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 $\rho$ with the following spectral decomposition:
:$\rho=\sum_p_( x) \vert x\rangle \langle x\vert .$
The weakly typical subspace is defined as the span of all vectors such that
the sample entropy $\overline( x^)$ of their classical
label is close to the true entropy $H( X)$ of the distribution
$p_( x)$:
:$T_^\equiv\text\left\ ,$
where
:$\overline( x^) \equiv-\frac\log( p_( x^) ) ,$
:$H( X) \equiv-\sum_p_( x) \log p_( x) .$
The projector $\Pi_^$ onto the typical subspace of $\rho$ is
defined as
:$\Pi_^\equiv\sum_\vert x^\rangle \langle x^\vert ,$
where we have "overloaded" the symbol
$T_^$ to refer also to the set of $\delta$-typical sequences:
:$T_^\equiv\left\ .$
The three important properties of the typical projector are as follows:
:$\text\left\ \geq1-\epsilon,$
:$\text\left\ \leq2^,$
:$2^\Pi_^ \leq\Pi_^\rho^\Pi_^\leq2^\Pi_^,$
where the first property holds for arbitrary $\epsilon,\delta>0$ and
sufficiently large $n$.

Conditional quantum typicality

Consider an ensemble $\left\ _$ of states. Suppose that each state $\rho_$ has the
following spectral decomposition:
:$\rho_=\sum_p_( y, x) \vert y_\rangle \langle y_\vert .$
Consider a density operator $\rho_$ which is conditional on a classical
sequence $x^\equiv x_\cdots x_$:
:$\rho_\equiv\rho_\otimes\cdots\otimes\rho_.$
We define the weak conditionally typical subspace as the span of vectors
(conditional on the sequence $x^$) such that the sample conditional entropy
$\overline( y^, x^)$ of their classical labels is close
to the true conditional entropy $H( Y, X)$ of the distribution
$p_( y, x) p_( x)$:
:$T_^\equiv\text\left\ ,$
where
:$\overline( y^, x^) \equiv-\frac\log\left( p_( y^, x^) \right) ,$
:$H( Y, X) \equiv-\sum_p_( x) \sum_ p_( y, x) \log p_( y, x) .$
The projector $\Pi_$ onto the weak conditionally typical
subspace of $\rho_$ is as follows:
:$\Pi_\equiv\sum_\vert y_^\rangle \langle y_^\vert ,$
where we have again overloaded the symbol $T_^$ to refer
to the set of weak conditionally typical sequences:
:$T_^\equiv\left\ .$
The three important properties of the weak conditionally typical projector are
as follows:
:$\mathbb_\left\ \geq1-\epsilon,$
:$\text\left\ \leq2^,$
:$2^\ \Pi_ \leq\Pi_\ \rho_\ \Pi_ \leq2^\ \Pi _,$
where the first property holds for arbitrary $\epsilon,\delta>0$ and
sufficiently large $n$, and the expectation is with respect to the
distribution $p_( x^)$.

See also

* Classical capacity
* Quantum information theory

References

* Wilde, Mark M., 2017,
Quantum Information Theory, Cambridge University Press
Also available a
eprint arXiv:1106.1145

{{Quantum computing

Quantum information theory