# 数学代写|信息论作业代写information theory代考|Bures distance

## 数学代写|信息论作业代写information theory代考|Bures distance

The Bures distance evaluates the distance between two complete positive maps, quantum channels and dual quantum channels, in particular, in terms of their dilations. We first discuss dual quantum channel $\Phi^*: \mathfrak{B}\left(\mathrm{H}_B\right) \rightarrow \mathfrak{B}\left(\mathrm{H}_A\right)$.

Recall from (6.22) that the Bures distance between quantum states $\rho$ and $\sigma$ is defined as
$$\beta(\rho, \sigma)=\sqrt{2(1-\sqrt{F(\rho, \sigma)})},$$
where $F(\rho, \sigma)=|\sqrt{\rho} \sqrt{\sigma}|_1$ is the fidelity of $\rho$ and $\sigma$.
Let $\Phi, \Psi \in \mathfrak{Q C}(A, B)$. We define the Bures distance between quantum channels $\Phi$ and $\Psi$ as
$$\beta(\Phi, \Psi):=\sup \beta\left(\left(\Phi \otimes \mathfrak{I}R\right)(\rho),\left(\Psi \otimes \mathfrak{I}_R\right)(\rho)\right), \quad \forall \rho \in \mathcal{S}\left(\mathrm{H}{A R}\right)$$
between quantum channels $\Phi$ and $\Psi$, where $\beta(\cdot, \cdot)$ in the right-hand side is the Bures distance between quantum states defined in equation (6.22) and $\mathrm{R}$ is any system.

Proposition 6.3.3. For any channels $\Phi$ and $\Psi$, we have
$$\frac{1}{2}|\Phi-\Psi|_{\diamond} \leq \beta(\Phi, \Psi) \leq \sqrt{|\Phi-\Psi|_{\diamond}} .$$
Proof. This follows immediately from the definition of diamond-norm $|\cdot|_{\diamond}(6.30)$, the definition of Bures distance $\beta(\cdot, \cdot)$ (6.34) and inequality (6.22),
$$\frac{1}{2}|\rho-\sigma|_1 \leq \beta(\rho, \sigma) \leq \sqrt{|\rho-\sigma|_1} .$$
This proves the proposition.
Inequality (6.35) indicates that the topology generated by the diamond norm is topologically equivalent to the Bures distance.

## 数学代写|信息论作业代写information theory代考|Norms on constrained channels

Due to infinite dimensionality of quantum systems whose Hamiltonian takes the form of an $\mathfrak{H}$-operator, situations often happen that the classical and quantum capacity (to be discussed in later chapters) become infinite when quantum channels are repeatedly used. To prevent this from happening, we consider energy constrained quantum channels that apply to a compact subset $\mathcal{K}_{\mathbf{H}}(E)$ of $\mathcal{S}(\mathbb{H})$.

In the following, let $\mathbf{H}$ be an $\mathfrak{H}$-operator on the input space $\mathrm{H}A$ (see (3.2) for the definition where $\lambda_n, n \geq 0$ were used for the point spectrum of $\mathbf{H}$ instead of $E_n$ used here for representing energy level $\mathbf{H}$ ) that represent a Hamiltonian of the quantum system $A$. That is, $\mathbf{H}$ is an unbounded positive linear operator on $\mathbb{H}_A$ with discrete spectrum (eigenvalues) of finite multiplicity: $$0 \leq E_0 \leq E_1 \leq \cdots \leq E_n \leq \cdots$$ with the smallest eigenvalue denoted by $E_0=\inf {|\phi|_{H_A}=1}\langle\phi|\mathbf{H}| \phi\rangle_{\mathrm{H}A}$. For each $E>E_0$, let $\mathcal{K}{\mathbf{H}}(E)$ be the compact subset of $\mathcal{S}\left(\mathrm{H}A\right)$ defined by $$\mathcal{K}{\mathbf{H}}(E)=\left{\rho \in \mathcal{S}\left(\mathbb{H}_A\right) \mid \operatorname{tr}[\mathbf{H} \rho] \leq E\right}$$
In this section, various energy-constrained norms/distances, including operator norm, diamond norm and Bures distance, for quantum channels are explored.

# 信息论代考

## 数学代写|信息论作业代写information theory代考|Bures distance

Bures 距离评估两个完整的正映射、量子通道和双量子通道之间的距离，特别是在它们 的膨胀方面。我们首先讨论双量子通道 $\Phi^*: \mathfrak{B}\left(\mathrm{H}B\right) \rightarrow \mathfrak{B}\left(\mathrm{H}_A\right)$. 回想一下 (6.22)，量子态之间的 Bures 距离 $\rho$ 和 $\sigma$ 定义为 $$\beta(\rho, \sigma)=\sqrt{2(1-\sqrt{F(\rho, \sigma)})}$$ 在哪里 $F(\rho, \sigma)=|\sqrt{\rho} \sqrt{\sigma}|_1$ 是保真度 $\rho$ 和 $\sigma$. 让 $\Phi, \Psi \in \mathfrak{Q C}(A, B)$. 我们定义量子通道之间的 Bures 距离 $\Phi$ 和 $\Psi$ 作为 $$\beta(\Phi, \Psi):=\sup \beta\left((\Phi \otimes \Im R)(\rho),\left(\Psi \otimes \Im_R\right)(\rho)\right), \quad \forall \rho \in \mathcal{S}(\mathrm{H} A R)$$ 量子通道之间 $\Phi$ 和 $\Psi$ ，在哪里 $\beta(\cdot, \cdot)$ 右侧是等式 (6.22) 中定义的量子态之间的 Bures 距 离，以及 $R$ 是任何系统。 提案 6.3.3。对于任何渠道 $\Phi$ 和 $\Psi$ ， 我们有 $$\frac{1}{2}|\Phi-\Psi|{\diamond} \leq \beta(\Phi, \Psi) \leq \sqrt{|\Phi-\Psi|_{\diamond}} .$$

$$\frac{1}{2}|\rho-\sigma|_1 \leq \beta(\rho, \sigma) \leq \sqrt{|\rho-\sigma|_1} .$$

## 数学代写|信息论作业代写information theory代考|Norms on constrained channels

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