## 物理代写|量子力学代写quantum mechanics代考|Isometric Extension of a Quantum Channel

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

## 物理代写|量子力学代写quantum mechanics代考|Isometric Extension of a Quantum Channel

We now give a general definition for an isometric extension of a quantum channel:
DEfinition 5.2.1 (Isometric Extension) Let $\mathcal{H}A$ and $\mathcal{H}_B$ be Hilbert spaces, and let $\mathcal{N}: \mathcal{L}\left(\mathcal{H}_A\right) \rightarrow \mathcal{L}\left(\mathcal{H}_B\right)$ be a quantum channel. Let $\mathcal{H}_E$ be a Hilbert space with dimension no smaller than the Choi rank of the channel $\mathcal{N}$. An isometric extension or Stinespring dilation $U: \mathcal{H}_A \rightarrow \mathcal{H}_B \otimes \mathcal{H}_E$ of the channel $\mathcal{N}$ is a linear isometry such that $$\operatorname{Tr}_E\left{U X_A U^{\dagger}\right}=\mathcal{N}{A \rightarrow B}\left(X_A\right),$$
for $X_A \in \mathcal{L}\left(\mathcal{H}A\right)$. The fact that $U$ is an isometry is equivalent to the following conditions: $$U^{\dagger} U=I_A, \quad U U^{\dagger}=\Pi{B E},$$
where $\Pi_{B E}$ is a projection of the tensor-product Hilbert space $\mathcal{H}B \otimes \mathcal{H}_E$. NOtATION 5.2.1 We often write a channel $\mathcal{N}: \mathcal{L}\left(\mathcal{H}_A\right) \rightarrow \mathcal{L}\left(\mathcal{H}_B\right)$ as $\mathcal{N}{A \rightarrow B}$ in order to indicate the input and output systems explicitly. Similarly, we often write an isometric extension $U: \mathcal{H}A \rightarrow \mathcal{H}_B \otimes \mathcal{H}_E$ of $\mathcal{N}$ as $U{A \rightarrow B E}^{\mathcal{N}}$ in order to indicate its association with $\mathcal{N}$ explicitly, as well the fact that it accepts an inputsystem $A$ and has output systems $B$ and $E$. The system $E$ is often referred to as an “environment” system. Finally, there is a quantum channel $\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}$ associated to an isometric extension $U{A \rightarrow B E}^{\mathcal{N}}$, which is defined by
$$\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}\left(X_A\right)=U X_A U^{\dagger}$$ for $X_A \in \mathcal{L}\left(\mathcal{H}_A\right)$. Note that $\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}$ is a quantum channel with a single Kraus operator $U$ given that $U^{\dagger} U=I_A$.

## 物理代写|量子力学代写quantum mechanics代考|Isometric Extension from Kraus Operators

It is possible to determine an isometric extension of a quantum channel directly from a set of Kraus operators. Consider a quantum channel $\mathcal{N}{A \rightarrow B}$ with the following Kraus representation: $$\mathcal{N}{A \rightarrow B}\left(\rho_A\right)=\sum_j N_j \rho_A N_j^{\dagger} .$$

An isometric extension of the channel $\mathcal{N}{A \rightarrow B}$ is the following linear map: $$U{A \rightarrow B E}^{\mathcal{N}} \equiv \sum_j N_j \otimes|j\rangle_E .$$
It is straightforward to verify that the above map is an isometry:
\begin{aligned} \left(U^{\mathcal{N}}\right)^{\dagger} U^{\mathcal{N}} & =\left(\sum_k N_k^{\dagger} \otimes\left\langle\left. k\right|E\right)\left(\sum_j N_j \otimes|j\rangle_E\right)\right. \ & =\sum{k, j} N_k^{\dagger} N_j\langle k \mid j\rangle \ & =\sum_k N_k^{\dagger} N_k \ & =I_A . \end{aligned}
The last equality follows from the completeness condition of the Kraus operators. As a consequence, we get that $U^{\mathcal{N}}\left(U^{\mathcal{N}}\right)^{\dagger}$ is a projector on the joint system $B E$, which follows by the same reasoning given in (4.259). Finally, we should verify that $U^{\mathcal{N}}$ is an extension of $\mathcal{N}$. Applying the channel $\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}$ to an arbitrary density operator $\rho_A$ gives the following map: \begin{aligned} \mathcal{U}{A \rightarrow B E}^{\mathcal{N}}\left(\rho_A\right) & \equiv U^{\mathcal{N}} \rho_A\left(U^{\mathcal{N}}\right)^{\dagger} \ & =\left(\sum_j N_j \otimes|j\rangle_E\right) \rho_A\left(\sum_k N_k^{\dagger} \otimes\left\langle\left. k\right|E\right)\right. \ & =\sum{j, k} N_j \rho_A N_k^{\dagger} \otimes|j\rangle\left\langle\left. k\right|E,\right. \end{aligned} and tracing out the environment system gives back the original quantum channel $\mathcal{N}{A \rightarrow B}$ :
$$\operatorname{Tr}E\left{\mathcal{U}{A \rightarrow B E}^{\mathcal{N}}\left(\rho_A\right)\right}=\sum_j N_j \rho_A N_j^{\dagger}=\mathcal{N}_{A \rightarrow B}\left(\rho_A\right)$$

# 量子力学代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|量子力学代写quantum mechanics代考|Local Density Operators and Partial Trace

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

## 物理代写|量子力学代写quantum mechanics代考|Local Density Operators and Partial Trace

A First Example
Consider the entangled Bell state $\left|\Phi^{+}\right\rangle_{A B}$ shared on systems $A$ and $B$. In the above analyses, we determined a local density operator description for both Alice and Bob. Now, we are curious if it is possible to determine such a local density operator description for Alice and Bob with respect to the state $\left|\Phi^{+}\right\rangle_{A B}$ or more general ones.

