Typical subspace


In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems. Its role is analogous to that of the typical set in classical information theory.

Unconditional quantum typicality

Consider a density operator with the following spectral decomposition:
The weakly typical subspace is defined as the span of all vectors such that
the sample entropy of their classical
label is close to the true entropy of the distribution
where
The projector onto the typical subspace of is
defined as
where we have "overloaded" the symbol
to refer also to the set of -typical sequences:
The three important properties of the typical projector are as follows:
where the first property holds for arbitrary and
sufficiently large.

Conditional quantum typicality

Consider an ensemble of states. Suppose that each state has the
following spectral decomposition:
Consider a density operator which is conditional on a classical
sequence :
We define the weak conditionally typical subspace as the span of vectors
such that the sample conditional entropy
of their classical labels is close
to the true conditional entropy of the distribution
where
The projector onto the weak conditionally typical
subspace of is as follows:
where we have again overloaded the symbol 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 and
sufficiently large, and the expectation is with respect to the
distribution.