# 数学代写|信息论作业代写information theory代考|Ensemble average and entanglement fidelities

## 数学代写|信息论作业代写information theory代考|Ensemble average and entanglement fidelities

For the finite-dimensional case, ensemble average fidelity and entanglement fidelity are two kinds of important fidelities connected to a quantum channel. In this subsection, we give the definitions of ensemble average fidelity and entanglement fidelity connected to a quantum channel for an infinite- dimensional system, and discuss their relationship.

The presentation in this subsection is largely based on results obtained by Hou and Qi $[90]$

Let $\mathbb{H}$ be an infinite-dimensional separable complex Hilbert space and let a quantum channel $\Phi: \mathcal{S}(\mathrm{H}) \rightarrow \mathcal{S}(\mathbb{H})$ be a quantum channel. As in the finite-dimensional case (see, e. g., Wilde [178]), for such quantum channel $\Phi$ and a given ensemble $\left{p_j, \rho_j\right}_{j=1}^{+\infty}$, one can define ensemble average fidelity $\bar{F}\left(\left{p_i, \rho_i\right}, \Phi\right)$ by
$$\bar{F}\left(\left{p_i, \rho_i\right}, \Phi\right)=\sum_i p_i F\left(\rho_i, \Phi\left(\rho_i\right)\right)^2$$
where $F\left(\rho_i, \Phi\left(\rho_i\right)\right)$ denotes the fidelity between the input state $\rho_i$ and its channel output state $\Phi\left(\rho_i\right)$, which are both on the same Hilbert space $\mathbb{H}$.

Similarly, for a state $\rho$, one can define the entanglement fidelity $F_{\text {ef }}(\cdot, \cdot): \mathcal{S}(\mathbb{H}) \times$ $\mathfrak{Q C}(\mathbb{H}) \rightarrow[0,1]$ by
\begin{aligned} F_{\mathrm{ef}}(\rho, \Phi) & =F\left(|\psi\rangle_{\mathbb{H} \otimes \mathrm{H}},(\Phi \otimes \mathfrak{I})\left(|\psi\rangle_{\mathbb{H} \otimes \mathrm{H}}\langle\psi|\right)\right)^2 \ & =\left\langle\psi\left|(\Phi \otimes \mathfrak{I})\left(|\psi\rangle_{\mathrm{H} \Perp \mathrm{H}}\langle\psi|\right)\right| \psi\right\rangle_{\mathrm{H} \otimes \mathrm{H}} \end{aligned}
where $|\psi\rangle \in \mathbb{H} \otimes \mathbb{H}$ is a purification of $\rho$. Note that the definition of $F_{\mathrm{ef}}(\rho, \Phi)$ does not depend on the choices of purifications. To see this, let $|\psi\rangle_{\mathrm{H} \otimes H}=\sum_j \sqrt{p_j}|j\rangle_{\mathbb{H}} \otimes\left|\mu_j\right\rangle_{\mathbf{H}}$ be any purification, where $\left{|j\rangle_{\mathbb{H}}\right}$ is an orthonormal basis and $\left{\left|\mu_j\right\rangle_{\mathcal{H}}\right}$ is an orthonormal set of $\mathbb{H}$. By Kraus representation (4.12), there exists a sequence of operators $\left(\mathbf{E}i\right){i=1}^{+\infty} \subset$ $\mathfrak{B}(\mathbb{H})$ with $\sum_i \mathbf{E}i^* \mathbf{E}_i=\mathbf{I}{\mathbb{H}}$ such that
$$\Phi(\sigma)=\sum_i \mathbf{E}_i \sigma \mathbf{E}_i^*, \quad \forall \sigma \in \mathcal{S}(\mathrm{H})$$

## 数学代写|信息论作业代写information theory代考|Complete boundedness norm

Recall from Proposition 4.1.5 that a linear map $Y$ from $C^$-algebra $\mathcal{A}$ to another $C^$ algebra $\mathcal{B}$ is completely positive if and only if $\Upsilon: \mathcal{A} \otimes \mathcal{M}_n \rightarrow \mathcal{B} \otimes \mathcal{M}_n$ is positive for each $n \in \mathbb{N}$.

