 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 isThere is no quantum decoherence, even for a very large bath. The device evolves as the pure state $a{e}^{it\sum g_k}|0⟩+b{e}^{-it\sum g_k}|1⟩$, 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.