Single-qubit environment dephasing model ($z(t)=\cos (2gt/\hslash )$)

A detector qubit with $H=g{Z}_D\otimes X_E$ and environment initially $|0{⟩}_E$ generates conditional states $|E_{\pm }⟩=\cos A|0⟩\pm i\sin A|1⟩$, where $A=gt/\hslash$. Their overlap is $\cos 2A$. Tracing out the environment multiplies detector coherences by this number while preserving the populations. Orthogonal records occur at $A=\pi /4$, but coherence returns later. For equal initial detector populations the reduced purity is $(1+{\cos }^{2}2A)/2$. Thus complete reduced dephasing at one time does not imply irreversible global collapse.