"Dynamical Quantum Systems: Controllability, Symmetries, and
Representation Theory"
We analyze the controllability of dynamical quantum systems. One can
decide controllability by computing the Lie closure [1] which is
sometimes cumbersome. These topics can be discussed likewise for
translationally invariant lattices [2]. Building on previous work [3,4],
we propose an additional method which utilizes the symmetry properties
of the considered system. We obtain as a necessary condition for
controllability that the system should not have any symmetries and
act therefore irreducibly. But this condition is not sufficient as
there exist irreducible subalgebras of the maximal possible system
Lie algebra. We classify the irreducible subalgebras and their inclusion
relations relying on results of Dynkin [5]. Using optimized computer
programs we can tabulate irreducible subalgebras up to dimension 215
(i.e. 15 qubits) complementing results of McKay and Patera [6]. For
concrete dynamical quantum systems many irreducible subalgebras can
be ruled out as obstructions for full controllability and we present
algorithms to this end. Our results provide an insight into the
question when spin, bosonic, and fermionic systems can simulate each
other. We will give a short introduction to the relevant
representation theory of Lie algebras.
[1] Jurdjevic/Sussmann, J. Diff. Eq. 12, 313 (1972)
[2] Kraus/Wolf/Cirac, Phys. Rev. A 75, 022303 (2007)
[3] Sander/Schulte-Herbrüggen, http://arxiv.org/abs/0904.4654
[4] Polack/Suchowski/Tannor, Phys. Rev. A 79, 053403 (2009)
[5] 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)
[6] McKay/Patera, Tables of Dimensions, Indices, and Branching Rules
for Representations of Simple Lie Algebras (1981)