As a first approach to this issue, recall that the density operator description arises from its usefulness in determining the probabilities of the outcomes of a particular measurement. We say that the density operator is “the state” of the system because it is a mathematical representation that allows us to compute the probabilities resulting from a physical measurement. So, if we would like to determine a “local density operator,” such a local density operator should predict the result of a local measurement.

Let us consider a local POVM $\left{\Lambda^j\right}_j$ that Alice can perform on her system. The global measurement operators for this local measurement are $\left{\Lambda_A^j \otimes I_B\right}_j$ because nothing (the identity) happens to Bob’s system. The probability of obtaining outcome $j$ when performing this measurement on the state $\left|\Phi^{+}\right\rangle_{A B}$ is
\begin{aligned} \left\langle\left.\Phi^{+}\right|{A B} \Lambda_A^j \otimes I_B \mid \Phi^{+}\right\rangle{A B} & =\frac{1}{2} \sum_{k, l=0}^1\left\langle\left. k k\right|{A B} \Lambda_A^j \otimes I_B \mid l l\right\rangle{A B} \ & =\frac{1}{2} \sum_{k, l=0}^1\left\langle\left. k\right|_A \Lambda_A^j \mid l\right\rangle_A\langle k \mid l\rangle_B \ & =\frac{1}{2}\left(\left\langle\left. 0\right|_A \Lambda_A^j \mid 0\right\rangle_A+\left\langle\left. 1\right|_A \Lambda_A^j \mid 1\right\rangle_A\right) \ & =\frac{1}{2}\left(\operatorname { T r } \left{\Lambda_A^j|0\rangle\left\langle\left. 0\right|_A\right}+\operatorname{Tr}\left{\Lambda_A^j|1\rangle\left\langle\left. 1\right|_A\right}\right)\right.\right. \ & =\operatorname{Tr}\left{\Lambda_A^j \frac{1}{2}\left(|0\rangle\left\langle\left. 0\right|_A+\mid 1\right\rangle\left\langle\left. 1\right|_A\right)\right}\right. \ & =\operatorname{Tr}\left{\Lambda_A^j \pi_A\right} \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Partial Trace

In general, we would like to determine a local density operator that predicts the outcomes of all local measurements. The general method for determining a local density operator is to employ the partial trace operation, which we motivate and define here, as a generalization of the example discussed at the beginning of Section 4.3.3.

Suppose that Alice and Bob share a bipartite state $\rho_{A B}$ and that Alice performs a local measurement on her system, described by a POVM $\left{\Lambda_A^j\right}$. Then the overall POVM on the joint system is $\left{\Lambda_A^j \otimes I_B\right}$ because we are assuming that Bob is not doing anything to his system. According to the Born rule, the probability for Alice to receive outcome $j$ after performing the measurement is given by the following expression:
$$p_J(j)=\operatorname{Tr}\left{\left(\Lambda_A^j \otimes I_B\right) \rho_{A B}\right}$$
In order to evaluate the trace, we can choose any orthonormal basis that we wish (see Definition 4.1.1 and subsequent statements). Taking $\left{|k\rangle_A\right}$ as an orthonormal basis for Alice’s Hilbert space and $\left{|l\rangle_B\right}$ as an orthonormal basis for Bob’s Hilbert space, the set $\left{|k\rangle_A \otimes|l\rangle_B\right}$ constitutes an orthonormal basis for the tensor product of their Hilbert spaces. So we can evaluate (4.138) as follows:
\begin{aligned} & \operatorname{Tr}\left{\left(\Lambda_A^j \otimes I_B\right) \rho_{A B}\right} \ & =\sum_{k, l}\left(\left\langle\left.k\right|A \otimes\left\langle\left. l\right|_B\right)\left[\left(\Lambda_A^j \otimes I_B\right) \rho{A B}\right]\left(|k\rangle_A \otimes|l\rangle_B\right)\right.\right. \end{aligned}

\begin{aligned} & =\sum_{k, l}\left\langlek | _ { A } \left( I_A \otimes\left\langle\left. l\right|B\right)\left[\left(\Lambda_A^j \otimes I_B\right) \rho{A B}\right]\left(I_A \otimes|l\rangle_B\right)|k\rangle_A\right.\right. \ & =\sum_{k, l}\left\langlek | _ { A } \Lambda _ { A } ^ { j } \left( I_A \otimes\left\langle\left. l\right|B\right) \rho{A B}\left(I_A \otimes|l\rangle_B\right)|k\rangle_A\right.\right. \ & =\sum_k\left\langlek | _ { A } \Lambda _ { A } ^ { j } \left[\sum_l\left(I_A \otimes\left\langle\left. l\right|B\right) \rho{A B}\left(I_A \otimes|l\rangle_B\right)\right]|k\rangle_A .\right.\right. \end{aligned}
The first equality follows from the definition of the trace in Definition 4.1.1 and using the orthonormal basis $\left{|k\rangle_A \otimes|l\rangle_B\right}$. The second equality follows because
$$|k\rangle_A \otimes|l\rangle_B=\left(I_A \otimes|l\rangle_B\right)|k\rangle_A$$
The third equality follows because
$$\left(I_A \otimes\left\langle\left. l\right|_B\right)\left(\Lambda_A^j \otimes I_B\right)=\Lambda_A^j\left(I_A \otimes\left\langle\left. l\right|_B\right) .\right.\right.$$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Local Density Operators and Partial Trace

