Abstract
We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to compare elements of mixed dimension, we use a classical embedding to send all fusion frame elements to points on a higher dimensional Euclidean sphere, where they are given “equal footing.” Over the embedded images—a compact subset in the higher dimensional embedded sphere—we define optimality in terms of the corresponding restricted coding problem. We then construct infinite families of solutions to the problem by using maximal sets of mutually unbiased bases and block designs. Finally, we show that using Hadamard 3-designs in this construction leads to infinitely many examples of maximal orthoplectic fusion frames of constant-rank. Moreover, any such fusion frames constructed by this method must come from Hadamard 3-designs.
In memory of John I. Haas.
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References
Bachoc, C., Bannai, E., and Coulangeon, R.: Codes and designs in Grassmannian spaces. Discrete Math. 277(1–3), 15–28 (2004)
Bandeira, A.S., Fickus, M., Mixon, D.G., and Wong, P.: The road to deterministic matrices with the restricted isometry property. J. Fourier Anal. Appl. 19(6), 1123–1149 (2013)
Benedetto, J.J. and Kolesar, J.D.: Geometric properties of Grassmannian frames for \(\mathbb {R}^2\) and \(\mathbb {R}^3\). EURASIP J. Appl. Signal Process. 1–17 (2006)
Bodmann, B.G. and Elwood, H.J.: Complex equiangular Parseval frames and Seidel matrices containing pth roots of unity. Proc. Amer. Math. Soc. 138(12), 4387–4404 (2010)
Bodmann, B.G. and Haas, J.: Frame potentials and the geometry of frames. J. Fourier Anal. Appl. 21(6), 1344–1383 (2015)
Bodmann, B.G. and Haas, J.: Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets. Linear Algebra Appl. 511, 54–71 (2016)
Bodmann, B.G. and Haas, J.: Maximal orthoplectic fusion frames from mutually unbiased bases and block designs. Proc. Amer. Math. Soc. 146(6), 2601–2616 (2018)
Calderbank, A.R., Hardin, R.H., Rains, E.M., Shor, P.W., and Sloane, N.J.A.: A group theoretic framework for the construction of packings in Grassmannian spaces. J. Algebraic Combin. 9(2), 129–140 (1999)
Cameron, P.J. and van Lint, J.H.: Designs, Graphs, Codes and their Links. London Mathematical Society Student Texts. Cambridge University Press. (1991)
Cameron, P.J. and Seidel, J.J.: Quadratic forms over GF(2). Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35, 1–8 (1973)
Casazza, P.G. and Haas, J.I.: On the rigidity of geometric and spectral properties of Grassmannian frames. ArXiv e-prints (May 2016)
Casazza, P.G. and Kutyniok, G. (editors).: Finite Frames. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York (2013)
Casazza, P.G., Haas IV, I.J., Stueck, J., and Tran, T.T.: Constructions and properties of optimally spread subspace packings via symmetric and affine block designs and mutually unbiased bases. ArXiv e-prints (June 2018)
Christensen, O.: An Introduction to Frames and Riesz Bases. An introduction to frames and Riesz bases. Second expanded edition, Birkhäuser, Boston, Basel, Berlin (2016)
Colbourn, C.J. and Dinitz, J.H. (editors): Handbook of Combinatorial Designs. Discrete Mathematics and its Applications (Boca Raton). Chapman Hall/CRC, Boca Raton, FL, second edition (2007)
Conway, J.H., Hardin, R.H., and Sloane, N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996)
Delsarte, P., Goethals, J.M., and Seidel, J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. 30, 91–105 (1975)
Dhillon, I.S., Heath Jr., R.W., Strohmer, T., and Tropp, J.A.: Constructing packings in Grassmannian manifolds via alternating projection. Exp. Math. 17(1), 9–35 (2008)
Et-Taoui, B.: Complex conference matrices, complex Hadamard matrices and equiangular tight frames. In: Convexity and Discrete Geometry including Graph Theory, 181–191, Springer Proc. Math. Stat., 148, Springer, [Cham] (2016)
Fickus, M., Mixon, D.G., and Tremain, J.C.: Steiner equiangular tight frames. Linear Algebra Appl. 436(5), 1014—1027 (2012)
Fickus, M., Jasper, J., Mixon, D.G., and Peterson, J.: Tremain equiangular tight frames. J. Combin. Theory Ser. A 153, 54–66 (2018)
Fickus, M., Jasper, J., and Mixon, D.G.: Packings in real projective spaces. SIAM J. Appl. Algebra Geom. 2(3), 377–409 (2018)
Hoffman, T.R. and Solazzo, J.P.: Complex equiangular tight frames and erasures. Linear Algebra Appl. 437(2), 549–558 (2012)
Holmes, R.B. and Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004)
Jasper, J., Mixon, D.G., and Fickus, M.: Kirkman equiangular tight frames and codes. IEEE Trans. Inf. Theory. 60(1), 170–181 (2014)
Kalra, D.: Complex equiangular cyclic frames and erasures. Linear Algebra Appl. 419(2–3), 373–399 (2006)
King, E.J.: New constructions and characterizations of flat and almost flat Grassmannian fusion frames. ArXiv e-prints, arXiv:1612.05784 (February 2019)
Kutyniok, G., Pezeshki, A., Calderbank, R., and Liu, T.: Robust dimension reduction, fusion frames, and Grassmannian packings. Appl. Comput. Harmon. Anal. 26(1), 64–76 (2009)
Oktay, O.: Frame quantization theory and equiangular tight frames. ProQuest LLC, Ann Arbor, MI. Dissertation (Ph.D.)-University of Maryland, College Park, MD (2007)
Rankin, R.A.: The closest packing of spherical caps in n dimensions. Proc. Glasgow Math. Assoc. 2, 139–144 (1955)
Renes, J.M., Blume-Kohout, R., Scott, A.J., and Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004)
Shor, P.W. and Sloane, N.J.A.: A family of optimal packings in Grassmannian manifolds. J. Algebraic Combin. 7(2), 157–163 (1998)
Strohmer, T. and Heath Jr., R.W.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003)
Sustik, M.A., Tropp, J.A., Dhillon, I.S., and Heath Jr., R.W.: On the existence of equiangular tight frames. Linear Algebra Appl. 426(2–3), 619–635 (2007)
Waldron, S.F.D.: An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York (2018)
Wootters, W.K. and Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191(2), 363–381 (1989)
Xia, P., Zhou, S., and Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory. 51(5), 1900–1907 (2005)
Zauner, G.: Quantendesigns - Grundz¨uge einer nichtkommutativen Designtheorie. University Wien (Austria). Dissertation (Ph.D.) (1999). English translation in International Journal of Quantum Information (IJQI). 9(1), 445–507 (2011)
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The authors were supported by NSF DMS 1609760, 1906725, and NSF ATD 1321779.
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Casazza, P.G., Stueck, J., Tran, T.T. (2021). A Notion of Optimal Packings of Subspaces with Mixed-Rank and Solutions. In: Hirn, M., Li, S., Okoudjou, K.A., Saliani, S., Yilmaz, Ö. (eds) Excursions in Harmonic Analysis, Volume 6. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-69637-5_7
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