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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.

For these three couplings, the decoherence factor becomesIt is periodic with period $2$. To locate its extrema, put $c=\cos \pi t$; then $z=c(1+c)/2$. Its maximum is $1$ at $t=0,2,4$, and its minimum is $-1/8$ where $c=-1/2$, namely $t=2/3,4/3,8/3,10/3$. The zeros on the displayed interval are $t=1/2,1,3/2,5/2,3,7/2$. The half-integer zeros are crossings; the zeros at $1$ and $3$ are double zeros, where the curve touches zero from below.

The finite spin-bath coherence recurrence returns the device to full coherence every two time units. Negative $z$ indicates a relative phase change and is not a negative probability. When $ab\ne 0$, the device reduced density matrix becomes diagonal at each zero, but quantum decoherence is not irreversible in this finite bath. The sketch explicitly displays both loss of coherence and its revival.

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 $|z|$.

Write $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. Before any interaction the system is in the pure state $|{\psi }_S⟩=\alpha |0⟩+\beta |1⟩$, soOne interaction leaves the environmental state $|+⟩$ unchanged on the system branch $|0⟩$, while on the branch $|1⟩$ it producesAfter interactions with $n$ distinct environment qubits, the joint state isand the decoherence factor is the product $(\cos ϵ{)}^n$. Taking the partial trace givesFor small $ϵ$, $(\cos ϵ{)}^n\simeq e^{-n{ϵ}^2/2}$: the populations remain fixed while the phase coherence decays. This is environmental decoherence, caused by entanglement with unobserved records rather than by a nonunitary evolution of the complete state.

A measurement in the computational basis cannot reveal the decay because its probabilities remain $|\alpha {|}^2$ and $|\beta {|}^2$. An interference measurement can. For example, measuring in the Hadamard basis givesChanging the measurement phase similarly accesses the imaginary part, so the shrinking interference visibility directly displays decoherence.

If the same environment qubit is reused, the conditional rotation accumulates coherently. After $n$ interactions the overlap is $\cos (nϵ)$ rather than $(\cos ϵ{)}^n$, and the off-diagonal entries oscillate. Coherence vanishes at some times but returns periodically: this finite environment exhibits quantum recoherence rather than effectively irreversible decay.