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
:
- :\left\vert {\overline {H}}(x^{n})-H(X)\right\vert \leq \delta \right\},}
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 .
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
(conditional on the sequence ) such that the sample conditional entropy
of their classical labels is close
to the true conditional entropy of the distribution
:
- :\left\vert {\overline {H}}(y^{n}|x^{n})-H(Y|X)\right\vert \leq \delta \right\},}
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 .