= Solution
\b[Neither monotonic decrease alone nor equality of the initial spectra guarantees convergence]. A nonnegative <Lyapunov function> can approach a positive limiting value, and the set where $\dot V=0$ can contain trajectories other than the target. Different initial spectra already obstruct exact tracking: <unitary time evolution> preserves the eigenvalues of each <density operator>, while two matrices approaching one another in norm must have the same limiting eigenvalues.
Even the spectral obstruction can be absent while the <Lyapunov quantum control> stalls. Take a <qubit> with $H_0=Z/2$, $H_1=X$, $\rho_d(0)=|0\rangle\langle0|$ and $\rho(0)=|1\rangle\langle1|$. Both <density operators> commute with $H_0$, and the <commutator> of $X$ with $\rho$ has zero diagonal, so $f=\operatorname{Tr}(\rho_d[-iX,\rho])=0$. Both states remain constant. They have the identical spectrum $(1,0)$, but $\boxed{\|\rho(t)-\rho_d(t)\|_{\mathrm{HS}}=\sqrt2\text{ for all }t}$. Moreover $iZ,iX$ generate the full <special unitary Lie algebra> $\mathfrak{su}(2)$, so the failure here is a limitation of this particular feedback law, not a lack of available state-transfer controls.
Back to article page
