Symmetry is a fundamentally important concept in many branches of physics. Inthis work, we discuss two types of symmetries, external symmetry and internalsymmetry, which appear frequently in controlled quantum spin chains and applythem to study various controllability problems. For spin chains under singlelocal end control when external symmetries exists, we can rigorously prove thatthe system is controllable in each of the invariant subspaces for both XXZ andXYZ chains, but not for XX or Ising chains. Such results have directapplications in controlling antiferromagnetic Heisenberg chains when thedynamics is naturally confined in the largest excitation subspace. We alsoaddress the theoretically important question of minimal control resources toachieve full controllability over the entire spin chain space. In the processwe establish a systematic way of evaluating the dynamical Lie algebras andusing known symmetries to help identify the dynamical Lie algebra.
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