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On the equivalence between real mutually unbiased bases and a certain class of association schemes. (English) Zbl 1195.81026

Summary: Mutually unbiased bases (MUBs) in complex vector spaces play several important roles in quantum information theory. At present, even the most elementary questions concerning the maximum number of such bases in a given dimension and their construction remain open. In an attempt to understand the complex case better, some authors have also considered real MUBs, mutually unbiased bases in real vector spaces. The main results of this paper establish an equivalence between sets of real mutually unbiased bases and 4-class cometric association schemes which are both \(Q\)-bipartite and \(Q\)-antipodal. We then explore the consequences of this equivalence, constructing new cometric association schemes and describing a potential method for the construction of sets of real MUBs.

MSC:

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
05E30 Association schemes, strongly regular graphs
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[1] Abdukhalikov, K.; Bannai, E.; Suda, S., Association schemes related to universally optimal configurations, kerdock codes and extremal Euclidean line-sets, J. combin. theory, ser. A, 116, 434-448, (2009) · Zbl 1250.05119
[2] Aschbacher, M.; Childs, A.M.; Wocjan, P., Limitations of Nice mutually unbiased bases, J. algebraic combin., 25, 2, 111-123, (2007) · Zbl 1109.81016
[3] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin-Cummings Menlo Park · Zbl 0555.05019
[4] E. Bannai, E. Bannai, On antipodal spherical \(t\)-designs of degree \(s\) with \(t \geq 2 s - 3\), J. Comb. Inf. Syst. Sci., special volume honoring the 75th birthday Prof. D. K. Ray-Chaudhuri (in press) · Zbl 1270.05017
[5] P.O. Boykin, M. Sitharam, M. Tarifi, P. Wocjan, Real mutually unbiased bases. Preprint. arXiv:quant-ph/0502024v2 [math.CO] (revised version dated Feb. 1, 2008)
[6] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin · Zbl 0747.05073
[7] P.J. Cameron, J.J. Seidel, Quadratic forms over \(G F(2)\), in: Proc. Koninkl. Nederl. Akademie van Wetenschappen, Series A, Vol. 76, Indag. Math., 35 1973, pp. 1-8
[8] Calderbank, A.R.; Kantor, W.M.; Seidel, J.J.; Sloane, N.J.A., \(\mathbb{Z}_4\)-kerdock codes, orthogonal spreads, and extremal Euclidean line-sets, Proc. London math. soc., 75, 3, 436-480, (1997) · Zbl 0916.94014
[9] ()
[10] Delsarte, P., An algebraic approach to the association schemes of coding theory, Philips res. reports suppl., 10, (1973) · Zbl 1075.05606
[11] Delsarte, P.; Goethals, J.-M.; Seidel, J.J., Bounds for systems of lines and Jacobi polynomials, Philips res. reports, 30, 91-105, (1975) · Zbl 0322.05023
[12] Delsarte, P.; Goethals, J.-M.; Seidel, J.J., Spherical codes and designs, Geom. ded., 6, 363-388, (1977) · Zbl 0376.05015
[13] Godsil, C.D., Algebraic combinatorics, (1993), Chapman and Hall New York · Zbl 0814.05075
[14] Godsil, C.; Roy, A., Equiangular lines, mutually unbiased bases, and spin models, European J. combin., 30, 246-262, (2009) · Zbl 1157.05014
[15] A Klappenecker, M. Rötteler, Constructions of mutually unbiased bases, in: Finite Fields and Appl., in: Proceedings of 7th International Conference, Fq7, Toulouse, France, May 2003, pp. 137-144
[16] Martin, W.J.; Muzychuk, M.; Williford, J., Imprimitive cometric association schemes: constructions and analysis, J. algebraic combin., 25, 399-415, (2007) · Zbl 1118.05100
[17] Mathon, R., The systems of linked 2-(16, 6, 2) designs, Ars combin., 11, 131-148, (1981) · Zbl 0468.05012
[18] Suzuki, H., Imprimitive \(Q\)-polynomial association schemes, J. algebraic combin., 7, 2, 165-180, (1998) · Zbl 0974.05083
[19] Suzuki, H., Association schemes with multiple \(Q\)-polynomial structures, J. algebraic combin., 7, 2, 181-196, (1998) · Zbl 0974.05082
[20] Wocjan, P.; Beth, T., New constructions of mutually unbiased bases in square dimensions, Quantum inf. comput., 5, 2, 93-101, (2005) · Zbl 1213.81108
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