existence problem, Quantum Inf. Comput. 15 (2015), 61–94.
• [4] M. Appleby, I. Bengtsson, I. Dumitru, S. Flammia, Dimension towers of SICs. I. Aligned SICs and embedded tight frames, J. Math. Phys. 58 (2017), 112201.
• [5] M. Appleby, T.-Y. Chien, S. Flammia, S. Waldron, Constructing exact symmetric informationally complete measurements from numerical solutions, J. Phys. A 51 (2018), 165302.
• [6] D. M. Appleby, Å. Ericsson, C.A. Fuchs, Properties of QBist state spaces, Found. Phys. 41 (2011), 564–579.
• [7] M. Appleby, S. Flammia, G. McConnell, J. Yard, Generating ray class fields of real quadratic fields via complex equiangular lines, Acta Arith. 192 (2020), 211–233.
• [8] M. Appleby, C.A. Fuchs, B. C. Stacey, H. Zhu, Introducing the Qplex: a novel arena for quantum theory, Eur. Phys. J. D 71 (2017), 197.
• [9] I. Bengtsson, SICs: Some Explanations, Found. Phys. 50 (2020), 1794–1808.
• [10] H. S. M. Coxeter, The polytope 221 whose twenty-seven vertices correspond to the lines to the general cubic surface, Amer. J. Math. 62 (1940), 457–486.
• [11] P. Delsarte, J. Goethals, J. Seidel, Bounds for systems of lines, and Jacobi polynomials, Philips Res. Rep. 30 (1975), 91–105.
• [12] S. Flammia, Exact SIC fiducial vectors, http://www.physics.usyd.edu.au/~sflammia/SIC/ (dostęp: 2020-12-20).
• [13] D. S. Freed, On Wigner’s theorem, Geom. Topol. Monogr. 18 (2012), 83–89.
• [14] C.A. Fuchs, M. C. Hoang, B.C. Stacey, Fe SIC Question: History and State of Play, Axioms 6 (2017), 21.
• [15] C.A. Fuchs, R. Schack, Quantum-Bayesian coherence, Rev. Modern Phys. 85 (2013), 16931715.
• [16] M. Grassl, On SIC-POVMs and MUBs in dimension 6, Proceedings ERATO Conference on Quantum Information Science (EQIS 2004), Tokyo, 2004, 60–61.
• [17] M. Grassl, Tomography of quantum states in small dimensions, Electron. Notes Discrete Math. 20 (2005), 151–164.
• [18] M. Grassl, Finding equiangular lines in complex space, MAGMA 2006 Conference (Technische Universität Berlin, July, 2006), http://magma.maths.usyd.edu.au/conferences/Magma2006/talks/Grassl_Berlin.pdf (dostęp: 2020-12-12).
• [19] M. Grassl, SIC-POVMs, http://sicpovm.markus-grassl.de/ (dostęp: 2020-12-20).
• [20] A. Grzesik, Proste równokątne, Wiad. Mat. 55 (2019), 377–381.
• [21] S. G. Hoggar, Two quaternionic 4-polytopes, [w:] The Geometric Vein. The Coxeter Festschrift (C. Davis, B. Grnbaum, F.A. Sherk, red.), Springer Verlag, New York 1981, 219–230.
• [22] S. G. Hoggar, t-designs in projective spaces, European J. Combin. 3 (1982), 233–254.
• [23] S. G. Hoggar, 64 Lines from a Quaternionic Polytope, Geom. Dedicata 69 (1998), 287–289.
• [24] P. Horodecki, Ł. Rudnicki, K. Życzkowski, Five open problems in quantum information, przyjęte do druku w PRX Quantum.
• [25] L. P. Hughston, S. M. Salamon, Surveying points in the complex projective plane, Adv. Math. 286 (2016), 1017–1052.
• [26] J. Jedwab, A. Wiebe, A simple construction of complex equiangular lines, [w:] Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics (C. Colbourn, red.), Springer, Cham 2015, 159–169.
• [27] M. Magsino, D. G. Mixon, Biangular Gabor frames and Zauner’s conjecture, Proceedings SPIE 11138, Wavelets and Sparsity XVIII, 111381G (9 September 2019), arXiv:1908.02801 [math.MG].
• [28] J. Mann, Equilateral dimension of Riemannian manifolds with bounded curvature, Rose-Hulman Undergrad. Math. J. 14 (2013), 101–111.
• [29] A. Neumaier, Combinatorial configurations in terms of distances, Memorandum 81-09 (Dept. of Mathematics), Eindhoven University of Technology 1981.
• [30] QBism Group UMass Boston, http://www.physics.umb.edu/Research/QBism/solutions.html (dostęp: 2020-12-20).
• [31] S. K. Pandey, V. I. Paulsen, J. Prakash, M. Rahaman, Entanglement breaking rank and the existence of SIC POVMs, J. Math. Phys. 61 (2020), 042203.
• [32] J. M. Renes, R. Blume-Kohout, A. J. Scott, C. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45 (2004), 2171–2180.
• [33] K. Scharnhorst, Angles in complex vectorspaces, Acta Appl. Math. 69 (2001), 95–103.
• [34] A. J. Scott, SICs: Extending the list of solutions, arXiv:1703.03993 [quant-ph].
• [35] A. J. Scott, M. Grassl, SICs: A new computer study, J. Math. Phys. 51 (2010), 042203.
• [36] W. Słomczyński, A. Szymusiak, Morphophoric POVMs, generalised qplexes, and 2-designs, Quantum 4 (2020), 338.
• [37] B.C. Stacey, Sporadic SICs and the Normed Division Algebras, Found. Phys. 47 (2017), 1060–1064.
• [38] A. Szymusiak, W. Słomczyński, Informational power of the Hoggar symmetric informationally complete positive operator-valued measure, Phys. Rev. A 94 (2016), 012122.
• [39] L. R. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Feory 20 (1974), 397–399.
• [40] R. F. Werner, Open quantum problems, https://oqp.iqoqi.univie.ac.at/ (dostęp:2020-12-20)
• [41] G. Zauner, Quantendesigns, Grundzüge einer nichtkommutativen Designtheorie, Ph.D. thesis, Universität Wien 1999.
• [42] G. Zauner, Quantum designs: foundations of a noncommutative design theory, Int. J. Quantum Inf. 9 (2011), 445–507.
• [43] H. Zhu, SIC-POVMs and Clifford groups in prime dimensions, J. Phys. A 43 (2010), 05305.
• [44] H. Zhu, Super-symmetric informationally complete measurements, Ann. Physics 362 (2015), 311–326.

Uwagi

PL

Opracowane ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)