Symmetry principles in quantum system theory of multi-qubit systems made simple

Controllability and observability of multi-spin systems in architectures of various symmetries of coupling type and topology are investigated. We complement recent work of explicitly determining the respective dynamic system Lie algebras and thereby also precise reachability sets under symmetry constraints. Here the focus is on the converse: under which conditions can the absence of symmetry be taken as an indicator of universality of the hardware architecture? More precisely, the absence of symmetry implies irreducibility and provides a convenient necessary condition for full controllability. Though much easier to assess than the well-established Lie-algebra rank condition, this is not sufficient unless in an n-qubit system with connected coupling topology the candidate dynamic simple Lie algebra can be identified uniquely as the full unitary algebra su(2ⁿ). - Here we discuss simple tests confined to solving homogeneous linear equations in order to filter irreducible unitary representations of other candidate algebras of classical type (orthogonal ones and unitary symplectic ones). Finally, we give an outlook under which conditions algebras of exceptional type can also be ruled out.