München 2009 – scientific programme
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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik
MP 3: Quantentheorie großer Systeme
MP 3.1: Talk
Tuesday, March 10, 2009, 14:00–14:20, M010
Viewing Markovian Quantum Channels as Lie Semigroups and GKS-Lindblad Generators as Lie Wedge: New Perspectives and Applications — •Thomas Schulte-Herbrüggen1, Gunther Dirr2, Indra Kurniawan2, and Uwe Helmke2 — 1Technical University Munich (TUM), Dept. Chemistry — 2University of Würzburg, Institute of Mathematics
Optimal control of Markovian and non-Markovian open quantum systems cuts errors typically by one order of magnitude [1] in realistic settings. Yet open systems require more intricate theoretical concepts of controllability than their closed counterparts [2,3]. We present such new concepts in terms of Lie semigroups and Lie semialgebras [3].
On a general scale, Markovian quantum channels (with det>0) recently characterised by their divisibility [4] can now be defined in the more general frame of invariant cones by their Lie-semigroup properties [3] with the GKS-Lindblad generators as Lie wedge. Its geometry proves powerful for addressing reachability as well as for numerical algorithms both in optimal control and in optimisation on various types of reachable sets within quantum state-space manifolds [2,3].
[1] Schulte-Herbrüggen, Spörl, Khaneja, Glaser, quant-ph/0609037;
Rebentrost, Serban, Schulte-H., Wilhelm, quant-ph/0612165
[2] Schulte-Herbrüggen, Dirr, Helmke, Glaser, arXiv:0802.4195
[3] Dirr, Helmke, Kurniawan, Schulte-Herbrüggen, arXiv:0811.3906
[4] Wolf and Cirac, Commun. Math. Phys. 279, 147 (2008);
Wolf, Eisert, Cubitt, Cirac, PRL 101, 150402 (2008)