Typical subspace

Consider a density operator $\backslash rho$ with the following spectral decomposition:

:$$

\rho=\sum_{x}p_{X}( x) \vert x\rangle \langle

x\vert .

The weakly typical subspace is defined as the span of all vectors such that

the sample entropy $\backslash overline\{H\}(\; x^\{n\})$ of their classical

label is close to the true entropy $H(\; X)$ of the distribution

$p\_\{X\}(\; x)$:

:$$

T_{\delta}^{X^{n}}\equiv\text{span}\left\{ \left\vert x^{n}\right\rangle

:\left\vert \overline{H}( x^{n}) -H( X) \right\vert

\leq\delta\right\} ,

where

:$$

\overline{H}( x^{n}) \equiv-\frac{1}{n}\log( p_{X^{n}

}( x^{n}) ) ,

:$H(\; X)\; \backslash equiv-\backslash sum\_\{x\}p\_\{X\}(\; x)\; \backslash log\; p\_\{X\}($

x) .

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

defined as

:$$

\Pi_{\rho,\delta}^{n}\equiv\sum_{x^{n}\in T_{\delta}^{X^{n}}}\vert

x^{n}\rangle \langle x^{n}\vert ,

where we have "overloaded" the symbol

$T\_\{\backslash delta\}^\{X^\{n\}\}$ to refer also to the set of $\backslash delta$-typical sequences:

:$$

T_{\delta}^{X^{n}}\equiv\left\{ x^{n}:\left\vert \overline{H}\left(

x^{n}\right) -H( X) \right\vert \leq\delta\right\} .

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

:$$

\text{Tr}\left\{ \Pi_{\rho,\delta}^{n}\rho^{\otimes n}\right\}

\geq1-\epsilon,

:$\backslash text\{Tr\}\backslash left\backslash \{\; \backslash Pi\_\{\backslash rho,\backslash delta\}^\{n\}\backslash right\backslash \}\; \backslash leq2^\{n\backslash left[\; H\backslash left($

X\right) +\delta\right] },

:$2^\{-n\backslash left[\; H(\; X)\; +\backslash delta\backslash right]\; \}\backslash Pi\_\{\backslash rho,\backslash delta\}^\{n\}$

\leq\Pi_{\rho,\delta}^{n}\rho^{\otimes n}\Pi_{\rho,\delta}^{n}\leq2^{-n\left[

H( X) -\delta\right] }\Pi_{\rho,\delta}^{n},

where the first property holds for arbitrary $\backslash epsilon,\backslash delta>0$ and

sufficiently large $n$.

Consider an ensemble $\backslash left\backslash \{\; p\_\{X\}(\; x)\; ,\backslash rho\_\{x\}\backslash right\backslash \}$

_{x\in\mathcal{X}} of states. Suppose that each state $\backslash rho\_\{x\}$ has the

following spectral decomposition:

:$$

\rho_{x}=\sum_{y}p_{Y|X}( y|x) \vert y_{x}\rangle

\langle y_{x}\vert .

Consider a density operator $\backslash rho\_\{x^\{n\}\}$ which is conditional on a classical

sequence $x^\{n\}\backslash equiv\; x\_\{1\}\backslash cdots\; x\_\{n\}$:

:$$

\rho_{x^{n}}\equiv\rho_{x_{1}}\otimes\cdots\otimes\rho_{x_{n}}.

We define the weak conditionally typical subspace as the span of vectors

(conditional on the sequence $x^\{n\}$) such that the sample conditional entropy

$\backslash overline\{H\}(\; y^\{n\}|x^\{n\})$ of their classical labels is close

to the true conditional entropy $H(\; Y|X)$ of the distribution

$p\_\{Y|X\}(\; y|x)\; p\_\{X\}(\; x)$:

:$$

T_{\delta}^{Y^{n}|x^{n}}\equiv\text{span}\left\{ \left\vert y_{x^{n}}

^{n}\right\rangle :\left\vert \overline{H}( y^{n}|x^{n})

-H( Y|X) \right\vert \leq\delta\right\} ,

where

:$$

\overline{H}( y^{n}|x^{n}) \equiv-\frac{1}{n}\log\left(

p_{Y^{n}|X^{n}}( y^{n}|x^{n}) \right) ,

:$H(\; Y|X)\; \backslash equiv-\backslash sum\_\{x\}p\_\{X\}(\; x)\; \backslash sum\_\{y\}$

p_{Y|X}( y|x) \log p_{Y|X}( y|x) .

The projector $\backslash Pi\_\{\backslash rho\_\{x^\{n\}\},\backslash delta\}$ onto the weak conditionally typical

subspace of $\backslash rho\_\{x^\{n\}\}$ is as follows:

:$$

\Pi_{\rho_{x^{n}},\delta}\equiv\sum_{y^{n}\in T_{\delta}^{Y^{n}|x^{n}}

}\vert y_{x^{n}}^{n}\rangle \langle y_{x^{n}}^{n}\vert ,

where we have again overloaded the symbol $T\_\{\backslash delta\}^\{Y^\{n\}|x^\{n\}\}$ to refer

to the set of weak conditionally typical sequences:

:$$

T_{\delta}^{Y^{n}|x^{n}}\equiv\left\{ y^{n}:\left\vert \overline{H}\left(

y^{n}|x^{n}\right) -H( Y|X) \right\vert \leq\delta\right\} .

The three important properties of the weak conditionally typical projector are

as follows:

:$$

\mathbb{E}_{X^{n}}\left\{ \text{Tr}\left\{ \Pi_{\rho_{X^{n}},\delta}

\rho_{X^{n}}\right\} \right\} \geq1-\epsilon,

:$\backslash text\{Tr\}\backslash left\backslash \{\; \backslash Pi\_\{\backslash rho\_\{x^\{n\}\},\backslash delta\}\backslash right\backslash \}\; \backslash leq2^\{n\backslash left[$

H( Y|X) +\delta\right] },

:$2^\{-n\backslash left[\; H(\; Y|X)\; +\backslash delta\backslash right]\; \}\backslash \; \backslash Pi\_\{\backslash rho\_\{x^\{n\}\}$

,\delta} \leq\Pi_{\rho_{x^{n}},\delta}\ \rho_{x^{n}}\ \Pi_{\rho_{x^{n}

},\delta} \leq2^{-n\left[ H( Y|X) -\delta\right] }\ \Pi

_{\rho_{x^{n}},\delta},

where the first property holds for arbitrary $\backslash epsilon,\backslash delta>0$ and

sufficiently large $n$, and the expectation is with respect to the

distribution $p\_\{X^\{n\}\}(\; x^\{n\})$.

• Classical capacity
• Quantum information theory

• Wilde, Mark M., 2017, [http://www.cambridge.org/us/academic/subjects/computer-science/cryptography-cryptology-and-coding/quantum-information-theory-2nd-edition Quantum Information Theory, Cambridge University Press], Also available at [https://arxiv.org/abs/1106.1445 eprint arXiv:1106.1145]

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Category:Quantum information theory