= Solution
The model factors into the <normal distribution> densities
$$
X_1\sim N(\theta,1)
$$
and, conditionally on $X_{i-1}$,
$$
X_i\mid X_{i-1}
\sim N\!\left(
\theta(1-\sqrt\gamma)+\sqrt\gamma X_{i-1},
1-\gamma
\right).
$$
Ignoring constants independent of $\theta$, the log likelihood is
$$
\ell_n(\theta)
=-\frac12(X_1-\theta)^2
-\frac1{2(1-\gamma)}
\sum_{i=2}^n
\left[
X_i-\sqrt\gamma X_{i-1}
-\theta(1-\sqrt\gamma)
\right]^2.
$$
Its score is
$$
S_n(\theta)
=(X_1-\theta)
+\frac{1-\sqrt\gamma}{1-\gamma}
\sum_{i=2}^n
\left[
X_i-\sqrt\gamma X_{i-1}
-\theta(1-\sqrt\gamma)
\right].
$$
In terms of the independent innovations,
$$
S_n(\theta)
=\varepsilon_1
+\frac{1-\sqrt\gamma}{\sqrt{1-\gamma}}
\sum_{i=2}^n\varepsilon_i.
$$
<Variance additivity for independent random variables> therefore yields
$$
\boxed{
I_n(\theta)
=1+(n-1)\frac{(1-\sqrt\gamma)^2}{1-\gamma}
=1+(n-1)\frac{1-\sqrt\gamma}{1+\sqrt\gamma}}.
$$
Since $I_1(\theta)=1$, the equality $I_n=nI_1$ holds for all $n$ exactly when
$$
\frac{1-\sqrt\gamma}{1+\sqrt\gamma}=1,
$$
namely when
$$
\boxed{\gamma=0}.
$$
Thus the information tensorizes only in the independent case.
Finally, the <Cramér-Rao bound> gives every unbiased estimator $\widehat\theta$ the lower bound
$$
\boxed{
\operatorname{Var}_\theta(\widehat\theta)
\geq
\frac1{I_n(\theta)}
=\left[
1+(n-1)\frac{1-\sqrt\gamma}{1+\sqrt\gamma}
\right]^{-1}}.
$$
This is the <Fisher information in a stationary Gaussian autoregressive location model>.
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