In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory.

Consider a density operator ${\displaystyle \rho }$ with the following spectral decomposition:

The weakly typical subspace is defined as the span of all vectors such that the sample entropy ${\displaystyle {\overline {H}}(x^{n})}$ of their classical label is close to the true entropy ${\displaystyle H(X)}$ of the distribution ${\displaystyle p_{X}(x)}$:

where

The projector ${\displaystyle \Pi _{\rho ,\delta }^{n}}$ onto the typical subspace of ${\displaystyle \rho }$ is defined as

where we have "overloaded" the symbol ${\displaystyle T_{\delta }^{X^{n}}}$ to refer also to the set of ${\displaystyle \delta }$-typical sequences:

The three important properties of the typical projector are as follows:

where the first property holds for arbitrary ${\displaystyle \epsilon ,\delta >0}$ and sufficiently large ${\displaystyle n}$.