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Let $B={\sum }_k{g}_k{Z}_k$. In the Zurek spin-bath model, $H=-Z_D\otimes B$, so unitary time evolution with $\hslash =1$ is $U(t)=e^{it{Z}_D\otimes B}$. The bath Hamiltonian terms commute, and device states $|0⟩,|1⟩$ have $Z_D$ eigenvalues $+1,-1$. Hencewhere the two normalized conditional bath states areTake each bath factor normalized, $|{\alpha }_k{|}^2+|{\beta }_k{|}^2=1$, and $|a{|}^2+|b{|}^2=1$. This entails no restriction: if only the product is initially normalized, divide each nonzero factor by its norm; the product of these norms is one.

Taking the partial trace over the bath gives the reduced density matrixThe orientation of this conditional environment overlap fixes the sign of the phase in the upper-right entry. Factorizing the overlap yields the decoherence factorThe populations are conserved because $[H,Z_D]=0$. Only phase coherence can be reduced. In particular,Quantum decoherence here results from distinguishable conditional bath states, although the complete system remains in a pure state under unitary time evolution.

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.

A pure spin-one-half state on the equator of the Bloch sphere has $⟨Z_k⟩=0$, so $|{\alpha }_k{|}^2=|{\beta }_k{|}^2=1/2$. Its azimuthal phase does not enter the conditional environment overlap, because the interaction is diagonal in $Z_k$. Substitution in the decoherence factor givesThe Zurek spin-bath model now has a real coherence factor: positive and negative values correspond to opposite relative phases, while suppression of coherence depends on $|z|$.

A device qubit couples diagonally to a finite bath of spins. Conditional bath states acquire opposite phase rotations, and their conditional environment overlap multiplies the device coherence. An initially aligned bath leaves the overlap's modulus at one. Initially equatorial spins yield $z_N(t)={\prod }_k\cos (2g_{k}t)$. Global unitary time evolution remains reversible, and a finite spin-bath coherence recurrence prevents a strict zero long-time limit at finite bath size.