Hannover 2010 – scientific programme

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Q: Fachverband Quantenoptik und Photonik

Q 52: Quantum Information: Concepts and Methods IV / Photons and Nonclassical Light I

Q 52.4: Talk

Thursday, March 11, 2010, 14:45–15:00, E 214

Geometry of Dynamical Quantum Systems — •Robert Zeier and Thomas Schulte-Herbrüggen — Technische Universität München, Department Chemie, Lichtenbergstr. 4, 85747 Garching

We relate the geometry of dynamical quantum systems to the broader context of classifying Lie algebras. We give an explicit description of all possible geometries and their inclusion relations relying on results of Dynkin [1] and complementing the work of McKay and Patera [2]. Building on previous work [3,4], we use the description of all possible geometries to present readily applicable conditions for the controllability of quantum systems. We compare our approach with the standard method of deciding controllability by computing the Lie closure [5]. We emphasize the importance of our methods for the universality of quantum computers and consider partial universality with respect to subsystems. We discuss computer implementations and present concrete examples.
[1] Borel/Siebenthal, Comment. Math. Helv. 23, 200 (1949); Dynkin, Trudy Mosov. Mat. Obsh. 1, 39 (1952), Amer. Math. Soc. Transl. (2) 6, 245 (1957); Dynkin, Mat. Sbornik (N.S.) 30(72), 349 (1952), Amer. Math. Soc. Transl. (2) 6, 111 (1957)

[2] McKay/Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras (1981)

[3] Sander/Schulte-Herbrüggen, http://arxiv.org/abs/0904.4654

[4] Polack/Suchowski/Tannor, Phys. Rev. A 79, 053403 (2009)

[5] Jurdjevic/Sussmann, J. Diff. Eq. 12, 313 (1972)