= Solution
With every bath spin up, the two conditional bath states in the <Zurek spin-bath model> differ only by global phases. The <decoherence factor> is
$$
\boxed{z(t)=e^{2it\sum_k g_k},\qquad |z(t)|^2=1}.
$$
There is \b[no quantum decoherence], even for a very large bath. The device evolves as the pure state $a e^{it\sum g_k}|0\rangle+b e^{-it\sum g_k}|1\rangle$, while the bath stays in its original product <eigenstate> up to phase. Since no device information is imprinted in distinguishable bath states, the <conditional environment overlap> has unit modulus. The phase rotation must not be mistaken for decay of off-diagonal magnitude.
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