## 物理代写|量子力学代写quantum mechanics代考|PHYS402

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|Minimal Coupling

In classical electrodynamics $\frac{55}{}$ the force on a particle of charge $q$ moving with velocity $\mathbf{v}$ through electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$ is given by the Lorentz force law:
$$\mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}) .$$
This force cannot be expressed as the gradient of a scalar potential energy function, and therefore the Schrödinger equation in its original form (Equation 1.1) cannot accommodate it. But in the more sophisticated form
$$i \hbar \frac{\partial \Psi}{\partial t}=\hat{H} \Psi$$
there is no problem. The classical Hamiltonian for a particle of charge $q$ and momentum p, in the presence of electromagnetic fields is $\frac{56}{}$
$$H=\frac{1}{2 m}(\mathbf{p}-q \mathbf{A})^2+q \varphi,$$
where $\mathbf{A}$ is the vector potential and $\varphi$ is the scalar potential:
$$\mathbf{E}=-\nabla \varphi-\partial \mathbf{A} / \partial t, \quad \mathbf{B}=\nabla \times \mathbf{A} .$$
Making the standard substitution $\mathbf{p} \rightarrow-i \hbar \nabla$, we obtain the Hamiltonian operator 57
$$\hat{H}=\frac{1}{2 m}(-i \hbar \nabla-q \mathbf{A})^2+q \varphi,$$
and the Schrödinger equation becomes
$$i \hbar \frac{\partial \Psi}{\partial t}=\left[\frac{1}{2 m}(-i \hbar \nabla-q \mathbf{A})^2+q \varphi\right] \Psi$$

## 物理代写|量子力学代写quantum mechanics代考|The Aharonov–Bohm Effect

In classical electrodynamics the potentials $\mathbf{A}$ and $\varphi$ are not uniquely determined; the physical quantities are the fields, $\mathrm{E}$ and $\mathrm{B} . 61$ Specifically, the potentials
$$\varphi^{\prime} \equiv \varphi-\frac{\partial \Lambda}{\partial t}, \quad \mathbf{A}^{\prime} \equiv \mathbf{A}+\nabla \Lambda$$
(where $\Lambda$ is an arbitrary real function of position and time) yield the same fields as $\varphi$ and $\mathrm{A}$. (Check that for yourself, using Equation $4.189$.) Equation $4.196$ is called a gauge transformation, and the theory is said to be gauge invariant.

In quantum mechanics the potentials play a more direct role (it is they, not the fields, that appear in the Equation 4.191 ), and it is of interest to ask whether the theory remains gauge invariant. It is easy to show (Problem 4.44) that
$$\Psi^{\prime} \equiv e^{i q \Lambda / \hbar} \Psi$$
satisfies Equation 4.191 with the gauge-transformed potentials $\varphi^{\prime}$ and $\mathbf{A}^{\prime}$ (Equation $4.196$ ). Since $\Psi^{\prime}$ differs from $\Psi$ only by a phase factor, it represents the same physical state, $\frac{62}{}$ and in this sense the theory is gauge invariant. For a long time it was taken for granted that there could be no electromagnetic influences in regions where $\mathrm{E}$ and $\mathrm{B}$ are zero-any more than there can be in the classical theory. But in 1959 Aharonov and Bohm 63 showed that the vector potential can affect the quantum behavior of a charged particle, even when the particle is confined to a region where the field itself is zero.

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Minimal Coupling

$$\mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}) .$$

$$i \hbar \frac{\partial \Psi}{\partial t}=\hat{H} \Psi$$

$$H=\frac{1}{2 m}(\mathbf{p}-q \mathbf{A})^2+q \varphi,$$

$$\mathbf{E}=-\nabla \varphi-\partial \mathbf{A} / \partial t, \quad \mathbf{B}=\nabla \times \mathbf{A} .$$

$$\hat{H}=\frac{1}{2 m}(-i \hbar \nabla-q \mathbf{A})^2+q \varphi,$$
Schrödinger方程变成了
$$i \hbar \frac{\partial \Psi}{\partial t}=\left[\frac{1}{2 m}(-i \hbar \nabla-q \mathbf{A})^2+q \varphi\right] \Psi$$

## 物理代写|量子力学代写quantum mechanics代考|The Aharonov–Bohm Effect

$$\varphi^{\prime} \equiv \varphi-\frac{\partial \Lambda}{\partial t}, \quad \mathbf{A}^{\prime} \equiv \mathbf{A}+\nabla \Lambda$$
(其中$\Lambda$是位置和时间的任意实函数)产生与$\varphi$和$\mathrm{A}$相同的字段。(使用公式$4.189$自己检查一下。)方程$4.196$称为规范变换，该理论称为规范不变量。

