×

On biunimodular vectors for unitary matrices. (English) Zbl 1330.15033

Summary: A biunimodular vector of a unitary matrix \(A \in U(n)\) is a vector \(v \in \mathbb{T}^n \subset \mathbb{C}^n\) such that \(\mathit{Av} \in \mathbb{T}^n\) as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences. Here, we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices. This search can be motivated in various ways. The main motivation is provided by the fact, that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices.

MSC:

15B05 Toeplitz, Cauchy, and related matrices
15A21 Canonical forms, reductions, classification
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alltop, W., Complex sequences with low periodic correlations, IEEE Trans. Inform. Theory, 26, 3, 350-354 (1980) · Zbl 0432.94011
[2] Backelin, J., Square Multiples \(n\) Give Infinitely Many Cyclic \(n\)-roots, Reports Matematiska Institutionen, vol. 8 (1989), Stockholms Universitet
[3] Backelin, J.; Fröberg, R., How we proved that there are exactly 924 7-roots, (Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation. Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC91 (1991), ACM Press) · Zbl 0925.13012
[4] Bauschke, H.; Combettes, P.; Luke, D. R., Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization, J. Opt. Soc. Amer. A, 19, 7, 1334-1345 (2002)
[5] Bauschke, H.; Borwein, J., On the convergence of von Neumann’s alternating projection algorithm for two sets, Set-Valued Anal., 1, 2, 185-212 (1993) · Zbl 0801.47042
[6] Benedetto, J. J.; Benedetto, R.; Woodworth, J., Optimal ambiguity functions and Weil’s exponential sum bound, J. Fourier Anal. Appl., 18, 3, 471-487 (2012) · Zbl 1250.42022
[7] Benedetto, J. J.; Konstantinidis, I.; Rangaswamy, M., Phase-coded waveforms and their design, IEEE Signal Process. Mag., 26, 1, 22-31 (2009)
[8] Bengtsson, I.; Bruzda, W.; Ericsson, Å.; Larsson, J.-Å; Tadej, W.; Życzkowski, K., Mutually unbiased bases and Hadamard matrices of order six, J. Math. Phys., 48, 5, 052106 (2007), 21 pp · Zbl 1144.81314
[9] Biran, P.; Entov, M.; Polterovich, L., Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math., 6, 5, 793-802 (2004) · Zbl 1076.53110
[10] Björck, G., Functions of modulus 1 on \(Z_n\) whose Fourier transforms have constant modulus, (A. Haar Memorial Conference, vols. I, II. A. Haar Memorial Conference, vols. I, II, Budapest, 1985. A. Haar Memorial Conference, vols. I, II. A. Haar Memorial Conference, vols. I, II, Budapest, 1985, Colloq. Math. Soc. János Bolyai, vol. 49 (1987), North-Holland: North-Holland Amsterdam), 193-197
[11] Björck, G., Functions of modulus 1 on \(Z_n\) whose Fourier transforms have constant modulus, and “cyclic \(n\)-roots”, (Recent Advances in Fourier Analysis and Its Applications, Proc. NATO/ASI. Recent Advances in Fourier Analysis and Its Applications, Proc. NATO/ASI, Il Ciocco, Italy, 1989. Recent Advances in Fourier Analysis and Its Applications, Proc. NATO/ASI. Recent Advances in Fourier Analysis and Its Applications, Proc. NATO/ASI, Il Ciocco, Italy, 1989, NATO ASI Ser. C, vol. 315 (1990)), 131-140
[12] Björck, G.; Fröberg, R., A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic \(n\)-roots, J. Symbolic Comput., 12, 3, 329-336 (1991) · Zbl 0751.12001
