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Every member of the typical set has probability at least $2^{-n(H(X)+\epsilon )}$. Summing these probabilities givessoThe upper bound itself holds at every block length. For sufficiently large $n$, the high-probability property also gives the companion lower boundhence $|T_{\epsilon }^{(n)}|\ge (1-\delta )2^{n(H(X)-\epsilon )}$. Together these are the typical-set cardinality bounds underlying Shannon source coding theorem.

Memorylessness means that the ensemble's average density operator is ${\rho }^{(n)}={\pi }^{\otimes n}$. Its spectral decomposition isThe product eigenvalues giveZero eigenvalues contribute zero to the Von Neumann entropy and are excluded from logarithmic typicality tests.

The quantum typical subspace is the span of eigenvectors whose eigenvalues satisfyIts orthogonal projection $P_{\epsilon }^{(n)}$ selects precisely the classical typical set for the eigenvalue distribution $q$. The typical-set cardinality bounds and the weak law of large numbers therefore givefor any fixed $\delta >0$ and all sufficiently large $n$.

For a density operator $\pi ={\sum }_i{q}_i|{\varphi }_i⟩⟨{\varphi }_i|$, the quantum typical subspace is spanned by product eigenvectors of ${\pi }^{\otimes n}$ whose eigenvalues $q_{i_1}\cdots q_{i_n}$ satisfyIt applies the classical weakly typical sequence definition to the spectrum. If $P_{\epsilon }^{(n)}$ is its orthogonal projection, thenThe 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 $S(\pi )<{\log }_{2}d$.