Abstract
It is known that if \(d^2\) vectors in a d-dimensional Hilbert space H form a symmetric, informationally complete, positive operator-valued measure (SIC-POVM), then the tensor squares of these vectors form an equiangular tight frame in the symmetric subspace of \(H\otimes H\). We prove that, for any SIC-POVM of the Weyl–Heisenberg group covariant type (WH-type), this frame can be obtained by projecting a WH-type basis in \(H\otimes H\) onto the symmetric subspace. We give a full description of the set of all WH-type bases, so that this set could be used as a search space for finding SIC solutions. Also, we show that a particular element of this set is close to a SIC solution in some structural sense. Finally, we give a geometric construction of SIC-related symmetric tight fusion frames that were discovered in odd dimensions in Appleby et al. (J Phys A 52(29):295301, 2019).
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References
Zauner, G.: Quantendesigns. Grundzüge einer nichtkommutativen Designtheorie, PhD thesis, Univ. Wien 1999. Available in English translation in Int. J. Quant. Inf. 9, 445 (2011)
Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004)
Fickus, M., Mixon, D.G.: Tables of the existence of equiangular tight frames. arXiv:1504.00253
Appleby, M., Bengtsson, I., Flammia, S., Goyeneche, D.: Tight frames, Hadamard matrices and Zauner’s conjecture. J. Phys. A 52(29), 295301 (2019)
Fuchs, C.A., Hoan, M.C., Stacey, B.C.: The SIC question: history and state of play. Axioms 6, 21 (2017)
Waldron, S.: An Introduction to Finite Tight Frames–Weyl–Heisenberg SICs, pp. 361–427. Birkhauser, Basel (2018)
Scott, A.J., Grassl, M.: SIC-POVMs: a new computer study. J. Math. Phys. 51, 042203 (2010)
Appleby, D.M.: SIC-POVMs and the extended Clifford group. J. Math. Phys. 46, 052107 (2005)
Hoggar, S.G.: 64 lines from a quaternionic polytope. Geometriae Dedicata 69, 287–289 (1998)
Harrow, A.W.: The Church of the Symmetric Subspace. arXiv:1308.6595
Szöllősi, F.: Complex Hadamard matrices and equiangular tight frames. Linear Algebra Appl. 438, 1962 (2013)
Casazza, P.G., Lynch, R.G.: A brief introduction to Hilbert space frame theory and its applications. arXiv:1509.07347
Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press (2012)
Davidson, K.R.: C*-Algebras by Example, Fields Institute monographs. Hindustan Book Agency (1996)
Ostrovskyi, V., Samoilenko, Y.S.: Introduction to the Theory of Representations of Finitely Presented *-Algebras. I. Representations by Bounded Operators. Harwood Academy Publishers (1999)
Saraceno, M., Ermann, L., Cormick, C.: Phase–space representations of SIC–POVM fiducial states. Phys. Rev. A 95, 032102 (2017)
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Ostrovskyi, V., Yakymenko, D. Geometric properties of SIC-POVM tensor square. Lett Math Phys 112, 7 (2022). https://doi.org/10.1007/s11005-021-01496-w
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DOI: https://doi.org/10.1007/s11005-021-01496-w