\begin{aligned} \left\langle\left.\Phi^{+}\right|{A B} \Lambda_A^j \otimes I_B \mid \Phi^{+}\right\rangle{A B} & =\frac{1}{2} \sum_{k, l=0}^1\left\langle\left. k k\right|{A B} \Lambda_A^j \otimes I_B \mid l l\right\rangle{A B} \ & =\frac{1}{2} \sum_{k, l=0}^1\left\langle\left. k\right|_A \Lambda_A^j \mid l\right\rangle_A\langle k \mid l\rangle_B \ & =\frac{1}{2}\left(\left\langle\left. 0\right|_A \Lambda_A^j \mid 0\right\rangle_A+\left\langle\left. 1\right|_A \Lambda_A^j \mid 1\right\rangle_A\right) \ & =\frac{1}{2}\left(\operatorname { T r } \left{\Lambda_A^j|0\rangle\left\langle\left. 0\right|_A\right}+\operatorname{Tr}\left{\Lambda_A^j|1\rangle\left\langle\left. 1\right|_A\right}\right)\right.\right. \ & =\operatorname{Tr}\left{\Lambda_A^j \frac{1}{2}\left(|0\rangle\left\langle\left. 0\right|_A+\mid 1\right\rangle\left\langle\left. 1\right|_A\right)\right}\right. \ & =\operatorname{Tr}\left{\Lambda_A^j \pi_A\right} \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Partial Trace

$$p_J(j)=\operatorname{Tr}\left{\left(\Lambda_A^j \otimes I_B\right) \rho_{A B}\right}$$

\begin{aligned} & \operatorname{Tr}\left{\left(\Lambda_A^j \otimes I_B\right) \rho_{A B}\right} \ & =\sum_{k, l}\left(\left\langle\left.k\right|A \otimes\left\langle\left. l\right|_B\right)\left[\left(\Lambda_A^j \otimes I_B\right) \rho{A B}\right]\left(|k\rangle_A \otimes|l\rangle_B\right)\right.\right. \end{aligned}

\begin{aligned} & =\sum_{k, l}\left\langlek | _ { A } \left( I_A \otimes\left\langle\left. l\right|B\right)\left[\left(\Lambda_A^j \otimes I_B\right) \rho{A B}\right]\left(I_A \otimes|l\rangle_B\right)|k\rangle_A\right.\right. \ & =\sum_{k, l}\left\langlek | _ { A } \Lambda _ { A } ^ { j } \left( I_A \otimes\left\langle\left. l\right|B\right) \rho{A B}\left(I_A \otimes|l\rangle_B\right)|k\rangle_A\right.\right. \ & =\sum_k\left\langlek | _ { A } \Lambda _ { A } ^ { j } \left[\sum_l\left(I_A \otimes\left\langle\left. l\right|B\right) \rho{A B}\left(I_A \otimes|l\rangle_B\right)\right]|k\rangle_A .\right.\right. \end{aligned}

$$|k\rangle_A \otimes|l\rangle_B=\left(I_A \otimes|l\rangle_B\right)|k\rangle_A$$

$$\left(I_A \otimes\left\langle\left. l\right|_B\right)\left(\Lambda_A^j \otimes I_B\right)=\Lambda_A^j\left(I_A \otimes\left\langle\left. l\right|_B\right) .\right.\right.$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|量子力学代写quantum mechanics代考|Noiseless Evolution of an Ensemble

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

## 物理代写|量子力学代写quantum mechanics代考|Noiseless Evolution of an Ensemble

Quantum states can evolve in a noiseless fashion either according to a unitary operator or a measurement. In this section, we determine the noiseless evolution of an ensemble and its corresponding density operator. We also show how density operators evolve under a quantum measurement.

Noiseless Unitary Evolution of a Noisy State
We first consider noiseless evolution according to some unitary $U$. Suppose we have the ensemble $\mathcal{E}$ in (4.2) with density operator $\rho$. Suppose without loss of generality that the state is $\left|\psi_x\right\rangle$. Then the evolution postulate of the noiseless quantum theory gives that the state after the unitary evolution is as follows: $U\left|\psi_x\right\rangle$. This result implies that the evolution leads to a new ensemble
$$\mathcal{E}U \equiv\left{p_X(x), U\left|\psi_x\right\rangle\right}{x \in \mathcal{X}}$$
The density operator of the evolved ensemble is
\begin{aligned} \sum_{x \in \mathcal{X}} p_X(x) U\left|\psi_x\right\rangle\left\langle\psi_x\right| U^{\dagger} & =U\left(\sum_{x \in \mathcal{X}} p_X(x)\left|\psi_x\right\rangle\left\langle\psi_x\right|\right) U^{\dagger} \ & =U \rho U^{\dagger} \end{aligned}
Thus, the above relation shows that we can keep track of the evolution of the density operator $\rho$, rather than worrying about keeping track of the evolution of every state in the ensemble $\mathcal{E}$. It suffices to keep track of only the density operator evolution because this operator is sufficient to determine probabilities when performing any measurement on the system.

