  Quantum speed limits provide upper bounds on the rate with which a quantum
system can move away from its initial state. Here, we provide a different kind
of speed limit, describing the divergence of a perturbed open system from its
unperturbed trajectory. In the case of weak coupling, we show that the
divergence speed is bounded by the quantum Fisher information under a
perturbing Hamiltonian, up to an error which can be estimated from system and
bath timescales. We give two applications of our speed limit. Firstly, it
enables experimental estimation of quantum Fisher information in the presence
of decoherence that is not fully characterised. Secondly, it implies that large
quantum work fluctuations are necessary for a thermal system to be driven
quickly out of equilibrium under a quench.