$$\Psi^{\prime} \equiv e^{i q \Lambda / \hbar} \Psi$$

## 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|PHYS518

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|Spin

In classical mechanics, a rigid object admits two kinds of angular momentum: orbital $(\mathbf{L}=\mathbf{r} \times \mathbf{p})$, associated with motion of the center of mass, and spin $(\mathbf{S}=I \omega)$, associated with motion about the center of mass. For example, the earth has orbital angular momentum attributable to its annual revolution around the sun, and spin angular momentum coming from its daily rotation about the north-south axis. In the classical context this distinction is largely a matter of convenience, for when you come right down to it, $\mathrm{S}$ is nothing but the sum total of the “orbital” angular momenta of all the rocks and dirt clods that go to make up the earth, as they circle around the axis. But a similar thing happens in quantum mechanics, and here the distinction is absolutely fundamental. In addition to orbital angular momentum, associated (in the case of hydrogen) with the motion of the electron around the nucleus (and described by the spherical harmonics), the electron also carries another form of angular momentum, which has nothing to do with motion in space (and which is not, therefore, described by any function of the position variables $r, \theta, \phi)$ but which is somewhat analogous to classical spin (and for which, therefore, we use the same word). It doesn’t pay to press this analogy too far: The electron (as far as we know) is a structureless point, and its spin angular momentum cannot be decomposed into orbital angular momenta of constituent parts (see Problem 4.28). 35 Suffice it to say that elementary particles carry intrinsic angular momentum $(\mathbf{S})$ in addition to their “extrinsic” angular momentum (L).

The algebraic theory of spin is a carbon copy of the theory of orbital angular momentum, beginning with the fundamental commutation relations: $\frac{36}{6}$
$$\left[S_x, S_y\right]=i \hbar S_z, \quad\left[S_y, S_z\right]=i \hbar S_x, \quad\left[S_z, S_x\right]=i \hbar S_y .$$
It follows (as before) that the eigenvectors of $S^2$ and $S_z$ satisfy ${ }^{37}$
$$s^2|s m\rangle=\hbar^2 s(s+1)|s m\rangle ; \quad S_z|s m\rangle=\hbar m|s m\rangle$$
and
$$S_{ \pm}|s m\rangle=\hbar \sqrt{s(s+1)-m(m \pm 1)}|s(m \pm 1)\rangle,$$
where $S_{ \pm} \equiv S_x \pm i S_y$. But this time the eigenvectors are not spherical harmonics (they’re not functions of $\theta$ and $\phi$ at all), and there is no reason to exclude the half-integer values of $s$ and $m$ :It so happens that every elementary particle has a specific and immutable value of $s$, which we call the spin of that particular species: $\pi$ mesons have spin 0 ; electrons have spin $1 / 2$; photons have spin 1 ; $\Delta$ baryons have spin 3/2; gravitons have spin 2; and so on. By contrast, the orbital angular momentum quantum number $l$ (for an electron in a hydrogen atom, say) can take on any (integer) value you please, and will change from one to another when the system is perturbed. But $s$ is fixed, for any given particle, and this makes the theory of spin comparatively simple. $\frac{38}{}$

## 物理代写|量子力学代写quantum mechanics代考|Spin 1/2

By far the most important case is $s=1 / 2$, for this is the spin of the particles that make up ordinary matter (protons, neutrons, and electrons), as well as all quarks and all leptons. Moreover, once you understand spin $1 / 2$, it is a simple matter to work out the formalism for any higher spin. There are just two eigenstates: $\left|\frac{1}{2} \frac{1}{2}\right\rangle$, which we call spin up (informally, $\uparrow$ ), and $\left|\frac{1}{2}\left(-\frac{1}{2}\right)\right\rangle$, spin down $(\downarrow)$. Using these as basis vectors, the general state $^{40}$ of a spin-1/2 particle can be represented by a two-element column matrix (or spinor):
$$\chi=\left(\begin{array}{l} a \ b \end{array}\right)=a \chi_{+}+b \chi_{-},$$
with
$$\chi_{+}=\left(\begin{array}{l} 1 \ 0 \end{array}\right)$$
representing spin up, and
$$\chi_{-}=\left(\begin{array}{l} 0 \ 1 \end{array}\right)$$
for spin down.
With respect to this basis the spin operators become $2 \times 2$ matrices, $\frac{41}{}$ which we can work out by noting their effect on $\chi_{+}$and $\chi_{-}$. Equation 4.135 says
$$\mathrm{S}^2 \chi_{+}=\frac{3}{4} \hbar^2 \chi_{+} \quad \text { and } \quad \mathrm{S}^2 \chi_{-}=\frac{3}{4} \hbar^2 \chi_{-}$$

If we write $\mathrm{S}^2$ as a matrix with (as yet) undetermined elements,
$$\mathrm{S}^2=\left(\begin{array}{ll} c & d \ e & f \end{array}\right)$$
then the first equation says
$$\left(\begin{array}{ll} c & d \ e & f \end{array}\right)\left(\begin{array}{l} 1 \ 0 \end{array}\right)=\frac{3}{4} \hbar^2\left(\begin{array}{l} 1 \ 0 \end{array}\right), \quad \text { or } \quad\left(\begin{array}{l} c \ e \end{array}\right)=\left(\begin{array}{c} \frac{3}{4} \hbar^2 \ 0 \end{array}\right),$$
so $c=(3 / 4) \hbar^2$ and $e=0$. The second equation says
$$\left(\begin{array}{ll} c & d \ e & f \end{array}\right)\left(\begin{array}{l} 0 \ 1 \end{array}\right)=\frac{3}{4} \hbar^2\left(\begin{array}{l} 0 \ 1 \end{array}\right), \quad \text { or } \quad\left(\begin{array}{l} d \ f \end{array}\right)=\left(\begin{array}{c} 0 \ \frac{3}{4} \hbar^2 \end{array}\right),$$
so $d=0$ and $f=(3 / 4) \hbar^2$. Conclusion:
$$\mathrm{S}^2=\frac{3}{4} \hbar^2\left(\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right) .$$