## 物理代写|量子力学代写quantum mechanics代考|Noiseless Measurement of a Noisy State

In a similar fashion, we can analyze the result of a measurement on a system with ensemble description $\mathcal{E}$ in (4.2). Suppose that we perform a projective measurement with projection operators $\left{\Pi_j\right}_j$ where $\sum_j \Pi_j=I$. The main result of this section is that two things happen after a measurement occurs. First, as shown in the development preceding (4.19), we receive the outcome $j$ with probability $p_J(j)=\operatorname{Tr}\left{\Pi_j \rho\right}$. Second, if the outcome of the measurement is $j$, then the state evolves as follows:
$$\rho \longrightarrow \frac{\Pi_j \rho \Pi_j}{p_J(j)}$$
To see the above, let us suppose that the state in the ensemble $\mathcal{E}$ is $\left|\psi_x\right\rangle$. Then the noiseless quantum theory predicts that the probability of obtaining outcome $j$ conditioned on the index $x$ is
$$p_{J \mid X}(j \mid x)=\left\langle\psi_x\left|\Pi_j\right| \psi_x\right\rangle$$
and the resulting state is
$$\frac{\Pi_j\left|\psi_x\right\rangle}{\sqrt{p_{J \mid X}(j \mid x)}}$$
Supposing that we receive outcome $j$, then we have a new ensemble:
$$\mathcal{E}j \equiv\left{p{X \mid J}(x \mid j), \frac{\Pi_j\left|\psi_x\right\rangle}{\sqrt{p_{J \mid X}(j \mid x)}}\right}_{x \in \mathcal{X}} .$$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Noiseless Evolution of an Ensemble

$$\mathcal{E}U \equiv\left{p_X(x), U\left|\psi_x\right\rangle\right}{x \in \mathcal{X}}$$

\begin{aligned} \sum_{x \in \mathcal{X}} p_X(x) U\left|\psi_x\right\rangle\left\langle\psi_x\right| U^{\dagger} & =U\left(\sum_{x \in \mathcal{X}} p_X(x)\left|\psi_x\right\rangle\left\langle\psi_x\right|\right) U^{\dagger} \ & =U \rho U^{\dagger} \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Noiseless Measurement of a Noisy State

$$\frac{\Pi_j\left|\psi_x\right\rangle}{\sqrt{p_{J \mid X}(j \mid x)}}$$

$$\mathcal{E}j \equiv\left{p{X \mid J}(x \mid j), \frac{\Pi_j\left|\psi_x\right\rangle}{\sqrt{p_{J \mid X}(j \mid x)}}\right}_{x \in \mathcal{X}} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|量子力学代写quantum mechanics代考|The Qudit Bell States

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

## 物理代写|量子力学代写quantum mechanics代考|The Qudit Bell States

Two-qudit states can be entangled as well. The maximally entangled qudit state is as follows:
$$|\Phi\rangle_{A B} \equiv \frac{1}{\sqrt{d}} \sum_{i=0}^{d-1}|i\rangle_A|i\rangle_B .$$
When Alice possesses the first qudit and Bob possesses the second qudit and they are also separated in space, the above state is a resource known as an edit (pronounced “ee · dit”). It is useful in the qudit versions of the teleportation protocol and the super-dense coding protocol discussed in Chapter 6. Throughout the book, we often find it convenient to make use of the unnormalized maximally entangled vector:
$$|\Gamma\rangle_{A B} \equiv \sum_{i=0}^{d-1}|i\rangle_A|i\rangle_B$$
Consider applying the operator $X(x) Z(z)$ to Alice’s share of the maximally entangled state $|\Phi\rangle_{A B}$. We use the following notation:
$$\left|\Phi^{x, z}\right\rangle_{A B} \equiv\left(X_A(x) Z_A(z) \otimes I_B\right)|\Phi\rangle_{A B}$$

The $d^2$ states $\left{\left|\Phi^{x, z}\right\rangle_{A B}\right}_{x, z=0}^{d-1}$ are known as the qudit Bell states and are important in qudit quantum protocols and in quantum Shannon theory. Exercise 3.7.11 asks you to verify that these states form a complete, orthonormal basis. Thus, one can measure two qudits in the qudit Bell basis. Similar to the qubit case, it is straightforward to see that the qudit state can generate a dit of shared randomness by extending the arguments in Section 3.6.1.

EXERCISE 3.7.11 Show that the set of states $\left{\left|\Phi^{x, z}\right\rangle_{A B}\right}_{x, z=0}^{d-1}$ forms a complete, orthonormal basis:
\begin{aligned} \left\langle\Phi^{x_1, z_1} \mid \Phi^{x_2, z_2}\right\rangle & =\delta_{x_1, x_2} \delta_{z_1, z_2} \ \sum_{x, z=0}^{d-1}\left|\Phi^{x, z}\right\rangle\left\langle\left.\Phi^{x, z}\right|{A B}\right. & =I{A B} \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Schmidt Decomposition

The Schmidt decomposition is one of the most important tools for analyzing bipartite pure states in quantum information theory, showing that it is possible to decompose any pure bipartite state as a superposition of coordinated orthonormal states. It is a consequence of the well known singular value decomposition theorem from linear algebra. We state this result formally as the following theorem:

