  In this paper, we study the synchronization of dissipative quantum harmonic
oscillators in the framework of quantum open system via the Active-Passive
Decomposition (APD) configuration. We show that two or more quantum systems may
be synchronized when the quantum systems of interest are embedded in
dissipative environments and influenced by a common classical system. Such a
classical system is typically termed as a controller, which (1) can drive
quantum systems to cross different regimes (e.g., from periodic to chaotic
motions) and (2) constructs the so-called Active-Passive Decomposition
configuration such that all the quantum objects under consideration may be
synchronized. The main findings of this paper is that we demonstrate that the
complete synchronizations measured by the standard quantum deviation may be
achieved for both stable regimes (quantum limit circles) and unstable regimes
(quantum chaotic motions). As an example, we numerically show in an
optomechanical setup that the complete synchronization can be realized in
quantum mechanical resonators.
