  We derive a thermodynamic uncertainty relation for general open quantum
dynamics, described by a joint unitary evolution on a composite system
comprising a system and an environment. By measuring the environmental state
after the system-environment interaction, we bound the counting observables in
the environment by the survival activity, which reduces to the dynamical
activity in classical Markov processes. Remarkably, the relation derived herein
holds for general open quantum systems with any counting observable and any
initial state. Therefore, our relation is satisfied for classical Markov
processes with arbitrary time-dependent transition rates and initial states. We
apply our relation to continuous measurement and the quantum walk to find that
the quantum nature of the system can enhance the precision. Moreover, we can
make the lower bound arbitrarily small by employing appropriate continuous
measurement.