THEOREm 3.8.1 (Schmidt Decomposition) Suppose that we have a bipartite pure state,
$$|\psi\rangle_{A B} \in \mathcal{H}A \otimes \mathcal{H}_B$$ where $\mathcal{H}_A$ and $\mathcal{H}_B$ are finite-dimensional Hilbert spaces, not necessarily of the same dimension, and $||\psi\rangle{A B} |_2=1$. Then it is possible to express this state as follows:
$$|\psi\rangle_{A B} \equiv \sum_{i=0}^{d-1} \lambda_i|i\rangle_A|i\rangle_B,$$
where the amplitudes $\lambda_i$ are real, strictly positive, and normalized so that $\sum_i \lambda_i^2=1$, the states $\left{|i\rangle_A\right}$ form an orthonormal basis for system $A$, and the states $\left{|i\rangle_B\right}$ form an orthonormal basis for the system $B$. The vector $\left[\lambda_i\right]_{i \in{0, \ldots, d-1}}$ is called the vector of Schmidt coefficients. The Schmidt rank $d$ of a bipartite state is equal to the number of Schmidt coefficients $\lambda_i$ in its Schmidt decomposition and satisfies
$$d \leq \min \left{\operatorname{dim}\left(\mathcal{H}_A\right), \operatorname{dim}\left(\mathcal{H}_B\right)\right}$$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|The Qudit Bell States

$$|\Phi\rangle_{A B} \equiv \frac{1}{\sqrt{d}} \sum_{i=0}^{d-1}|i\rangle_A|i\rangle_B .$$

$$|\Gamma\rangle_{A B} \equiv \sum_{i=0}^{d-1}|i\rangle_A|i\rangle_B$$

$$\left|\Phi^{x, z}\right\rangle_{A B} \equiv\left(X_A(x) Z_A(z) \otimes I_B\right)|\Phi\rangle_{A B}$$

$d^2$状态$\left{\left|\Phi^{x, z}\right\rangle_{A B}\right}_{x, z=0}^{d-1}$被称为qudit Bell状态，在qudit量子协议和量子香农理论中很重要。练习3.7.11要求您验证这些状态是否构成一个完整的标准正交基。因此，可以在qudit Bell基中测量两个qudit。与量子位的情况类似，通过扩展3.6.1节中的参数，很容易看出量子位状态可以生成共享随机性的dit。

## 物理代写|量子力学代写quantum mechanics代考|Schmidt Decomposition

Schmidt分解是量子信息论中分析二部纯态最重要的工具之一，它表明任何纯二部态都可以分解为协调正交态的叠加。它是线性代数中著名的奇异值分解定理的一个结果。我们将这一结果形式化地表述为以下定理:

$$|\psi\rangle_{A B} \in \mathcal{H}A \otimes \mathcal{H}B$$其中$\mathcal{H}_A$和$\mathcal{H}_B$是有限维希尔伯特空间，不一定是相同的维数，还有$||\psi\rangle{A B} |_2=1$。那么，可以将这种状态表示为: $$|\psi\rangle{A B} \equiv \sum_{i=0}^{d-1} \lambda_i|i\rangle_A|i\rangle_B,$$

$$d \leq \min \left{\operatorname{dim}\left(\mathcal{H}_A\right), \operatorname{dim}\left(\mathcal{H}_B\right)\right}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|量子力学代写quantum mechanics代考|Quantum Strategies

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

## 物理代写|量子力学代写quantum mechanics代考|Quantum Strategies

What does a quantum strategy of Alice and Bob look like? Here the parameter $\lambda$ can correspond to a shared quantum $|\phi\rangle_{A B}$. Alice and Bob perform local measurements depending on the values of the inputs $x$ and $y$ that they receive. We can write Alice’s $x$-dependent measurement as $\left{\Pi_a^{(x)}\right}$ where for each $x, \Pi_a^{(x)}$ is a projector and $\sum_a \Pi_a^{(x)}=I$. Similarly, we can write Bob’s $y$-dependent measurement as $\left{\Pi_b^{(y)}\right}$. Then we instead employ the Born rule to determine the conditional probability distribution $p_{A B \mid X Y}(a, b \mid x, y)$ :
$$p_{A B \mid X Y}(a, b \mid x, y)=\left\langle\left.\phi\right|{A B} \Pi_a^{(x)} \otimes \Pi_b^{(y)} \mid \phi\right\rangle{A B}$$
so that the winning probability with a particular quantum strategy is as follows:
$$\frac{1}{4} \sum_{a, b, x, y} V(x, y, a, b)\left\langle\left.\phi\right|{A B} \Pi_a^{(x)} \otimes \Pi_b^{(y)} \mid \phi\right\rangle{A B} .$$

Interestingly, if Alice and Bob share a maximally entangled state, they can achieve a higher winning probability than if they share classical correlations only. This is one demonstration of the power of entanglement, and we leave it as an exercise to prove that the following quantum strategy achieves a winning probability of $\cos ^2(\pi / 8) \approx 0.85$ in the $\mathrm{CHSH}$ game.

## 物理代写|量子力学代写quantum mechanics代考|Maximum Quantum Winning Probability

Given that classical strategies cannot win with probability any larger than $3 / 4$, it is natural to wonder if there is a bound on the winning probability of a quantum strategy. It turns out that $\cos ^2(\pi / 8)$ is the maximum probability with which Alice and Bob can win the $\mathrm{CHSH}$ game using a quantum strategy, a result known as Tsirelson’s bound. To establish this result, let us go back to the CHSH game. Conditioned on the inputs $x$ and $y$ being equal to 00,01 , or 10 , we know that Alice and Bob win if they report back the same results. The probability for this to happen with a given quantum strategy is
$$\left\langle\left.\phi\right|{A B} \Pi_0^{(x)} \otimes \Pi_0^{(y)} \mid \phi\right\rangle{A B}+\left\langle\left.\phi\right|{A B} \Pi_1^{(x)} \otimes \Pi_1^{(y)} \mid \phi\right\rangle{A B}$$
and the probability for it not to happen is
$$\left\langle\left.\phi\right|{A B} \Pi_0^{(x)} \otimes \Pi_1^{(y)} \mid \phi\right\rangle{A B}+\left\langle\left.\phi\right|{A B} \Pi_1^{(x)} \otimes \Pi_0^{(y)} \mid \phi\right\rangle{A B}$$
So, conditioned on $x$ and $y$ being equal to 00,01 , or 10 , the probability of winning minus the probability of losing is
$$\left\langle\left.\phi\right|{A B} A^{(x)} \otimes B^{(y)} \mid \phi\right\rangle{A B}$$
where we define the observables $A^{(x)}$ and $B^{(y)}$ as follows:
\begin{aligned} & A^{(x)} \equiv \Pi_0^{(x)}-\Pi_1^{(x)}, \ & B^{(y)} \equiv \Pi_0^{(y)}-\Pi_1^{(y)} . \end{aligned}
If $x$ and $y$ are both equal to one, then Alice and Bob should report back different results, and similar to the above, one can work out that the probability of winning minus the probability of losing is equal to
$$-\left\langle\left.\phi\right|{A B} A^{(1)} \otimes B^{(1)} \mid \phi\right\rangle{A B}$$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Quantum Strategies

