  Laser spectroscopy of cold and trapped molecular ions is a powerful tool for
fundamental physics, including the determination of fundamental constants, the
laboratory test for their possible variation, and the search for a possible
electric dipole moment of the electron. Optical clocks based on molecular ions
sensitive to some of these effects are expected to achieve uncertainties
approaching the $10^{-18}$ level. While the complexity of molecular structure
facilitates these applications, the absence of cycling transitions poses a
challenge for direct laser cooling, quantum state control, and detection.
Previously employed state detection techniques based on photo-dissociation or
chemical reactions are destructive and therefore inefficient. Here we
experimentally demonstrate non-destructive state detection of a single trapped
molecular ion through its strong Coulomb coupling to a well-controlled
co-trapped atomic ion. An algorithm based on a state-dependent optical dipole
force(ODF) changes the internal state of the atom conditioned on the internal
state of the molecule. We show that individual states in the molecule can be
distinguished by their coupling strength to the ODF and observe black-body
radiation-induced quantum jumps between rotational states. Using the detuning
dependence of the state detection signal, we implement a variant of quantum
logic spectroscopy and improve upon a previous measurement of the
$\mathrm{X}^1\Sigma^+(J=1)\rightarrow\mathrm{A}^1\Sigma^+(J=0)$ transition in
MgH, finding a frequency of 1067.74752(53)THz. We estimate that non-destructive
state detection with near 100% efficiency could take less than 10 ms. The
technique we demonstrate is applicable to a wide range of molecular ions,
enabling further applications in state-controlled quantum chemistry and
spectroscopic investigations of molecules serving as probes for interstellar
clouds.
