## 数学代写|信息论作业代写information theory代考|Definitions and properties

Consider the quantum systems $A$ and $B$ represented by separable complex Hilbert spaces $\mathbb{H}_A$ and $\mathbb{H}_B$, respectively.

Recall that a linear map $Y: \mathfrak{B}\left(\mathbb{H}A\right) \rightarrow \mathfrak{B}\left(\mathbb{H}_B\right)$ is said to be positive if $Y\left(\mathbf{A}^* \mathbf{A}\right) \geq \mathbf{0}$ in $\mathfrak{B}\left(\mathrm{H}_B\right)$ for every $\mathbf{A} \in \mathfrak{B}\left(\mathbb{H}_A\right)$, where $\mathbf{A}^*$ is the adjoint of $\mathbf{A}$. While a map being positive is useful in describing many relevant quantities in commutative physical systems including classical ones, it however fails to do so in quantum physics. Specifically, the map $Y$ just being positive on $\mathfrak{B}\left(\mathrm{H}_A\right)$ is not sufficient in describing the transformation of quantum states from system $A$ to system $B$, because the open quantum system often interacts with an external quantum system. In the following, we first show the inadequacy of positivity of $Y$. Imagine that outside of the system $A$, there is often another quantum system, that is often referred to environment $E$, which is represented by Hilbert space $\mathbb{H}_E$ that interacts with $\mathbb{H}_A$. Consider the composite system $\mathrm{H}{A E}:=\mathrm{H}A \otimes \mathrm{H}_E$ and the extended transformation $\hat{Y}$, which consists in applying the transformation $\mathrm{T}$ to $\mathbb{H}_A$ and ignoring $\mathbb{H}_E$. That is, applying the transformation $Y$ to the $\mathrm{H}_A$-part and the identity to $\mathrm{H}_E$-part. In other words, this means considering the mapping $$\hat{\Upsilon}=Y \otimes \mathcal{I}_E: \mathfrak{B}\left(\mathbb{H}_A \otimes \mathbb{H}_E\right) \rightarrow \mathfrak{B}\left(\mathbb{H}_A \otimes \mathbb{H}_E\right)$$ defined by $$\hat{Y}(\mathbf{A} \otimes \mathbf{E})=\left(\mathrm{Y} \otimes \mathfrak{I}_E\right)(\mathbf{A} \otimes \mathbf{E})=\mathrm{Y}(\mathbf{A}) \otimes \mathfrak{I}_E(\mathbf{E})=Y(\mathbf{A}) \otimes \mathbf{E}$$ for all $\mathbf{A} \in \mathfrak{B}\left(\mathbb{H}_A\right)$ and $\mathbf{E} \in \mathfrak{B}\left(\mathrm{H}_E\right)$, where $\mathfrak{I}_E:=\mathfrak{I}{\mathrm{H}_E}$ is the identity operator on $\mathfrak{B}\left(\mathbb{H}_E\right)$, i. e., $\mathfrak{I}_E(\mathbf{E})=\mathbf{E}$ for all $\mathbf{E} \in \mathfrak{B}\left(\mathbb{H}_E\right)$. Even though the mapping $Y$ is positive, but surprisingly enough, the extended mapping $\hat{Y}=\mathrm{Y} \otimes \mathfrak{I}_E$ does not necessarily preserve positivity as shown in the following example.

## 数学代写|信息论作业代写information theory代考|Some technical results

We now present some technical results regarding the implications of various convergence of an infinite series, $\sum_{n=1}^{+\infty} \mathbf{M}_n^* \mathbf{M}_n$, of operators. These results will be used to en-sure the good convergence of the series of operators associated with a specific type of mappings that are important in the sections that follow and in Chapter 5.

The proofs of the technical results presented in this section can be found in At$\operatorname{tal}[3]$.

• It is recommended that readers skip the proofs of these technical results at the first reading and revisit them when they are needed at a later time. The reader can consult the lecture notes by Attal [3] for proofs omitted.

Lemma 4.2.1. Let $\mathbf{M}$ and $\mathbf{X}$ be any bounded linear operators on separable Hilbert space $\mathbb{H}$, where $\mathbf{X}$ is self-adjoint. Then we have
$$\mathbf{M}^* \mathbf{X M} \leq|\mathbf{X}|_{\infty} \mathbf{M}^* \mathbf{M}$$
Proof. Since $\mathbf{X}$ is self-adjoint, $\langle\phi, \mathbf{X} \phi\rangle_{\mathbb{H}}$ is real for all $\phi \in \mathbb{H}$. By the Cauchy-Schwarz inequality (see equation (1.2)), we have
$$\langle\phi, \mathbf{X} \phi\rangle_{\mathrm{H}} \leq|\mathbf{X}|_{\infty}|\phi|_{\mathrm{H}}^2$$
for all $\phi \in \mathbb{H}$. Let $\phi=\mathrm{M} \psi$, we have from the above inequality
\begin{aligned} \left\langle\psi, \mathbf{M}^* \mathbf{X M} \psi\right\rangle_{\mathbf{H}} & =\langle\mathbf{M} \psi, \mathbf{X} \mathbf{M} \psi\rangle_{\mathrm{H}}=\langle\phi, \mathbf{X} \phi\rangle_{\mathbb{H}{\mathrm{H}}} \ & \leq|\mathbf{X}|{\infty}|\phi|_{\mathrm{H}}^2=|\mathbf{X}|_{\infty}|\mathbf{M} \psi|_{\mathbb{H}}^2=|\mathbf{X}|_{\infty}\left\langle\psi, \mathbf{M}^* \mathbf{M} \psi\right\rangle_{\mathrm{H}^*} \end{aligned}

# 信息论代考

## 数学代写|信息论作业代写information theory代考|Definitions and properties

$$\hat{\Upsilon}=Y \otimes \mathcal{I}_E: \mathfrak{B}\left(\mathbb{H}_A \otimes \mathbb{H}_E\right) \rightarrow \mathfrak{B}\left(\mathbb{H}_A \otimes \mathbb{H}_E\right)$$

$$\hat{Y}(\mathbf{A} \otimes \mathbf{E})=\left(\mathrm{Y} \otimes \mathfrak{I}_E\right)(\mathbf{A} \otimes \mathbf{E})=\mathrm{Y}(\mathbf{A}) \otimes \mathfrak{I}_E(\mathbf{E})=Y(\mathbf{A}) \otimes \mathbf{E}$$

## 数学代写|信息论作业代写information theory代考|Some technical results

$$\mathbf{M}^* \mathbf{X M} \leq|\mathbf{X}|{\infty} \mathbf{M}^* \mathbf{M}$$ 不等式 (见方程 (1.2))，我们有 $$\langle\phi, \mathbf{X} \phi\rangle{\mathrm{H}} \leq|\mathbf{X}|{\infty}|\phi|{\mathrm{H}}^2$$

$$\left\langle\psi, \mathbf{M}^* \mathbf{X M} \psi\right\rangle_{\mathbf{H}}=\langle\mathbf{M} \psi, \mathbf{X M} \psi\rangle_{\mathrm{H}}=\langle\phi, \mathbf{X} \phi\rangle_{\mathrm{HH}} \quad \leq|\mathbf{X}| \infty|\phi|{\mathrm{H}}^2=|\mathbf{X}|{\infty}|\mathbf{M} \psi|_{\mathrm{H}}^2$$

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