An application of positive definite functions to the problem of MUBs

Kolountzakis, Mihail; Matolcsi, Máté; Weiner, Mihály

doi:10.1090/proc/13829

An application of positive definite functions to the problem of MUBs
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by Mihail N. Kolountzakis, Máté Matolcsi and Mihály Weiner PDF

Proc. Amer. Math. Soc. 146 (2018), 1143-1150 Request permission

Abstract:

We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most $d+1$ MUBs in $\mathbb {C}^d$, and it may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.

References

Somshubhro Bandyopadhyay, P. Oscar Boykin, Vwani Roychowdhury, and Farrokh Vatan, A new proof for the existence of mutually unbiased bases, Algorithmica 34 (2002), no. 4, 512–528. Quantum computation and quantum cryptography. MR 1943521, DOI 10.1007/s00453-002-0980-7
Kyle Beauchamp and Remus Nicoara, Orthogonal maximal abelian $\ast$-subalgebras of the $6\times 6$ matrices, Linear Algebra Appl. 428 (2008), no. 8-9, 1833–1853. MR 2398121, DOI 10.1016/j.laa.2007.10.023
Ingemar Bengtsson, Wojciech Bruzda, Åsa Ericsson, Jan-Åke Larsson, Wojciech Tadej, and Karol Życzkowski, Mutually unbiased bases and Hadamard matrices of order six, J. Math. Phys. 48 (2007), no. 5, 052106, 21. MR 2326331, DOI 10.1063/1.2716990
P. Oscar Boykin, Meera Sitharam, Pham Huu Tiep, and Pawel Wocjan, Mutually unbiased bases and orthogonal decompositions of Lie algebras, Quantum Inf. Comput. 7 (2007), no. 4, 371–382. MR 2363400
Stephen Brierley and Stefan Weigert, Maximal sets of mutually unbiased quantum states in dimension 6, Phys. Rev. A (3) 78 (2008), no. 4, 042312, 8. MR 2491052, DOI 10.1103/PhysRevA.78.042312
Stephen Brierley and Stefan Weigert, Constructing mutually unbiased bases in dimension six, Phys. Rev. A (3) 79 (2009), no. 5, 052316, 13. MR 2550430, DOI 10.1103/PhysRevA.79.052316
Stephen Brierley, Stefan Weigert, and Ingemar Bengtsson, All mutually unbiased bases in dimensions two to five, Quantum Inf. Comput. 10 (2010), no. 9-10, 803–820. MR 2731441
P. Butterley and W. Hall, Numerical evidence for the maximum number of mutually unbiased bases in dimension six, Physics Letters A 369 (2007) 5–8.
Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. MR 1973059, DOI 10.4007/annals.2003.157.689
Benoît Collins and Piotr Śniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys. 264 (2006), no. 3, 773–795. MR 2217291, DOI 10.1007/s00220-006-1554-3
Monique Combescure, Circulant matrices, Gauss sums and mutually unbiased bases. I. The prime number case, Cubo 11 (2009), no. 4, 73–86 (English, with English and Spanish summaries). MR 2571796
Monique Combescure, Block-circulant matrices with circulant blocks, Weil sums, and mutually unbiased bases. II. The prime power case, J. Math. Phys. 50 (2009), no. 3, 032104, 12. MR 2510895, DOI 10.1063/1.3078420
P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep. 27 (1972), 272–289. MR 314545
T. Durt, B. G. Englert, I. Bengtsson and K. Życzkowski, On mutually unbiased bases, International Journal of Quantum Information, 8 (2010), 535–640.
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
D. Goyeneche, Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6, J. Phys. A 46 (2013), no. 10, 105301, 15. MR 3030161, DOI 10.1088/1751-8113/46/10/105301
Uffe Haagerup, Orthogonal maximal abelian $*$-subalgebras of the $n\times n$ matrices and cyclic $n$-roots, Operator algebras and quantum field theory (Rome, 1996) Int. Press, Cambridge, MA, 1997, pp. 296–322. MR 1491124, DOI 10.1215/s0012-7094-97-09004-9
I. D. Ivanović, Geometrical description of quantal state determination, J. Phys. A 14 (1981), no. 12, 3241–3245. MR 639558
Philippe Jaming, Máté Matolcsi, Péter Móra, Ferenc Szöllősi, and Mihály Weiner, A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6, J. Phys. A 42 (2009), no. 24, 245305, 25. MR 2515542, DOI 10.1088/1751-8113/42/24/245305
Bengt R. Karlsson, Three-parameter complex Hadamard matrices of order 6, Linear Algebra Appl. 434 (2011), no. 1, 247–258. MR 2737247, DOI 10.1016/j.laa.2010.08.020
Andreas Klappenecker and Martin Rötteler, Constructions of mutually unbiased bases, Finite fields and applications, Lecture Notes in Comput. Sci., vol. 2948, Springer, Berlin, 2004, pp. 137–144. MR 2092627, DOI 10.1007/978-3-540-24633-6_{1}0
Máté Matolcsi, A Fourier analytic approach to the problem of mutually unbiased bases, Studia Sci. Math. Hungar. 49 (2012), no. 4, 482–491. MR 3098293, DOI 10.1556/SScMath.49.2012.4.1221
M. Matolcsi, I. Z. Ruzsa, and M. Weiner, Systems of mutually unbiased Hadamard matrices containing real and complex matrices, Australas. J. Combin. 55 (2013), 35–47. MR 3058325
Máté Matolcsi and Ferenc Szöllősi, Towards a classification of $6\times 6$ complex Hadamard matrices, Open Syst. Inf. Dyn. 15 (2008), no. 2, 93–108. MR 2451755, DOI 10.1142/S1230161208000092
F. M. Oliveira de Filho and F. Vallentin, Mathematical optimization for packing problems, SIAG/OPT Views and News 23 (2015), 5–14.
I. Z. Ruzsa, Difference sets without squares, Period. Math. Hungar. 15 (1984), no. 3, 205–209. MR 756185, DOI 10.1007/BF02454169
A. J. Skinner, V. A. Newell, and R. Sanchez, Unbiased bases (Hadamards) for six-level systems: four ways from Fourier, J. Math. Phys. 50 (2009), no. 1, 012107, 7. MR 2492587, DOI 10.1063/1.3059166
Ferenc Szöllősi, Complex Hadamard matrices of order 6: a four-parameter family, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 616–632. MR 2927801, DOI 10.1112/jlms/jdr052
Mihály Weiner, A gap for the maximum number of mutually unbiased bases, Proc. Amer. Math. Soc. 141 (2013), no. 6, 1963–1969. MR 3034423, DOI 10.1090/S0002-9939-2013-11487-5
Pawel Wocjan and Thomas Beth, New construction of mutually unbiased bases in square dimensions, Quantum Inf. Comput. 5 (2005), no. 2, 93–101. MR 2132048
William K. Wootters and Brian D. Fields, Optimal state-determination by mutually unbiased measurements, Ann. Physics 191 (1989), no. 2, 363–381. MR 1003014, DOI 10.1016/0003-4916(89)90322-9
G. Zauner, Quantendesigns-Grundzüge einer nichtkommutativen Designtheorie, Ph.D. thesis, Universität Wien, 1999. (available at http://www.mat.univie.ac.at/$\sim$neum/ms/zauner.pdf)