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Associated to an Hadamard matrix H∈MN(C) is the spectral measure μ∈P[0,N] of the corresponding Hopf image algebra, A=C(G) with G⊂S+N. We study a certain family of discrete measures μr∈P[0,N], coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type ∫N0(x/N)pdμr(x)=∫N0(x/N)rdνp(x), where μr,νr are the truncations of the spectral measures μ,ν associated to H,Ht. We also prove, using these truncations μr,νr, that for any deformed Fourier matrix H=FM⊗QFN we have μ=ν.
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Tom
Strony
161--179
Opis fizyczny
Bibliogr. 12 poz.
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autor
- Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise, France
Bibliografia
- [1] T. Banica, Compact Kac algebras and commuting squares, J. Funct. Anal. 176 (2000), 80{99.
- [2] T. Banica and J. Bichon, Hopf images and inner faithful representations, Glasgow Math. J. 52 (2010), 677{703.
- [3] T. Banica, U. Franz and A. Skalski, Idempotent states and the inner linearity property, Bull. Polish Acad. Sci. Math. 60 (2012), 123{132.
- [4] I. Bengtsson, Three ways to look at mutually unbiased bases, in: Foundations of Probability in Physics|4, AIP Conf. Proc. 889, Amer. Inst. Phys., Melville, NY, 2007, 40{51.
- [5] P. Dita, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (2004), 5355{5374.
- [6] U. Franz and A. Skalski, On idempotent states on quantum groups, J. Algebra 322 (2009), 1774{1802.
- [7] V. F. R. Jones and V. S. Sunder, Introduction to Subfactors, Cambridge Univ. Press, 1997.
- [8] W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Systems Information Dynam. 13 (2006), 133{177.
- [9] S. Wang, Quantum symmetry groups of _nite spaces, Comm. Math. Phys. 195 (1998), 195{211.
- [10] R. F. Werner, All teleportation and dense coding schemes, J. Phys. A 34 (2001), 7081{7094.
- [11] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613{665.
- [12] S. L.Woronowicz, Tannaka{Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35{76.
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Bibliografia
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