  In quantum mechanics, measurements are dynamical processes and thus they
should be capable of inducing currents. The symmetries of the Hamiltonian and
measurement operator provide an organizing principle for understanding the
conditions for such currents to emerge. The central role is played by the
inversion and time-reversal symmetries. We classify the distinct behaviors that
emerge from single and repeated measurements, with and without coupling to a
dissipative bath. While the breaking of inversion symmetry alone is sufficient
to generate currents through measurements, the breaking of time-reversal
symmetry by the measurement operator leads to a dramatic increase in the
magnitude of the currents. We consider the dependence on the measurement rate
and find that the current is non-monotonic. Furthermore, nondegenerate
measurements can lead to current loops within the steady state even in the Zeno
limit.
