A universal gap for non-spin quantum control systems

Gauthier, Jean-Paul; Rossi, Francesco

doi:10.1090/proc/15301

A universal gap for non-spin quantum control systems
HTML articles powered by AMS MathViewer

by Jean-Paul Gauthier and Francesco Rossi PDF

Proc. Amer. Math. Soc. 149 (2021), 1203-1214 Request permission

Abstract:

We prove the existence of a universal gap for minimum time controllability of finite dimensional quantum systems, except for some basic representations of spin groups.

This is equivalent to the existence of a gap in the diameter of orbit spaces of the corresponding compact connected Lie group unitary actions on the Hermitian spheres.

References

Francesca Albertini and Domenico D’Alessandro, Notions of controllability for bilinear multilevel quantum systems, IEEE Trans. Automat. Control 48 (2003), no. 8, 1399–1403. MR 2004373, DOI 10.1109/TAC.2003.815027
Arthur L Besse, Manifolds all of whose geodesics are closed, vol. 93, Springer Science & Business Media, 2012.
Ugo Boscain, Jean-Paul Gauthier, Francesco Rossi, and Mario Sigalotti, Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems, Comm. Math. Phys. 333 (2015), no. 3, 1225–1239. MR 3302633, DOI 10.1007/s00220-014-2195-6
Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344, DOI 10.1007/978-3-662-12918-0
Robert N Cahn, Semi-simple lie algebras and their representations, Courier Corporation, 2014.
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
William D. Dunbar, Sarah J. Greenwald, Jill McGowan, and Catherine Searle, Diameters of 3-sphere quotients, Differential Geom. Appl. 27 (2009), no. 2, 307–319. MR 2503981, DOI 10.1016/j.difgeo.2008.10.011
Evgenii Borisovich Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Transl. (2) 6 (1957), 245–378. 666
Jürgen Fuchs and Christoph Schweigert, Symmetries, lie algebras and representations: A graduate course for physicists, Cambridge University Press, 2003.
Roe Goodman and Nolan R. Wallach, Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol. 255, Springer, Dordrecht, 2009. MR 2522486, DOI 10.1007/978-0-387-79852-3
Claudio Gorodski, Christian Lange, Alexander Lytchak, and Ricardo A. E. Mendes, A diameter gap for quotients of the unit sphere, 2019.
Claudio Gorodski and Alexander Lytchak, On orbit spaces of representations of compact Lie groups, J. Reine Angew. Math. 691 (2014), 61–100. MR 3213548, DOI 10.1515/crelle-2012-0084
Claudio Gorodski and Alexander Lytchak, Isometric actions on spheres with an orbifold quotient, Math. Ann. 365 (2016), no. 3-4, 1041–1067. MR 3521081, DOI 10.1007/s00208-015-1304-y
Claudio Gorodski and Alexander Lytchak, The curvature of orbit spaces, Geom. Dedicata 190 (2017), 135–142. MR 3704815, DOI 10.1007/s10711-017-0231-3
Ben Green, On the width of transitive sets: bounds on matrix coefficients of finite groups, Duke Math. J. 169 (2020), no. 3, 551–578. MR 4065149, DOI 10.1215/00127094-2019-0074
Sarah J. Greenwald, Diameters of spherical Alexandrov spaces and curvature one orbifolds, Indiana Univ. Math. J. 49 (2000), no. 4, 1449–1479. MR 1836537, DOI 10.1512/iumj.2000.49.1860
Jill McGowan, The diameter function on the space of space forms, Compositio Math. 87 (1993), no. 1, 79–98. MR 1219453
Mazyar Mirrahimi, Lyapunov control of a quantum particle in a decaying potential, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 5, 1743–1765. MR 2566708, DOI 10.1016/j.anihpc.2008.09.006
Clément Sayrin, Igor Dotsenko, Xingxing Zhou, Bruno Peaudecerf, Théo Rybarczyk, Sébastien Gleyzes, Pierre Rouchon, Mazyar Mirrahimi, Hadis Amini, Michel Brune, Jean-Michel Raimond, and Serge Haroche, Real-time quantum feedback prepares and stabilizes photon number states, Nature 477 (2011), no. 7362, 73–77.