Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 50 / 1 / 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 $H_0=Z/2$, $H_1=X$, ${\rho }_d(0)=|0⟩⟨0|$ and $\rho (0)=|1⟩⟨1|$. Both density operators commute with $H_0$, and the commutator of $X$ with $\rho$ has zero diagonal, so $f=\mathrm{Tr}({\rho }_d[-iX,\rho ])=0$. Both states remain constant. They have the identical spectrum $(1,0)$, but $\boxed{\Vert \rho (t)-{\rho }_d(t){\Vert }_{\mathrm{HS}}=\sqrt{2}\text{ for all }t}$. 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.