 Solution

Use the high-temperature state consistently to first order in $ϵ$. Applying the lower-target controlled-transfer gate with $\cos {\theta }_1=1/3$ givesThe upper-target gate with $\cos {\theta }_2=0$ then equalizes the second and fourth populations. The result isThis is a pseudo-pure state; the identity part is unchanged by unitary gates and does not contribute to traceless NMR signals.

For $n$ equal, weakly polarized spins, the high-temperature expansion is $\rho \simeq 2^{-n}[I+(ϵ/2){\sum }_j{Z}_j]$. The all-zero population is $a_0=2^{-n}(1+nϵ/2)$. Preserving this largest population and averaging the other $2^n-1$ populations givesThus the signal-bearing effective purity falls approximately as $n/2^n$ at fixed small per-spin polarization. The linear expansion requires $n|ϵ|\ll 1$. More generally, if the exact single-spin polarization is $r=\tanh (ϵ/2)$, the same population-preserving averaging gives $p_n=[(1+r{)}^n-1]/(2^n-1)$, which still decreases exponentially for fixed $0<r<1$. This scaling concerns high-temperature ensemble preparation, not an impossibility theorem for every initialization method.