 Solution

For fixed $\epsilon >0$, let the quantum typical subspace ${\mathcal{T}}_{\epsilon }^{(n)}$ be spanned by the product eigenvectors with nonzero eigenvalues satisfyingEquivalently their spectral information $-n^{-1}{\log }_2{\lambda }_{\mathbf{j}}$ lies within $\epsilon$ of $S(\pi )$. Denote its orthogonal projection by $P_{\epsilon }^{(n)}$.

The typical subspace theorem states that for every $\delta >0$ there is $n_0$ such that for $n\ge n_0$,Also the operator bounds on its range areTo 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 $1$.