$$p_{A B \mid X Y}(a, b \mid x, y)=\left\langle\left.\phi\right|{A B} \Pi_a^{(x)} \otimes \Pi_b^{(y)} \mid \phi\right\rangle{A B}$$

$$\frac{1}{4} \sum_{a, b, x, y} V(x, y, a, b)\left\langle\left.\phi\right|{A B} \Pi_a^{(x)} \otimes \Pi_b^{(y)} \mid \phi\right\rangle{A B} .$$

## 物理代写|量子力学代写quantum mechanics代考|Maximum Quantum Winning Probability

$$\left\langle\left.\phi\right|{A B} \Pi_0^{(x)} \otimes \Pi_0^{(y)} \mid \phi\right\rangle{A B}+\left\langle\left.\phi\right|{A B} \Pi_1^{(x)} \otimes \Pi_1^{(y)} \mid \phi\right\rangle{A B}$$

$$\left\langle\left.\phi\right|{A B} \Pi_0^{(x)} \otimes \Pi_1^{(y)} \mid \phi\right\rangle{A B}+\left\langle\left.\phi\right|{A B} \Pi_1^{(x)} \otimes \Pi_0^{(y)} \mid \phi\right\rangle{A B}$$

$$\left\langle\left.\phi\right|{A B} A^{(x)} \otimes B^{(y)} \mid \phi\right\rangle{A B}$$

\begin{aligned} & A^{(x)} \equiv \Pi_0^{(x)}-\Pi_1^{(x)}, \ & B^{(y)} \equiv \Pi_0^{(y)}-\Pi_1^{(y)} . \end{aligned}

$$-\left\langle\left.\phi\right|{A B} A^{(1)} \otimes B^{(1)} \mid \phi\right\rangle{A B}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|量子力学代写quantum mechanics代考|Probability Amplitudes for Composite Systems

statistics-lab™ 为您的留学生涯保驾护航 在代写量子力学quantum mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量子力学quantum mechanics代写方面经验极为丰富，各种代写量子力学quantum mechanics相关的作业也就用不着说。

## 物理代写|量子力学代写quantum mechanics代考|Probability Amplitudes for Composite Systems

We relied on the orthogonality of the two-qubit computational basis states for evaluating amplitudes such as $\langle 00 \mid 10\rangle$ or $\langle 00 \mid 00\rangle$ in the above matrix representation. It turns out that there is another way to evaluate these amplitudes that relies only on the orthogonality of the single-qubit computational basis states.

Suppose that we have four single-qubit states $\left|\phi_0\right\rangle,\left|\phi_1\right\rangle,\left|\psi_0\right\rangle,\left|\psi_1\right\rangle$, and we make the following two-qubit states from them:
$$\left|\phi_0\right\rangle \otimes\left|\psi_0\right\rangle, \quad\left|\phi_1\right\rangle \otimes\left|\psi_1\right\rangle$$
We may represent these states equally well as follows:
$$\left|\phi_0, \psi_0\right\rangle, \quad\left|\phi_1, \psi_1\right\rangle$$
because the Dirac notation is versatile (virtually anything can go inside a ket as long as its meaning is not ambiguous). The bra $\left\langle\phi_1, \psi_1\right|$ is dual to the ket $\left|\phi_1, \psi_1\right\rangle$, and we can use it to calculate the following amplitude:
$$\left\langle\phi_1, \psi_1 \mid \phi_0, \psi_0\right\rangle$$
This amplitude is equal to the multiplication of the single-qubit amplitudes:
$$\left\langle\phi_1, \psi_1 \mid \phi_0, \psi_0\right\rangle=\left\langle\phi_1 \mid \phi_0\right\rangle\left\langle\psi_1 \mid \psi_0\right\rangle$$