[13] Björck, G.; Fröberg, R., Methods to divide out certain solutions from systems of algebraic equations, applied to find all cyclic 8 roots, (Analysis, Algebra, and Computers in Mathematical Research. Analysis, Algebra, and Computers in Mathematical Research, Lulea, 1992. Analysis, Algebra, and Computers in Mathematical Research. Analysis, Algebra, and Computers in Mathematical Research, Lulea, 1992, Lecture Notes in Pure and Appl. Math., vol. 156 (1994), Dekker: Dekker New York), 57-70 · Zbl 0811.12002
[14] Björck, G.; Saffari, B., New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries, C. R. Acad. Sci. Paris, Sér. I, 320, 3, 319-324 (1995) · Zbl 0846.11016
[15] Brierley, S.; Weigert, S., Constructing mutually unbiased bases in dimension six, Phys. Rev. A (3), 79, 5, 052316 (2009), 13 pp
[16] Brierley, S.; Weigert, S.; Bengtsson, I., All mutually unbiased bases in dimensions two to five, Quantum Inf. Comput., 10, 9-10, 803-820 (2010) · Zbl 1237.81032
[17] Cabrera, R.; Strohecker, T.; Rabitz, H., The canonical coset decomposition of unitary matrices through Householder transformations, J. Math. Phys., 51, 8, 082101 (2010), 7 pp · Zbl 1312.81044
[18] Candès, E.; Eldar, Y.; Strohmer, T.; Voroninski, V., Phase retrieval via matrix completion, SIAM J. Imaging Sci., 6, 1, 199-225 (2013) · Zbl 1280.49052
[19] Candès, E.; Li, X.; Soltanolkotabi, M., Phase retrieval via Wirtinger flow: theory and algorithms (2014)
[20] Cho, C.-H., Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not. IMRN, 2004, 35, 1803-1843 (2004) · Zbl 1079.53133
[21] Combettes, P.; Trussell, H., Method of successive projections for finding a common point of sets in metric spaces, J. Optim. Theory Appl., 67, 3, 487-507 (1990) · Zbl 0696.90052
[22] Davenport, J., Looking at a set of equations (1987), Bath Computer Science, Technical Report 87-06
[23] De Vos, A.; De Baerdemacker, S., Scaling a unitary matrix, Open Syst. Inf. Dyn., 21, 4 (2014) · Zbl 1306.65198
[24] Diţă, P., Parametrisation of unitary matrices, J. Phys. A, 15, 11, 3465-3473 (1982) · Zbl 0497.15015
[25] Diţă, P., Factorization of unitary matrices, J. Phys. A, 36, 11, 2781-2789 (2003) · Zbl 1057.15012
[26] Escalante, R.; Raydan, M., Alternating Projection Methods (2011), SIAM · Zbl 1275.65031
[27] Faugère, J.-C., Finding all the solutions of cyclic 9 using Gröbner basis techniques, (Computer Mathematics. Computer Mathematics, Matsuyama, 2001. Computer Mathematics. Computer Mathematics, Matsuyama, 2001, Lecture Notes Ser. Comput., vol. 9 (2001), World Sci. Publ.), 1-12 · Zbl 1030.68112
[28] Faugère, J.-C., A new efficient algorithm for computing Gröbner bases without reduction to zero (F5), (Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (ISSAC) (2002), ACM Press), 75-83 · Zbl 1072.68664
[29] Fogel, F.; Waldspurger, I.; d’Aspremont, A., Phase retrieval for imaging problems (2013)
[30] Gabidulin, E.; Shorin, V., New sequences with zero autocorrelation, Probl. Inf. Transm., 38, 4, 255-267 (2002) · Zbl 1027.11019
[31] Gilbert, J.; Rzeszotnik, Z., The norm of the Fourier transform on finite abelian groups, Ann. Inst. Fourier (Grenoble), 60, 4, 1317-1346 (2010) · Zbl 1202.42065
[32] Gerchberg, R.; Saxton, W., A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik, 35, 237-246 (1972)
[33] Grassl, M., On SIC-POVMs and MUBs in dimension 6, (Proc. ERATO Conf. on Quant. Inf. Science. Proc. ERATO Conf. on Quant. Inf. Science, EQIS (2004)), 60-61
