Abstract
We apply methods of the representation theory, combinatorial algebra, and noncommutative geometry to various problems of quantum tomography. We introduce the algebra of projectors that satisfy a certain commutation relation, examine this relation by combinatorial methods, and develop the representation theory of this algebra. We also present a geometrical interpretation of our problem and apply the results obtained to the description of the Petrescu family of mutually unbiased bases in dimension 7.
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References
A. Bondal and I. Zhdanovskiy, Representation theory for system of projectors and discrete Laplace operator, Preprint IPMU13-0001.
A. Bondal and I. Zhdanovskiy, “Orthogonal pairs and mutually unbiased bases,” J. Math. Sci., 216, No. 1, 23–40 (2016).
A. Bondal and I. Zhdanovskiy, “Symplectic geometry of unbiasedness and critical points of a potential,” Adv. Stud. Pure Math. (to appear); arXiv:1507.00081.
T. Durt, B.-G. Englert, I. Bengtsson, and K. Zyczkowski, “On mutually unbiased bases,” Int. J. Quantum Inform., 8, 535 (2010).
U. Haagerup, “Orthogonal maximal abelian *-subalgebras of the n × n matrices and cyclic nroots,” in: Operator Algebras and Quantum Field Theory (S. Doplicher et al., eds.), International Press (1997), pp. 296–322.
A. I. Kostrikin, I. A. Kostrikin, and V. A. Ufnarovski, “Orthogonal decompositions of simple Lie algebras (type A n),” Tr. Mat. Inst. Steklova, 158, 105–120 (1981).
A. I. Kostrikin and Pham Huu Tiep, Orthogonal Decompositions and Integral Lattices, Walter de Gruyter (1994).
M. Matolcsi and F. Szollosi, “Towards a classification of 6 × 6 complex Hadamard matrices,” Open. Syst. Inf. Dyn., 15, 93 (2008).
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Berlin (1955).
R. Nicoara, “A finiteness result for commuting squares of matrix algebras,” J. Operator Theory, 55, No. 2, 295–310 ((2006)); arXiv:math/0404301.
M. Petrescu, Existence of continuous families of complex Hadamard matrices of certain prime dimensions and related results, Ph.D. Thesis, University of California Los Angeles (1997).
S. Popa, “Orthogonal pairs of *-subalgebras in finite von Neumann algebras,” J. Operator Theory, 9, 253–268 (1983).
M. B. Ruskai, “Some connections between frames, mutually unbiased bases, and POVM’s in quantum information theory,” Acta Appl. Math., 108, No. 3, 709–719 (2009).
S. Sakai, C ∗ -Algebras and W ∗ -Algebras, Springer-Verlag, New York–Heidelberg–Berlin (1971).
F. Szollosi, “Complex Hadamard matrices of order 6: A four-parameter family,” J. Lond. Math. Soc., 85, 616–632 (2012); arXiv:http:1008.0632.
W. Tadej and K. Zyczkowski, “Defect of a unitary matrix,” Linear Algebra Appl., 429, 447–481 (2008).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 138, Quantum Computing, 2017.
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Zhdanovskiy, I.Y., Kocherova, A.S. Algebras of Projectors and Mutually Unbiased Bases in Dimension 7. J Math Sci 241, 125–157 (2019). https://doi.org/10.1007/s10958-019-04413-8
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DOI: https://doi.org/10.1007/s10958-019-04413-8