## 物理代写|量子力学代写quantum mechanics代考|Controlled Gates

An important two-qubit unitary evolution is the controlled-NOT (CNOT) gate. We consider its classical version first. The classical gate acts on two cbits. It does nothing if the first bit is equal to zero, and flips the second bit if the first bit is equal to one:
$$00 \rightarrow 00, \quad 01 \rightarrow 01, \quad 10 \rightarrow 11, \quad 11 \rightarrow 10$$
We turn this gate into a quantum gate ${ }^5$ by demanding that it act in the same way on the two-qubit computational basis states:
$$|00\rangle \rightarrow|00\rangle, \quad|01\rangle \rightarrow|01\rangle, \quad|10\rangle \rightarrow|11\rangle, \quad|11\rangle \rightarrow|10\rangle .$$
By linearity, this behavior carries over to superposition states as well:
$$\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle \quad \stackrel{\text { CNOT }}{\longrightarrow} \alpha|00\rangle+\beta|01\rangle+\gamma|11\rangle+\delta|10\rangle .$$
A useful operator representation of the CNOT gate is
$$\mathrm{CNOT} \equiv|0\rangle\langle 0|\otimes I+| 1\rangle\langle 1| \otimes X$$
The above representation truly captures the coherent quantum nature of the CNOT gate. In the classical CNOT gate, we can say that it is a conditional gate, in the sense that the gate applies to the second bit conditioned on the value of the first bit. In the quantum CNOT gate, the second operation is controlled on the basis state of the first qubit (hence the choice of the name “controlled-NOT”). That is, the gate acts on superpositions of quantum states and maintains these superpositions, shuffling the probability amplitudes around while it does so. The one case in which the gate has no effect is when the first qubit is prepared in the state $|0\rangle$ and the state of the second qubit is arbitrary.

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Probability Amplitudes for Composite Systems

$$\left|\phi_0\right\rangle \otimes\left|\psi_0\right\rangle, \quad\left|\phi_1\right\rangle \otimes\left|\psi_1\right\rangle$$

$$\left|\phi_0, \psi_0\right\rangle, \quad\left|\phi_1, \psi_1\right\rangle$$

$$\left\langle\phi_1, \psi_1 \mid \phi_0, \psi_0\right\rangle$$

$$\left\langle\phi_1, \psi_1 \mid \phi_0, \psi_0\right\rangle=\left\langle\phi_1 \mid \phi_0\right\rangle\left\langle\psi_1 \mid \psi_0\right\rangle$$

## 物理代写|量子力学代写quantum mechanics代考|Controlled Gates

$$00 \rightarrow 00, \quad 01 \rightarrow 01, \quad 10 \rightarrow 11, \quad 11 \rightarrow 10$$

$$|00\rangle \rightarrow|00\rangle, \quad|01\rangle \rightarrow|01\rangle, \quad|10\rangle \rightarrow|11\rangle, \quad|11\rangle \rightarrow|10\rangle .$$

$$\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle \quad \stackrel{\text { CNOT }}{\longrightarrow} \alpha|00\rangle+\beta|01\rangle+\gamma|11\rangle+\delta|10\rangle .$$
CNOT门的一个有用的算子表示是
$$\mathrm{CNOT} \equiv|0\rangle\langle 0|\otimes I+| 1\rangle\langle 1| \otimes X$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|量子力学代写quantum mechanics代考|Bases in Hilbert Space

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## 物理代写|量子力学代写quantum mechanics代考|Bases in Hilbert Space

Imagine an ordinary vector $\mathbf{A}$ in two dimensions (Fig. 3.3(a)). How would you describe this vector to someone? You might tell them “It’s about an inch long, and it points $20^{\circ}$ clockwise from straight up, with respect to the page.” But that’s pretty awkward. A better way would be to introduce cartesian axes, $x$ and $y$, and specify the components of $\mathbf{A}: A_x=\hat{\imath} \cdot \mathbf{A}, A_y=\hat{\jmath} \cdot \mathbf{A}$ (Fig. 3.3(b)). Of course, your sister might draw a different set of axes, $x^{\prime}$ and $y^{\prime}$, and she would report different components: $A_x^{\prime}=\hat{l}^{\prime} \cdot \mathbf{A}, A_y^{\prime}=\hat{J}^{\prime} \cdot \mathbf{A}$ (Fig. 3.3(c)) …but it’s all the same vector-we’re simply expressing it with respect to two different bases $\left({\hat{\imath}, \hat{\jmath}}\right.$ and $\left.\left{\hat{l}^{\prime}, \hat{\jmath}^{\prime}\right}\right)$. The vector itself lives “out there in space,” independent of anybody’s (arbitrary) choice of coordinates.

The same is true for the state of a system in quantum mechanics. It is represented by a vector, $\mid \mathcal{S}(t))$, that lives “out there in Hilbert space,” but we can express it with respect to any number of different bases. The wave function $\Psi(x, t)$ is actually the $x$ “component” in the expansion of $|\mathcal{S}(t)\rangle$ in the basis of position eigenfunctions:
$$\Psi(x, t)=\langle x \mid \mathcal{S}(t)\rangle$$
(the analog to $\hat{\imath} \cdot \mathbf{A}$ ) with $|x\rangle$ standing for the eigenfunction of $\hat{x}$ with eigenvalue $x .^{27}$ The momentum space wave function $\Phi(p, t)$ is the $p$ component in the expansion of $|\mathcal{S}(t)\rangle$ in the basis of momentum eigenfunctions:
$$\Phi(p, t)=\langle p \mid \mathcal{S}(t)\rangle$$
(with $|p\rangle$ standing for the eigenfunction of $\hat{p}$ with eigenvalue $p$ ). $\frac{28}{}$ Or we could expand $|\mathcal{S}(t)\rangle$ in the basis of energy eigenfunctions (supposing for simplicity that the spectrum is discrete):
$$c_n(t)=\langle n \mid \mathcal{S}(t)\rangle$$
(with $|n\rangle$ standing for the $n$th eigenfunction of $\hat{H}$-Equation 3.46). But it’s all the same state; the functions $\Psi$ and $\Phi$, and the collection of coefficients $\left{c_n\right}$, contain exactly the same information-they are simply three different ways of identifying the same vector:
\begin{aligned} |\mathcal{S}(t)\rangle & \rightarrow \int \Psi(y, t) \delta(x-y) d y=\int \Phi(p, t) \frac{1}{\sqrt{2 \pi \hbar}} e^{i p x / \hbar} d p \ & =\sum c_n e^{-i E_n t / \hbar} \psi_n(x) \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Dirac Notation

