  Precision spectroscopy has been the driving force for progress of our
physical understanding and still is a promising tool for the investigation of
new physics. Molecules offer transitions which allow tests that are not
possible in atomic systems. However, usually precision spectroscopy of
molecules is challenging due to the lack of cycling transitions for state
preparation and state detection. For molecular ions, this obstacle can be
overcome by quantum logic spectroscopy, where dissipation for state preparation
and detection is provided by a co-trapped atomic ion exploiting the shared
eigenstates of motion. Here, we propose a full quantum logic spectroscopy
scheme for molecular oxygen ions and theoretically investigate the feasibility
of quantum logic assisted state detection and preparation. Furthermore, we
provide coupling rates for a direct single-photon quadrupole excitation of a
vibrational overtone transition that can serve as a sensitive transition for
tests of a possible variation of the proton-to-electron mass ratio.
