Geodesic
Some Algebraic and Geometric Aspects of Quantum Measurements
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 109-123.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study positive operator-valued measures by algebraic and geometric methods. We prove that positive operator-valued measures are parametrized by a Poisson manifold. Also, we show how to obtain symplectic leaves of this Poisson manifold in terms of parameters of the measures. In addition, we study the interaction of two projection-valued measures by the methods of algebraic geometry. 
@article{TM_2021_313_a9,
     author = {A. S. Kocherova and I. Yu. Zhdanovskiy},
     title = {Some {Algebraic} and {Geometric} {Aspects} of {Quantum} {Measurements}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {109--123},
     publisher = {mathdoc},
     volume = {313},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_313_a9/}
}
TY  - JOUR
AU  - A. S. Kocherova
AU  - I. Yu. Zhdanovskiy
TI  - Some Algebraic and Geometric Aspects of Quantum Measurements
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2021
SP  - 109
EP  - 123
VL  - 313
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2021_313_a9/
LA  - ru
ID  - TM_2021_313_a9
ER  - 
%0 Journal Article
%A A. S. Kocherova
%A I. Yu. Zhdanovskiy
%T Some Algebraic and Geometric Aspects of Quantum Measurements
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2021
%P 109-123
%V 313
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2021_313_a9/
%G ru
%F TM_2021_313_a9
A. S. Kocherova; I. Yu. Zhdanovskiy. Some Algebraic and Geometric Aspects of Quantum Measurements. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 109-123. http://geodesic.mathdoc.fr/item/TM_2021_313_a9/

[1] Belmans P., Presotto D., Van den Bergh M., Comparison of two constructions of noncommutative surfaces with exceptional collections of length 4, E-print, 2018, arXiv: 1811.08810v1 [math.AG] | MR | Zbl

[2] Bocklandt R., Le Bruyn L., “Necklace Lie algebras and noncommutative symplectic geometry”, Math. Z., 240:1 (2002), 141–167 | DOI | MR | Zbl

[3] Bondal A., Zhdanovskiy I., Representation theory for system of projectors and discrete Laplace operator, Preprint IPMU13-0001, Kavli Inst. Phys. Math. Universe, Kashiwa, 2013 http://research.ipmu.jp/ipmu/sysimg/ipmu/1012.pdf | MR

[4] A. Bondal and I. Zhdanovskiy, “Orthogonal pairs and mutually unbiased bases”, J. Math. Sci., 216:1 (2016), 23–40 | DOI | MR | Zbl

[5] Bondal A., Zhdanovskiy I., “Symplectic geometry of unbiasedness and critical points of a potential”, Primitive forms and related subjects—Kavli IPMU 2014, Adv. Stud. Pure Math., 83, Math. Soc. Japan, Tokyo, 2019, 1–18 | Zbl

[6] Castelnuovo G., “Su certi gruppi associati di punti”, Rend. Circ. Mat. Palermo, 3 (1889), 179–192 | DOI

[7] Chekhov L.O., Mazzocco M., Rubtsov V.N., “Painlevé monodromy manifolds, decorated character varieties, and cluster algebras”, Int. Math. Res. Not., 2017:24 (2017), 7639–7691 | MR | Zbl

[8] Crawley-Boevey W., “Geometry of the moment map for representations of quivers”, Compos. math., 126:3 (2001), 257–293 | DOI | MR | Zbl

[9] Crawley-Boevey W., “On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero”, Duke Math. J., 118:2 (2003), 339–352 | DOI | MR | Zbl

[10] D'Ariano G.M., Lo Presti P., Perinotti P., “Classical randomness in quantum measurements”, J. Phys. A: Math. Gen., 38:26 (2005), 5979–5991 | DOI | MR | Zbl

[11] Dolgachev I., Ortland D., Point sets in projective spaces and theta functions, Astérisque, 165, Soc. math. France, Paris, 1988 | MR | Zbl

[12] Eisenbud D., Popescu S., “The projective geometry of the Gale transform”, J. Algebra, 230:1 (2000), 127–173 | DOI | MR | Zbl

[13] Filippov S.N., Heinosaari T., Leppäjärvi L., “Necessary condition for incompatibility of observables in general probabilistic theories”, Phys. Rev. A, 95:3 (2017), 032127 | DOI | MR

[14] Gelfand I., Neumark M., “On the imbedding of normed rings into the ring of operators in Hilbert space”, Mat. sb., 12:2 (1943), 197–217 | MR

[15] Haapasalo E., Heinosaari T., Pellonpää J.-P., “Quantum measurements on finite dimensional systems: Relabeling and mixing”, Quantum Inf. Process., 11:6 (2012), 1751–1763 | DOI | MR | Zbl

[16] Heinosaari T., Jivulescu M.A., Nechita I., “Random positive operator valued measures”, J. Math. Phys., 61:4 (2020), 042202 | DOI | MR | Zbl

[17] Heinosaari T., Ziman M., The mathematical language of quantum theory: From uncertainty to entanglement, Cambridge Univ. Press, Cambridge, 2012 | MR | Zbl

[18] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982 | MR | Zbl

[19] Kac V.G., “Infinite root systems, representations of graphs and invariant theory”, Invent. math., 56 (1980), 57–92 | DOI | MR | Zbl

[20] Karzhemanov I.V., Maximum likelihood degree of surjective rational maps, E-print, 2018, arXiv: 1811.05176v1 [math.AG]

[21] Karzhemanov I., Zhdanovskiy I., “Some properties of surjective rational maps”, Eur. J. Math., 4:1 (2018), 326–329 | DOI | MR | Zbl

[22] Kocherova A.S., Zhdanovskiy I.Yu., “On the algebra generated by projectors with commutator relation”, Lobachevskii J. Math., 38:4 (2017), 670–687 | DOI | MR | Zbl

[23] Kocherova A.S., Zhdanovskiy I.Yu., Partial abelianization of free product of algebras, E-print, 2019, arXiv: 1912.05256 [math.RT] | Zbl

[24] Kostant B., “On convexity, the Weyl group and the Iwasawa decomposition”, Ann. sci. Éc. norm. supér., 6 (1973), 413–455 | DOI | MR | Zbl

[25] Kraus K., States, effects, and operations: Fundamental notions of quantum theory, Lect. Notes Phys., 190, Springer, Berlin, 1983 | DOI | MR | Zbl

[26] Simpson C.T., “Products of matrices”, Differential geometry, global analysis, and topology, Proc. Spec. Session Can. Math. Soc. Summer Meet. (Halifax, 1990), CMS Conf. Proc., 12, Amer. Math. Soc., Providence, RI, 1991, 157–185 | MR

[27] Zhdanovskiy I.Yu., “Homotopes of finite-dimensional algebras”, Commun. Algebra, 49:1 (2021), 43–57 | DOI | MR | Zbl

[28] I. Yu. Zhdanovskiy and A. S. Kocherova, “Algebras of projectors and mutually unbiased bases in dimension 7”, J. Math. Sci., 241:2 (2019), 125–157 | DOI | MR