Controllability and Observability of Multi-Spin Systems: Constraints by Symmetry and by Relaxation

We investigate the universality of multi-spin systems in architectures of various symmetries of coupling type and topology. Explicit reachability sets under symmetry constraints are provided. Thus for a given (possibly symmetric) experimental coupling architecture several decision problems can be solved in a unified way: (i) can a target Hamiltonian be simulated? (ii) can a target gate be synthesised? (iii) to which extent is the system observable by a given set of detection operators? and, as a special case of the latter, (iv) can an underlying system Hamiltonian be identified with a given set of detection operators? Finally, in turn, lack of symmetry provides a convenient necessary condition for full controllability. Though often easier to assess than the well-established Lie-algebra rank condition, this is not sufficient unless the candidate dynamic simple Lie algebra can be pre-identified uniquely, which is fortunately less complicated than expected.