= Symplectic dynamical symmetry in quantum control
{title2=$X^TJ+JX=0\Rightarrow U^TJU=J$}
For a nonsingular skew-symmetric unitary $J$, if each skew-Hermitian control generator obeys $X^TJ+JX=0$, then so do their real linear combinations and <commutators>. Differentiating $U^TJU$ shows that it stays equal to $J$. The reachable group is therefore contained in a unitary conjugate of the <compact symplectic group>. This is preservation of an alternating bilinear form, not commutation with $J$.
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