Higher order traps for some strongly degenerate quantum control systems

Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor, "Are there traps in quantum control landscapes?", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.