Collective dephasing ($U_{\varphi }=\exp [-i\varphi {\sum }_j{Z}_j/2]$)

Correlated dephasing caused by the same fluctuating $z$ rotation on all qubits. Averaging $U_{\varphi }\rho U_{\varphi }^{†}$ over the unknown angle suppresses quantum coherence between sectors with different total $Z$ eigenvalues. Within one eigenspace, however, every state acquires the same global phase, so its density matrix is preserved. Such an eigenspace is a decoherence-free subspace for this noise model. The assumption concerns the shared rotation; independent local phase noise generally destroys this protection.