= Symplectic invariant of a density operator
{title2=$I_J(\rho)=\operatorname{Tr}(\rho J\rho^TJ^\dagger)$}
For $U^TJU=J$ with real orthogonal skew-symmetric $J$, the dual $\widetilde\rho=J\rho^TJ^\dagger$ transforms as $\widetilde{U\rho U^\dagger}=U\widetilde\rho U^\dagger$. 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 $2(a^2+b^2)$.
Back to article page