[34] Haagerup, U., Orthogonal maximal abelian ⁎-subalgebras of the \(n \times n\) matrices and cyclic \(n\)-roots, (Doplicher, S.; etal., Operator Algebras and Quantum Field Theory (1997), Accademia Nazionale dei Lincei/International Press: Accademia Nazionale dei Lincei/International Press Roma, Italy/Cambridge, MA), 296-322 · Zbl 0914.46045
[35] Haagerup, U., Cyclic \(p\)-roots of prime length \(p\) and related complex Hadamard matrices
[36] Idel, M.; Wolf, M., Sinkhorn normal form for unitary matrices, Linear Algebra Appl., 471, 76-84 (2015) · Zbl 1307.15014
[37] Jaming, P., Phase retrieval techniques for radar ambiguity problems, J. Fourier Anal. Appl., 5, 4, 309-329 (1999) · Zbl 0940.94003
[38] Jarlskog, C., A recursive parametrization of unitary matrices, J. Math. Phys., 46, 10, 103508 (2005), 4 pp · Zbl 1111.15027
[39] Langevin, P.; Leander, G., Counting all bent functions in dimension eight designs, Des. Codes Cryptogr., 59, 1-3, 193-205 (2011) · Zbl 1215.94059
[40] Li, T.-Y.; Tsai, C.-H., HOM4PS-2.0para: parallelization of HOM4PS-2.0 for solving polynomial systems, Parallel Comput., 4, 226-238 (2009)
[41] Mezzadri, F., How to generate random matrices from the classical compact groups, Notices Amer. Math. Soc., 54, 592-604 (2007) · Zbl 1156.22004
[42] Murnaghan, F., On a convenient system of parameters for the unitary group, Proc. Natl. Acad. Sci. USA, 38, 127-129 (1952) · Zbl 0047.25903
[43] Nemoto, K., Generalized coherent states for \(SU(N)\) systems, J. Phys. A: Math. Gen., 33, 3493-3506 (2000) · Zbl 0948.81567
[44] Saffari, B., Some polynomial extremal problems which emerged in the twentieth century, (Byrnes, James S., Twentieth Century Harmonic Analysis - A Celebration. Proceedings of the NATO Advanced Study Institute. Twentieth Century Harmonic Analysis - A Celebration. Proceedings of the NATO Advanced Study Institute, Il Ciocco, Italy, July 2-15, 2000. Twentieth Century Harmonic Analysis - A Celebration. Proceedings of the NATO Advanced Study Institute. Twentieth Century Harmonic Analysis - A Celebration. Proceedings of the NATO Advanced Study Institute, Il Ciocco, Italy, July 2-15, 2000, NATO Sci. Ser. II, Math. Phys. Chem., vol. 33 (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 201-223 · Zbl 0996.42001
[45] Reck, M.; Zeilinger, A.; Bernstein, H.; Bertani, P., Experimental realization of any discrete unitary operator, Phys. Rev. Lett., 73, 1, 58-61 (1994)
[46] Rowe, D.; Sanders, B.; de Guise, H., Representations of the Weyl group and Wigner functions for \(SU(3)\), J. Math. Phys., 40, 7, 3604-3615 (1999) · Zbl 1063.81589
[47] Schwinger, J., Unitary operator bases, Proc. Natl. Acad. Sci. USA, 46, 570-579 (1960) · Zbl 0090.19006
[48] Spengler, C.; Huber, M.; Hiesmayr, B., A composite parameterization of unitary groups, density matrices and subspaces, J. Phys. A, 43, 38, 385306 (2010), 11 pp · Zbl 1198.81058
[49] Tadej, W.; Życzkowski, K., Defect of a unitary matrix, Linear Algebra Appl., 429, 447-481 (2008) · Zbl 1143.15024
[50] Tilma, T.; Sudarshan, E., Generalized Euler angle parametrization for \(SU(N)\), J. Phys. A, 35, 48, 10467-10501 (2002) · Zbl 1047.22012
[51] Tilma, T.; Sudarshan, E., Generalized Euler angle parametrization for \(U(N)\) with applications to \(SU(N)\) coset volume measures, J. Geom. Phys., 52, 263-283 (2004) · Zbl 1069.22012
[52] von Neumann, J., Functional Operators, vol. II (1950), Princeton University Press: Princeton University Press Princeton, NJ, Reprint of mimeographed lecture notes first distributed in 1933 · Zbl 0039.11701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.