  Phase synchronization was proved to be unbounded in quantum level, but the
witness of phase synchronization is always expensive in terms of the quantum
resource and non-local measurements involved. Based on the quantum uncertainty
relation, we construct two local criterions for the phase synchronization in
this paper. The local criterions indicate that the phase synchronization in the
quantum level can be witnessed only by the local measurements, and the
deduction has been verified in the optomechanics system in numerical way.
Besides, by analyzing the physical essence of the phase synchronization in
quantum level, we show that one can prepare a state, which describes two
synchronized oscillators with no entanglement between them. Thus, the
entanglement resource is not necessary in the occurrence of the ideal phase
synchronization, and also the reason for this phenomenon is discussed.
