Geodesic
Algebras of projectors and mutually unbiased bases in dimension~7
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Quantum computing, Tome 138 (2017), pp. 19-49.

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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 $7$.
Keywords: mutually unbiased bases, orthogonal pairs
Mots-clés : commutation relation, algebra of observables. 
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     author = {I. Yu. Zhdanovskii and A. S. Kocherova},
     title = {Algebras of projectors and mutually unbiased bases in dimension~7},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {19--49},
     publisher = {mathdoc},
     volume = {138},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2017_138_a2/}
}
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I. Yu. Zhdanovskii; A. S. Kocherova. Algebras of projectors and mutually unbiased bases in dimension~7. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Quantum computing, Tome 138 (2017), pp. 19-49. http://geodesic.mathdoc.fr/item/INTO_2017_138_a2/