  A genuine feature of projective quantum measurements is that they inevitably
alter the mean energy of the observed system if the measured quantity does not
commute with the Hamiltonian. Compared to the classical case, Jacobs proved
that this additional energetic cost leads to a stronger bound on the work
extractable after a single measurement from a system initially in thermal
equilibrium [Phys. Rev. A 80, 012322 (2009)]. Here, we extend this bound to a
large class of feedback-driven quantum engines operating periodically and in
finite time. The bound thus implies a natural definition for the efficiency of
information to work conversion in such devices. For a simple model consisting
of a laser-driven two-level system, we maximize the efficiency with respect to
the observable whose measurement is used to control the feedback operations. We
find that the optimal observable typically does not commute with the
Hamiltonian and hence would not be available in a classical two level system.
This result reveals that periodic feedback engines operating in the quantum
realm can exploit quantum coherences to enhance efficiency.
