= Single-qubit environment dephasing model
{title2=$z(t)=\cos(2gt/\hbar)$}
A detector qubit with $H=gZ_D\otimes X_E$ and environment initially $|0\rangle_E$ generates conditional states $|E_\pm\rangle=\cos A|0\rangle\pm i\sin A|1\rangle$, where $A=gt/\hbar$. Their overlap is $\cos2A$. 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 $(1+\cos^22A)/2$. Thus complete reduced dephasing at one time does not imply irreversible global collapse.
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