Geodesic
On moduli space of the Wigner quasiprobability distributions for $N$-dimensional quantum systems
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 177-201 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A mapping between operators on the Hilbert space of $N$-dimensional quantum system and the Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich–Weyl kernel. It is shown that the moduli space of the Stratonovich–Weyl kernel is given by an intersection of the coadjoint orbit space of the $SU(N)$ group and a unit $(N-2)$-dimensional sphere. The general consideration is exemplified by a detailed description of the moduli space of $2,3$ and $4$-dimensional systems. 
@article{ZNSL_2018_468_a12,
     author = {V. Abgaryan and A. Khvedelidze and A. Torosyan},
     title = {On moduli space of the {Wigner} quasiprobability distributions for $N$-dimensional quantum systems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {177--201},
     year = {2018},
     volume = {468},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a12/}
}
TY  - JOUR
AU  - V. Abgaryan
AU  - A. Khvedelidze
AU  - A. Torosyan
TI  - On moduli space of the Wigner quasiprobability distributions for $N$-dimensional quantum systems
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2018
SP  - 177
EP  - 201
VL  - 468
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a12/
LA  - en
ID  - ZNSL_2018_468_a12
ER  - 
%0 Journal Article
%A V. Abgaryan
%A A. Khvedelidze
%A A. Torosyan
%T On moduli space of the Wigner quasiprobability distributions for $N$-dimensional quantum systems
%J Zapiski Nauchnykh Seminarov POMI
%D 2018
%P 177-201
%V 468
%U http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a12/
%G en
%F ZNSL_2018_468_a12
V. Abgaryan; A. Khvedelidze; A. Torosyan. On moduli space of the Wigner quasiprobability distributions for $N$-dimensional quantum systems. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 177-201. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a12/

[1] Johann v. Neumann, Mathematische Grundlagen der Quantenmechanik, Verlang von Julius Springer, Berlin, 1932 | MR

[2] H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel-Verlag, Leipzig, 1928 | MR | Zbl

[3] E. P. Wigner, “On the quantum correction for thermodynamic equilibrium”, Phys. Rev., 40 (1932), 749–759 | DOI | Zbl

[4] K. Husimi, “Some formal properties of the density matrix”, Proc. Phys. Math. Soc. Japan, 22 (1940), 264–314 | Zbl

[5] H. J. Groenewold, “On the principles of elementary quantum mechanics”, Physica, 12 (1946), 405–460 | DOI | MR | Zbl

[6] J. E. Moyal, “Quantum mechanics as a statistical theory”, Mathematical Proceedings of the Cambridge Philosophical Society, 45 (1949), 99–124 | DOI | MR | Zbl

[7] G. Dito, D. Sternheimer, Deformation quantization: genesis, developments and metamorphoses, 2002, arXiv: math/0201168 | MR | Zbl

[8] E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams”, Phys. Rev. Lett., 10 (1963), 277–279 | DOI | MR | Zbl

[9] R. J. Glauber, “Coherent and incoherent states of the radiation field”, Phys. Rev., 131 (1963), 2766–2788 | DOI | MR | Zbl

[10] M. Hillery, R. F. O'Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: Fundamentals”, Phys. Rep., 106 (1984), 121–167 | DOI | MR

[11] D. J. Rowe, B. C. Sanders, H. de Guise, “Representations of the Weyl group and Wigner functions for SU(3)”, J. Math. Phys., 40 (1999), 3604 | DOI | MR | Zbl

[12] S. Chumakov, A. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems”, Phys. Rev. A, 61 (2000), 034101 | DOI

[13] M. A. Alonso, G. S. Pogosyan, K. B. Wolf, “Wigner functions for curved spaces”, J. Math. Phys., 43 (2002), 5857 | DOI | MR | Zbl

[14] A. I. Lvovsky, M. G. Raymer, “Continuous-variable optical quantum-state tomography”, Rev. Mod. Phys., 81 (2009), 299–332 | DOI

[15] A. B. Klimov, H. de Guise, “General approach to $\mathfrak{GU}(n)$ quasi-distribution functions”, J. Phys. A, 43 (2010), 402001 | DOI | MR | Zbl

[16] I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Rehacek, Z. Hradil, “Orbital angular momentum in phase space”, Ann. Phys., 326 (2011), 426–439 | DOI | MR | Zbl

[17] T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, K. Nemoto, “Wigner functions for arbitrary quantum systems”, Phys. Rev. Lett., 117 (2016), 180401 | DOI

[18] A. B. Klimov, J. L. Romero, H. de Guise, “Generalized SU(2) covariant Wigner functions and some of their applications”, J. Phys. A, 50 (2017), 323001 | DOI | MR | Zbl

[19] M. O. Scully, M. S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, 2006

[20] I. Bengtsson, K. E. Życzkowski, Geometry of Quantum States. An Introduction to Quantum Entanglement, 2nd Edition, Cambridge University Press, Cambridge, 2017 | MR | Zbl

[21] R. P. Feynman, “Negative Probaility”, Quantum Implications, Essays in Honour of David Bohm, eds. Basil J. Hiley, D. Peat, Methuen, 1987, 235–248 | MR

[22] R. L. Stratonovich, “On distributions in representation space”, Sov. Phys. JETP, 4:6 (1957), 891–898 | MR

[23] C. Briff, A. Mann, “Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries”, Phys. Rev., 59 (1999), 971 | DOI | MR

[24] A. Khvedelidze, V. Abgaryan, On the Family of Wigner Functions for $N$-Level Quantum System, 2018, arXiv: 1708.05981

[25] A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, 64, Amer. Math. Soc., 2004 | DOI | MR | Zbl

[26] D. Weingarten, “Asymptotic behavior of group integrals in the limit of infinite rank”, J. Mat. Phys., 19 (1978), 999–1001 | DOI | MR | Zbl

[27] B. Collins, “Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability”, Int. Math. Res. Not., 17 (2003), 953–982 | DOI | MR | Zbl

[28] B. Collins, P. Sniady, “Integration with respect to the Haar measure on unitary, orthogonal and symplectic group”, Comm. Math. Phys., 264:3 (2006), 773–795 | DOI | MR | Zbl

[29] A. Luis, “A SU(3) Wigner function for three-dimensional systems”, J. Phys. A, 41 (2008), 495302 | DOI | MR | Zbl

[30] V. I. Arnold, “Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect”, Selecta Mathematica, New Series, 1:1 (1995), 1–19 | DOI | MR | Zbl