The concept of complete boundedness to be defined below is closely related to completely positivity characterized in Proposition 4.1.5. The norm of complete boundedness for a linear map between two $C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$ is defined as follows.

Definition 6.3.1. Let $Y: \mathcal{A} \rightarrow \mathcal{B}$ be a linear map between $C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$. The norm of complete boundedness $|Y|_{\text {cb }}$ of $Y$ is defined as
$$|Y|_{\mathrm{cb}}=\sup {n \in \mathbb{N}}\left|Y \otimes \mathbf{I}{(n \times n)}\right|_{\infty}$$ where $\mathbf{I}{(n \times n)}$ is the identity map of $n \times n$ complex matrices (i. e., $\mathbf{I}{(n \times n)}$ is an $n \times n$ identity matrix). The map $Y$ is called a completely bounded map if $|Y|_{c b}<\infty$.

Equivalently, $Y: \mathcal{A} \rightarrow \mathcal{B}$ is completely bounded if and only if the linear map $Y_n$ that maps $n \times n$ matrices in $\mathcal{A} \otimes \mathcal{M}n$ to $n \times n$ matrices in $\mathcal{B} \otimes \mathcal{M}_n$ defined by $$\left(\begin{array}{cccc} \mathbf{a}{11} & \mathbf{a}{12} & \ldots & \mathbf{a}{1 n} \ \mathbf{a}{21} & \mathbf{a}{22} & \ldots & \mathbf{a}{2 n} \ \vdots & \vdots & \ddots & \vdots \ \mathbf{a}{n 1} & \mathbf{a}{n 2} & \ldots & \mathbf{a}{n n} \end{array}\right) \mapsto\left(\begin{array}{cccc} \Upsilon\left(\mathbf{a}{11}\right) & Y\left(\mathbf{a}{12}\right) & \ldots & Y\left(\mathbf{a}{1 n}\right) \ \Upsilon\left(\mathbf{a}{21}\right) & Y\left(\mathbf{a}{22}\right) & \ldots & Y\left(\mathbf{a}{2 n}\right) \ \vdots & \vdots & \ddots & \vdots \ Y\left(\mathbf{a}{n 1}\right) & Y\left(\mathbf{a}{n 2}\right) & \ldots & Y\left(\mathbf{a}{n n}\right) \end{array}\right)$$ is uniformly bounded for all $n \in \mathbb{N}$. That is, $\sup {n \in \mathbb{N}}\left|\mathrm{Y}n\right|{\infty}<+\infty$.

# 信息论代考

## 数学代写|信息论作业代写information theory代考|Ensemble average and entanglement fidelities

$$F_{\text {ef }}(\rho, \Phi)=F\left(|\psi\rangle_{\mathrm{H} \otimes \mathrm{H}},(\Phi \otimes \mathfrak{I})\left(|\psi\rangle_{\mathrm{H} \otimes \mathrm{H}}\langle\psi|\right)\right)^2 \quad=\left\langle\psi\left|(\Phi \otimes \mathfrak{I})\left(|\psi\rangle_{\mathrm{H} \backslash \operatorname{PerpH}}\langle\psi|\right)\right| \psi\right\rangle$$

$$\Phi(\sigma)=\sum_i \mathbf{E}_i \sigma \mathbf{E}_i^*, \quad \forall \sigma \in \mathcal{S}(\mathrm{H})$$

## 数学代写|信息论作业代写information theory代考|Complete boundedness norm

$$|Y|{\mathrm{cb}}=\sup n \in \mathbb{N}|Y \otimes \mathbf{I}(n \times n)|{\infty}$$

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