= Solution
For fixed $\varepsilon>0$, let the <quantum typical subspace> $\mathcal T_\varepsilon^{(n)}$ be spanned by the product eigenvectors with nonzero eigenvalues satisfying
$$
2^{-n(S(\pi)+\varepsilon)}\leq\lambda_{\mathbf j}\leq2^{-n(S(\pi)-\varepsilon)}.
$$
Equivalently their spectral information $-n^{-1}\log_2\lambda_{\mathbf j}$ lies within $\varepsilon$ of $S(\pi)$. Denote its <orthogonal projection> by $P_\varepsilon^{(n)}$.
The <typical subspace theorem> states that for every $\delta>0$ there is $n_0$ such that for $n\geq n_0$,
$$
\boxed{\operatorname{Tr}(\pi^{\otimes n}P_\varepsilon^{(n)})\geq1-\delta,\qquad
(1-\delta)2^{n(S(\pi)-\varepsilon)}\leq\dim\mathcal T_\varepsilon^{(n)}\leq2^{n(S(\pi)+\varepsilon)}.}
$$
Also the operator bounds on its range are
$$
2^{-n(S(\pi)+\varepsilon)}P_\varepsilon^{(n)}\leq P_\varepsilon^{(n)}\pi^{\otimes n}P_\varepsilon^{(n)}\leq2^{-n(S(\pi)-\varepsilon)}P_\varepsilon^{(n)}.
$$
To see the concentration statement, sample $j_s$ independently with probabilities $\lambda_j$. The mean of $-\log_2\lambda_{j_s}$ is $S(\pi)$, so the <weak law of large numbers> gives typical probability tending to one. Zero-eigenvalue letters have probability zero and are excluded. The dimension bounds follow by summing the upper and lower typical eigenvalue bounds and using the typical mass between $1-\delta$ and $1$.
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