# A Query about an article on quantum synchronization

#### Danny Boy

Summary
Query in attached paper on quantum synchronization.
I am currently studying this paper on quantum synchronization. The first page gives an introduction to synchronization and the basic setup of the ensembles in the cavity. My query is on the second page where the following statements are made.

Because of the $U(1)$ symmetry, $\langle \hat{\sigma}^{\pm}_{(A,B)j} \rangle = 0$. Therefore, all nonzero observables can be expressed in terms of $\langle \hat{\sigma}^{z}_{(A,B)j} \rangle, \langle \hat{\sigma}^{+}_{(A,B)i}\hat{\sigma}^{-}_{(A,B)j} \rangle$ and $\langle \hat{\sigma}^{z}_{(A,B)i}\hat{\sigma}^{z}_{(A,B)j} \rangle.$
Can anyone see why the implication is that all observables can be expressed by these terms?

Thanks for any assistance.

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