= Orthogonal dynamical symmetry in quantum control
{title2=$x^TJ+Jx=0\Rightarrow U^TJU=J$}
Let $J$ be a symmetric unitary matrix and every skew-Hermitian control generator obey $x^TJ+Jx=0$. The condition is closed under real linear combinations and <commutators>, and evolution preserves $U^TJU=J$. In a unitary basis where $J$ becomes the identity, the generators are real antisymmetric, so the <Dynamical Lie algebra> is contained in $\mathfrak{so}(N)$. The state quantity $|\psi^TJ\psi|$ is invariant under the reachable evolution and insensitive to global phase, obstructing unrestricted <pure-state controllability>. This is a bilinear-form symmetry; it does not assert that $J$ commutes with the Hamiltonians.
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