The Appendix to a recent paper Graph States as a Resource for Quantum Metrology states:
We model an $n$ qubit graph state $G$ undergoing iid dephasing via $$G \to G^{\text{dephasing}} = \sum_{\vec{k}}p^{k}(1-p)^{n-k}Z_{\vec{k}}GZ_{\vec{k}}$$ where $p$ is the probability that a qubit undergoes a phase flip. This effectively maps the graph state onto the orthonormal basis $\{Z_{\vec{k}}|G\rangle\}_{\vec{k}}$.
Question: Is it clear why this expression is given in this form (in particular what does the $Z_{\vec{k}}$ denote and how do we know that $\{Z_{\vec{k}}|G\rangle\}_{\vec{k}}$ is an orthonormal basis)?
These might be trivial questions but I am new to this area hence some of the conventions I am still unfamiliar with.