Abstract:

A dynamical quantum system describes the time evolution of a controlled quantum systems which is governed by drift and control Hamiltonians. We relate the symmetry properties of a dynamical quantum systems to the question of full controllability, i.e., if any unitary transformation can be implemented using the available controls.
The absence of symmetry implies irreducibility and provides a convenient necessary condition for full controllability much easier to assess than the well-established Lie-algebra rank condition. Curiously enough, there exist irreducible subalgebras without any symmetry that are not fully controllable. Thereby, the question of full controllability is linked to the classification of irreducible simple subalgebras of special unitary algebras. We present the complete lattice of irreducible simple subalgebras of su(2^n) for up to n=15 qubits.
This complements the symmetry condition by allowing for easy tests solving homogeneous linear equations to filter irreducible representations of candidate algebras of classical and exceptional type.
In particular, we provide a single necessary and sufficient symmetry condition for full controllability. We present a plethora of examples showing under which conditions spin systems and fermionic systems with pair or higher many-body interactions can mutually simulate eachother.

(joint work with Thomas Schulte-Herbrüggen, see J. Math. Phys. 52, 113510 (2011); http://dx.doi.org/10.1063/1.3657939 )