Quantum speed limit for thermal states

Quantum speed limits are rigorous estimates on how fast a state of quantum system can depart from the initial in course evolution. Most known limits, including celebrated Mandelstam-Tamm and Margolus-Levitin ones, general bounds applicable to arbitrary states. However, when applied mixed states many-body systems, they, as rule, dramatically overestimate evolution fail provide meaningful thermodynamic limit. Here we derive limit for closed initially prepared thermal evolving under time-dependent Hamiltonian. This exploits structure and, particular, explicitly depends temperature. In broad class setups it proves be drastically stronger than limits.