# 量子力学代考

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

$$\left[S_x, S_y\right]=i \hbar S_z, \quad\left[S_y, S_z\right]=i \hbar S_x, \quad\left[S_z, S_x\right]=i \hbar S_y .$$

$$s^2|s m\rangle=\hbar^2 s(s+1)|s m\rangle ; \quad S_z|s m\rangle=\hbar m|s m\rangle$$

$$S_{ \pm}|s m\rangle=\hbar \sqrt{s(s+1)-m(m \pm 1)}|s(m \pm 1)\rangle,$$

## 物理代写|量子力学代写quantum mechanics代考|Spin 1/2

$$\chi=\left(\begin{array}{l} a \ b \end{array}\right)=a \chi_{+}+b \chi_{-},$$

$$\chi_{+}=\left(\begin{array}{l} 1 \ 0 \end{array}\right)$$

$$\chi_{-}=\left(\begin{array}{l} 0 \ 1 \end{array}\right)$$

$$\mathrm{S}^2 \chi_{+}=\frac{3}{4} \hbar^2 \chi_{+} \quad \text { and } \quad \mathrm{S}^2 \chi_{-}=\frac{3}{4} \hbar^2 \chi_{-}$$

## 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|PHYS662

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 Radial Equation

Notice that the angular part of the wave function, $Y(\theta, \phi)$, is the same for all spherically symmetric potentials; the actual shape of the potential, $V(r)$, affects only the radial part of the wave function, $R(r)$, which is determined by Equation 4.16:
$$\frac{d}{d r}\left(r^2 \frac{d R}{d r}\right)-\frac{2 m r^2}{\hbar^2}[V(r)-E] R=\ell(\ell+1) R .$$
This simplifies if we change variables: Let
$$u(r) \equiv r R(r),$$
so that $R=u / r, d R / d r=[r(d u / d r)-u] / r^2,(d / d r)\left[r^2(d R / d r)\right]=r d^2 u / d r^2$, and hence
$$-\frac{\hbar^2}{2 m} \frac{d^2 u}{d r^2}+\left[V+\frac{\hbar^2}{2 m} \frac{\ell(\ell+1)}{r^2}\right] u=E u .$$
This is called the radial equation; ${ }^9$ it is identical in form to the one-dimensional Schrödinger equation (Equation 2.5), except that the effective potential,
$$V_{\mathrm{eff}}=V+\frac{\hbar^2}{2 m} \frac{\ell(\ell+1)}{r^2},$$
contains an extra piece, the so-called centrifugal term, $\left(\hbar^2 / 2 m\right)\left[\ell(\ell+1) / r^2\right]$. It tends to throw the particle outward (away from the origin), just like the centrifugal (pseudo-)force in classical mechanics. Meanwhile, the normalization condition (Equation 4.31) becomes
$$\int_0^{\infty}|u|^2 d r=1$$

## 物理代写|量子力学代写quantum mechanics代考|The Hydrogen Atom

The hydrogen atom consists of a heavy, essentially motionless proton (we may as well put it at the origin), of charge $e$, together with a much lighter electron (mass $m_e$, charge $-e$ ) that orbits around it, bound by the mutual attraction of opposite charges (see Figure 4.4). From Coulomb’s law, the potential energy of the electron $\frac{13}{}$ (in SI units) is
$$V(r)=-\frac{e^2}{4 \pi \epsilon_0} \frac{1}{r},$$
and the radial equation (Equation $4.37$ ) says
$$-\frac{\hbar^2}{2 m_e} \frac{d^2 u}{d r^2}+\left[-\frac{e^2}{4 \pi \epsilon_0} \frac{1}{r}+\frac{\hbar^2}{2 m_e} \frac{\ell(\ell+1)}{r^2}\right] u=E u .$$
(The effective potential-the term in square brackets—is shown in Figure 4.5.) Our problem is to solve this equation for $u(r)$, and determine the allowed energies. The hydrogen atom is such an important case that $\mathrm{I} m$ not going to hand you the solutions this time-we’ll work them out in detail, by the method we used in the analytical solution to the harmonic oscillator. (If any step in this process is unclear, you may want to refer back to Section 2.3 .2 for a more complete explanation.) Incidentally, the Coulomb potential (Equation 4.52) admits continuum states (with $E>0$ ), describing electron-proton scattering, as well as discrete bound states, representing the hydrogen atom, but we shall confine our attention to the latter. ${ }^{14}$

# 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|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。