  Quantum-logic techniques used to manipulate quantum systems are now
increasingly being applied to molecules. Previous experiments on single trapped
diatomic species have enabled state detection with excellent fidelities and
highly precise spectroscopic measurements. However, for complex molecules with
a dense energy-level structure improved methods are necessary. Here, we
demonstrate an enhanced quantum protocol for molecular state detection using
state-dependent forces. Our approach is based on interfering a reference and a
signal force applied to a single atomic and molecular ion, respectively, in
order to extract their relative phase. We use this phase information to
identify states embedded in a dense molecular energy-level structure and to
monitor state-to-state inelastic scattering processes. This method can also be
used to exclude a large number of states in a single measurement when the
initial state preparation is imperfect and information on the molecular
properties is incomplete. While the present experiments focus on N$_2^+$, the
method is general and is expected to be of particular benefit for polyatomic
systems.
