= Quantum typical subspace
{title2=$\mathcal T_\varepsilon^{(n)}$}
For a <density operator> $\pi=\sum_iq_i|\phi_i\rangle\langle\phi_i|$, the quantum typical subspace is spanned by product eigenvectors of $\pi^{\otimes n}$ whose eigenvalues $q_{i_1}\cdots q_{i_n}$ satisfy
$$
\left|-\frac1n\log_2(q_{i_1}\cdots q_{i_n})-S(\pi)\right|\leq\varepsilon.
$$
It applies the classical <weakly typical sequence> definition to the spectrum. If $P_\varepsilon^{(n)}$ is its <orthogonal projection>, then
$$
\dim\mathcal T_\varepsilon^{(n)}\leq2^{n(S(\pi)+\varepsilon)},\qquad
\operatorname{Tr}(\pi^{\otimes n}P_\varepsilon^{(n)})\longrightarrow1.
$$
The first statement follows from the <typical-set cardinality bounds>, and the second from the <weak law of large numbers>. This subspace holds nearly all the source probability while using exponentially fewer dimensions than the whole space when $S(\pi)<\log_2d$.
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