In this work, given an initial state \(\rho \) and a time-independent Hamiltonian H, we investigate the problem of identifying the farthest state in the unitary evolution orbit of \(\rho \) with respect to the Hilbert–Schmidt fidelity. Our results generalize previous work on the qubit case Zhang (Phys Lett A 382:2599–2604, 2018) to higher-dimensional systems. We demonstrate that, under specific conditions on the Hamiltonian’s spectrum, both the farthest state and the minimal time required for \(\rho \) to evolve into this state can be determined explicitly. Moreover, by applying Kronecker’s theorem, we uncover a novel phenomenon in higher dimensions: the farthest state may fail to exist. In this case, the Hilbert–Schmidt inner product between the evolving state and the initial state asymptotically approaches, but never attains, its infimum value as \(t\rightarrow \infty . \qquad \qquad \)
