Berlin 2015 – wissenschaftliches Programm
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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik
MP 2: Quanteninformation
MP 2.2: Vortrag
Dienstag, 17. März 2015, 11:15–11:35, HFT-FT 101
Tensor square representations of Lie algebras and quantum control theory — •Zoltán Zimborás1, Robert Zeier2, Thomas Schulte-Herbrüggen2, and Daniel Burgarth3 — 1University College London, UK — 2Technische Universität München, Germany — 3Aberystwyth University, UK
We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicites and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. The sum of squares of multiplicities is equal to the dimension of the commutant of all complex matrices commuting with the tensor square representation. Hence, our results offer a test whether a subalgebra of a compact semisimple Lie algebra is a proper one, which uses only linear-algebra computations on sets of generators without calculating the relevant Lie closures. At the end of the talk, we show that this test can be naturally applied in the control theory of quantum systems.