Query about an article on quantum synchronization

[Danny Boy]

TL;DR

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.

 

[azntoon]

The statement is referring to the fact that, due to the $U(1)$ symmetry, the expectation values of the raising and lowering operators are zero. This means that any observables that can be expressed in terms of these operators will also have zero expectation values. As such, the only nonzero observables are those that can be expressed in terms of the $\langle \hat{\sigma}^{z}_{(A,B)j} \rangle, \langle \hat{\sigma}^{+}_{(A,B)i}\hat{\sigma}^{-}_{(A,B)j} \rangle$ and $\langle \hat{\sigma}^{z}_{(A,B)i}\hat{\sigma}^{z}_{(A,B)j} \rangle$ operators, as these are the only ones which do not contain raising or lowering operators.