  Polyatomic molecular ions contain many desirable attributes of a useful
quantum system, including rich internal degrees of freedom and highly
controllable coupling to the environment. To date, the vast majority of
state-specific experimental work on molecular ions has concentrated on diatomic
species. The ability to prepare and readout polyatomic molecules in single
quantum states would enable diverse experimental avenues not available with
diatomics, including new applications in precision measurement, sensitive
chemical and chiral analysis at the single molecule level, and precise studies
of Hz-level molecular tunneling dynamics. While cooling the motional state of a
polyatomic ion via sympathetic cooling with a laser cooled atomic ion is
straightforward, coupling this motional state to the internal state of the
molecule has proven challenging. Here we propose a new method for readout and
projective measurement of the internal state of a trapped polyatomic ion. The
method exploits the rich manifold of technically accessible rotational states
in the molecule to realize robust state-preparation and readout with far less
stringent engineering than quantum logic methods recently demonstrated on
diatomic molecules. The method can be applied to any reasonably small (less
than about 10 atoms) polyatomic ion with an anisotropic polarizability.
