  Bounds to the speed of evolution of a quantum system are of fundamental
interest in quantum metrology, quantum chemical dynamics and quantum
computation. We derive a time-energy uncertainty relation for open quantum
systems undergoing a general, completely positive and trace preserving (CPT)
evolution which provides a bound to the quantum speed limit. When the evolution
is of the Lindblad form, the bound is analogous to the Mandelstam-Tamm relation
which applies in the unitary case, with the role of the Hamiltonian being
played by the adjoint of the generator of the dynamical semigroup. The utility
of the new bound is exemplified in different scenarios, ranging from the
estimation of the passage time to the determination of precision limits for
quantum metrology in the presence of dephasing noise.
