Typical subspace

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.
Unconditional quantum typicality.
Consider a density operator formula_1 with the following spectral decomposition:
The weakly typical subspace is defined as the span of all vectors such that
the sample entropy formula_3 of their classical
label is close to the true entropy formula_4 of the distribution
formula_5:
where
The projector formula_9 onto the typical subspace of formula_1 is
defined as
where we have "overloaded" the symbol
formula_12 to refer also to the set of formula_13-typical sequences:
The three important properties of the typical projector are as follows:
where the first property holds for arbitrary formula_18 and
sufficiently large formula_19.
Conditional quantum typicality.
Consider an ensemble formula_20 of states. Suppose that each state formula_21 has the
following spectral decomposition:
Consider a density operator formula_23 which is conditional on a classical
sequence formula_24:
We define the weak conditionally typical subspace as the span of vectors
(conditional on the sequence formula_26) such that the sample conditional entropy
formula_27 of their classical labels is close
to the true conditional entropy formula_28 of the distribution
formula_29:
where
The projector formula_33 onto the weak conditionally typical
subspace of formula_23 is as follows:
where we have again overloaded the symbol formula_36 to refer
to the set of weak conditionally typical sequences:
The three important properties of the weak conditionally typical projector are
as follows:
where the first property holds for arbitrary formula_18 and
sufficiently large formula_19, and the expectation is with respect to the
distribution formula_43.