  We introduce a general framework of phase reduction theory for quantum
nonlinear oscillators. By employing the quantum trajectory theory, we define
the limit-cycle trajectory and the phase according to a stochastic
Schr\"{o}dinger equation. Because a perturbation is represented by unitary
transformation in quantum dynamics, we calculate phase response curves with
respect to generators of a Lie algebra. Our method shows that the continuous
measurement yields phase clusters and alters the phase response curves. The
observable clusters capture the phase dynamics of individual quantum
oscillators, unlike indirect indicators obtained from density operators.
Furthermore, our method can be applied to finite-level systems that lack
classical counterparts.