Dirac proposed to chop the bracket notation for the inner product, $\langle\alpha \mid \beta\rangle$, into two pieces, which he called bra, $\langle\alpha|$, and ket, $|\beta\rangle$ (I don’t know what happened to the $\mathrm{c}$ ). The latter is a vector, but what exactly is the former? It’s a linear function of vectors, in the sense that when it hits a vector (to its right) it yields a (complex) number-the inner product. (When an operator hits a vector, it delivers another vector; when a bra hits a vector, it delivers a number.) In a function space, the bra can be thought of as an instruction to integrate:
$$\langle f|=\int f^[\cdots] d x$$ with the ellipsis $[\cdots]$ waiting to be filled by whatever function the bra encounters in the ket to its right. In a finite-dimensional vector space, with the kets expressed as columns (of components with respect to some basis), $$|\alpha\rangle \rightarrow\left(\begin{array}{c} a_1 \ a_2 \ \vdots \ a_n \end{array}\right),$$ the bras are rows: $$\langle\beta| \rightarrow\left(b_1^ b_2^* \ldots b_n^\right)$$ and $\langle\beta \mid \alpha\rangle=b_1^ a_1+b_2^* a_2+\cdots+b_n^* a_n$ is the matrix product. The collection of all bras constitutes another vector space-the so-called dual space.

The license to treat bras as separate entities in their own right allows for some powerful and pretty notation. For example, if $|\alpha\rangle$ is a normalized vector, the operator
$$\hat{P} \equiv|\alpha\rangle\langle\alpha|$$
picks out the portion of any other vector that “lies along” $\mid \alpha)$ :
$$\hat{P}|\beta\rangle=(\langle\alpha \mid \beta\rangle)|\alpha\rangle$$
we call it the projection operator onto the one-dimensional subspace spanned by $|\alpha\rangle$. If $\left{\left|e_n\right\rangle\right}$ is a discrete orthonormal basis,
$$\left\langle e_m \mid e_n\right\rangle=\delta_{m n}$$
then
$$\sum_n\left|e_n\right\rangle\left\langle e_n\right|=1$$
(the identity operator). For if we let this operator act on any vector $\mid \alpha)$, we recover the expansion of $|\alpha\rangle$ in the $\left{\left|e_n\right\rangle\right}$ basis:
$$\sum_n\left(\left\langle e_n \mid \alpha\right\rangle\right)\left|e_n\right\rangle=|\alpha\rangle$$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Bases in Hilbert Space

$$\Psi(x, t)=\langle x \mid \mathcal{S}(t)\rangle$$
(与$\hat{\imath} \cdot \mathbf{A}$类似)，$|x\rangle$代表$\hat{x}$的本征函数，本征值为$x .^{27}$。动量空间波函数$\Phi(p, t)$是基于动量本征函数展开$|\mathcal{S}(t)\rangle$中的$p$分量:
$$\Phi(p, t)=\langle p \mid \mathcal{S}(t)\rangle$$
(其中$|p\rangle$代表$\hat{p}$的特征函数，特征值为$p$)。$\frac{28}{}$或者我们可以在能量特征函数的基础上展开$|\mathcal{S}(t)\rangle$(为了简单起见，假设频谱是离散的):
$$c_n(t)=\langle n \mid \mathcal{S}(t)\rangle$$
($|n\rangle$表示$\hat{H}$的第$n$个特征函数-方程3.46)。但它们都是同一个状态;函数$\Psi$和$\Phi$以及系数集合$\left{c_n\right}$包含完全相同的信息——它们只是标识同一向量的三种不同方式:
\begin{aligned} |\mathcal{S}(t)\rangle & \rightarrow \int \Psi(y, t) \delta(x-y) d y=\int \Phi(p, t) \frac{1}{\sqrt{2 \pi \hbar}} e^{i p x / \hbar} d p \ & =\sum c_n e^{-i E_n t / \hbar} \psi_n(x) \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Dirac Notation

$$\langle f|=\int f^[\cdots] d x$$与省略号$[\cdots]$等待由胸罩在其右侧的ket中遇到的任何函数填充。在有限维的向量空间中，矩阵表示为列(关于某些基的分量)，$$|\alpha\rangle \rightarrow\left(\begin{array}{c} a_1 \ a_2 \ \vdots \ a_n \end{array}\right),$$胸罩是行:$$\langle\beta| \rightarrow\left(b_1^ b_2^* \ldots b_n^\right)$$和$\langle\beta \mid \alpha\rangle=b_1^ a_1+b_2^* a_2+\cdots+b_n^* a_n$是矩阵乘积。所有胸罩的集合构成了另一个向量空间——所谓的对偶空间。

$$\hat{P} \equiv|\alpha\rangle\langle\alpha|$$

$$\hat{P}|\beta\rangle=(\langle\alpha \mid \beta\rangle)|\alpha\rangle$$

$$\left\langle e_m \mid e_n\right\rangle=\delta_{m n}$$

$$\sum_n\left|e_n\right\rangle\left\langle e_n\right|=1$$
(单位运算符)。因为如果我们让这个算子作用于任意向量$\mid \alpha)$，我们可以在$\left{\left|e_n\right\rangle\right}$基中恢复$|\alpha\rangle$的展开:
$$\sum_n\left(\left\langle e_n \mid \alpha\right\rangle\right)\left|e_n\right\rangle=|\alpha\rangle$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。