Quantum speed limit for states with a bounded energy spectrum

Quantum speed limits set the maximal pace of state evolution. Two well-known limits exist for a unitary time-independent Hamiltonian: the Mandelstam-Tamm and Margolus-Levitin bounds. The former restricts the rate according to the state energy uncertainty, while the latter depends on the mean energy relative to the ground state. Here we report on an additional bound that exists for states with a bounded energy spectrum. This bound is dual to the Margolus-Levitin one in the sense that it depends on the difference between the state's mean energy and the energy of the highest occupied eigenstate. Each of the three bounds can become the most restrictive one, depending on the spread and mean of the energy, forming three dynamical regimes which are accessible in a multi-level system. The new bound is relevant for quantum information applications, since in most of them, information is stored and manipulated in a Hilbert space with a bounded energy spectrum.