Symplectic dynamical symmetry in quantum control ($X^{T}J+JX=0\Rightarrow U^{T}JU=J$)

For a nonsingular skew-symmetric unitary $J$, if each skew-Hermitian control generator obeys $X^{T}J+JX=0$, then so do their real linear combinations and commutators. Differentiating $U^{T}JU$ 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 $J$.

For $U^{T}JU=J$ with real orthogonal skew-symmetric $J$, the dual $\tilde{\rho }=J{\rho }^T{J}^{†}$ transforms as $\widetilde{U\rho U^{†}}=U\tilde{\rho }{U}^{†}$. Hence $I_J(\rho )$ is constant on the reachable unitary orbit of a density operator. Different values can obstruct density operator controllability even when the ordinary spectra agree. For a four-dimensional $J$ pairing indices $1,4$ and $2,3$, the two diagonal arrangements $(a,a,b,b)$ and $(a,b,b,a)$ have invariants $4ab$ and $2(a^2+b^2)$.