# 数学代写|信息论作业代写information theory代考|Constrained diamond norms and Bures distances

## 数学代写|信息论作业代写information theory代考|Constrained diamond norms and Bures distances

Let $\mathbf{H}$ be any $\mathfrak{H}$-operator. The energy-constrained diamond norm of quantum channel $\Phi: \mathcal{S}\left(\mathrm{H}A\right) \rightarrow \mathcal{S}\left(\mathrm{H}_B\right)$ is defined by $$|\Phi|{\diamond}^E=\sup {\rho \in \mathcal{S}\left(\mathrm{H}{A R}\right), \mathrm{tr}\left[\mathrm{H} \rho_A\right] \leq E}\left|\left(\Phi \otimes \mathfrak{I}R\right) \rho\right|_1, \quad E>E_0,$$ where $R$ is any quantum system and $\rho_A=\operatorname{tr}_R\left[\rho{A R}\right]$.
We define the energy-constrained Bures distance between the quantum channels $\Phi$ and $\Psi: \mathcal{S}\left(\mathbb{H}A\right) \rightarrow \mathcal{S}\left(\mathbb{H}_B\right)$ as $$\beta_E(\Phi, \Psi)=\sup {\rho \in \mathcal{S}\left(\mathrm{H}{A R}\right), \mathrm{tr}\left[\mathrm{H} \rho_A\right] \leq E} \beta\left(\left(\Phi \otimes \mathfrak{I}_R\right) \rho,\left(\Psi \otimes \mathfrak{I}_R\right) \rho\right), \quad E>E_0$$ where $\beta(\cdot, \cdot)$ is the Bures distance defined in equation (6.34). Proposition 6.4.3. The following inequalities hold: $$\frac{1}{2}|\Phi-\Psi|{\diamond}^E \leq \beta_E(\Phi, \Psi) \leq \sqrt{|\Phi-\Psi|_{\diamond}^E}$$
Proof. The inequality follows easily from the definition energy-constrained diamond norm $|\cdot|_{\diamond}^E$ (see equation (6.40)), the definition of energy-constrained Bures distance $\beta_E(\cdot, \cdot)$ (see equation (6.41)) and inequality (6.35). This proves the proposition.

Lemma 6.4.4. Let $\mathbf{H}$ be an $\mathfrak{H}$-operator on $\mathrm{H}A$ and let $E>E_0$, where $E_0:=$ $\inf {|\varphi|_{H_A}=1}\langle\varphi|\mathbf{H}| \varphi\rangle_A$ is the smallest eigenvalue of $\mathbf{H}$. For any arbitrary quantum channel $\Phi$ from $A$ to $B$, there exist a separable Hilbert space $\mathbb{H}E$ and a Stinespring isometry $\mathbf{V}{\Phi}: \mathrm{H}A \rightarrow \mathbb{H}{B E}$ of the channel $\Phi$ with the following property: for any quantum channel $\Psi$ from $A$ to $B$, there is a Stinespring isometry $\mathbf{V}{\Psi}: \mathbb{H}_A \rightarrow \mathbb{H}{B E}$ of $\Psi$ such that
$$\left|\mathbf{V}{\Psi}-\mathbf{V}{\Phi}\right|_E=\beta_E(\Psi, \Phi) .$$

## 数学代写|信息论作业代写information theory代考|Approximations of quantum channels

In the following, we have an approximation result of the infinite-dimensional quantum channel via finite-dimensional projections. The result is due originally to Shirokov and Holevo [158].

Proposition 6.5.1. Let $\Phi: \mathfrak{T}{+}\left(\mathbb{H}_A\right) \rightarrow \mathfrak{T}{+}\left(\mathbb{H}B\right)$ be an extended quantum channel from input system $A$ to output system $B$, and let $\left(\mathbf{P}_n\right){n=1}^{+\infty}$ be an increasing sequence of finitedimensional projections on $\mathrm{H}A$ that converges strongly to the identity operator $\mathbf{I}_A$. Then there is a family $\left(\Phi_n\right){n=1}^{+\infty}$ of completely positive maps such that $\Phi_n$ is trace preserving on $\mathbf{P}n\left(\mathbb{H}_A\right)$ and $\Phi_n(\mathbf{A}) \rightarrow \Phi(\mathbf{A})$ uniformly for all $\mathbf{A} \in \mathfrak{T}{+}\left(\mathbb{H}_A\right)$

Proof. First note that, for each $\mathbf{A} \in \mathfrak{B}\left(\mathbb{H}A\right), \mathbf{A}_n \equiv \mathbf{P}_n \mathbf{A P}_n^* \rightarrow \mathbf{A}$ in operator norm $|\cdot|{\infty}$, when $n \rightarrow+\infty$. We can write $\mathbf{A}-\mathbf{A}n=\left(\mathbf{A}-\mathbf{A}_n\right){+}-\left(\mathbf{A}-\mathbf{A}n\right){-}$, where $\left(\mathbf{A}-\mathbf{A}n\right){+}$and $\left(\mathbf{A}-\mathbf{A}n\right){-}$are in $\tau_{+}\left(\mathbb{H}A\right)$ and $\left(\mathbf{A}-\mathbf{A}_n\right){+}\left(\mathbf{A}-\mathbf{A}n\right){-}=\left(\mathbf{A}-\mathbf{A}n\right){-}\left(\mathbf{A}-\mathbf{A}n\right){+}=\mathbf{0}$. Note that $\left(\mathbf{A}-\mathbf{A}n\right){+}$and $\left(\mathbf{A}-\mathbf{A}n\right){-}$are, respectively, the positive and negative part of $\mathbf{A}-\mathbf{A}n$. Obviously $$\Phi\left(\left(\mathbf{A}-\mathbf{A}_n\right){+}\right), \Phi\left(\left(\mathbf{A}-\mathbf{A}n\right){-}\right) \in \mathfrak{T}{+}\left(\mathrm{H}_B\right)$$ and $$\lim {n \rightarrow+\infty} \operatorname{tr}\left[\Phi\left(\left(\mathbf{A}-\mathbf{A}n\right){+}\right)\right]=\lim {n \rightarrow+\infty} \operatorname{tr}\left[\Phi\left(\left(\mathbf{A}-\mathbf{A}_n\right){-}\right)\right]=0 .$$
Consequently,
$$\lim {n \rightarrow+\infty}\left|\Phi\left(\left(\mathbf{A}-\mathbf{A}_n\right){+}\right)\right|_1=\lim {n \rightarrow+\infty}\left|\Phi\left(\left(\mathbf{A}-\mathbf{A}_n\right){-}\right)\right|_1=0$$
and
$$\lim _{n \rightarrow+\infty}\left|\Phi\left(\mathbf{A}-\mathbf{A}_n\right)\right|_1=0$$

# 信息论代考

## 数学代写|信息论作业代写information theory代考|Constrained diamond norms and Bures distances

$\inf |\varphi|_{H_A}=1\langle\varphi|\mathbf{H}| \varphi\rangle_A$ 是的最小特征值H. 对于任意任意量子通道 $\Phi$ 从 $A$ 到 $B$ ，存在一

$$|\mathbf{V} \Psi-\mathbf{V} \Phi|_E=\beta_E(\Psi, \Phi) .$$

## 数学代写|信息论作业代写information theory代考|Approximations of quantum channels

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