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