### 物理代写|量子计算代写Quantum computer代考|CS583

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|量子计算代写Quantum computer代考|Pauli Matrices and Dirac Matrices

For the imaginary unit $\sqrt{-1}$, we will use the designation $i \equiv \sqrt{-1}$. The matrices $\sigma_{1}, \sigma_{2}$, and $\sigma_{3}$ :
$$\sigma_{1}=\left[\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}\right], \quad \sigma_{2}=\left[\begin{array}{cc} 0 & -\mathrm{i} \ \mathrm{i} & 0 \end{array}\right], \quad \sigma_{3}=\left[\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right],$$
are called the Pauli matrices. They are widely used in quantum theory for describing half-integer spin particles, for example, an electron. (Spin is a quantum property of an elementary particle, its intrinsic angular momentum [1]. So, electrons, protons, and neutrinos have half-integer spin; the spin of photons and gravitons is an integer.).
The following properties are valid for the Pauli matrices.
(1) The Pauli matrices are Hermitian ${ }^{1}$ and unitary (see Appendix B on page 103):
$$\forall k \in{1,2,3} \quad \sigma_{k}=\sigma_{k}^{\dagger}=\sigma_{k}^{-1} .$$
(2) $\forall k \in{1,2,3}$, the square of the Pauli matrix is equal to the identity matrix:
$$\sigma_{i}^{2}=\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right]$$
(3) $\forall i, j \in{1,2,3}$, the equalities
$$\sigma_{i} \sigma_{j}+\sigma_{j} \sigma_{i}=2 \delta_{i j}\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right]$$
are valid. Here, is used a notation for the Kronecker symbol (see definition on page 3 ).
Note that the value ${A, B}=A B+B A$ is usually called anticommutator of matrices $A$ and $B$, and the value $[A, B]=A B-B A$ is called commutator. Next,for the identity matrix, we will apply the notation $I$, and for the zero one $O$. In particular, the formula (2.4) can be written as follows:
$$\left{\sigma_{i}, \sigma_{j}\right}=2 \delta_{i j} I .$$
The matrices $A$ and $B$ are called commutative, if $A B=B A$. Commutative matrices are always square and have the same order.

By definition, the condition $[A, B]=O$ is met for the commutative matrices [2]. It is clear that the Pauli matrices do not commutate with each other (see Exercise 1).

## 物理代写|量子计算代写Quantum computer代考|Let us prove the Jacobi2 identity

$$[[P, Q], R]+[[Q, R], P]+[[R, P], Q] \equiv O$$
that is valid for commutators of any matrices of size $n \times n$.
Proof.
We use the definition of the commutator $[P, Q]=P Q-Q P$, then
\begin{aligned} {[[P, Q], R] } &=[P Q-Q P, R]=(P Q-Q P) R-R(P Q-Q P) \ &=P Q R-Q P R-R P Q+R Q P \end{aligned}
Next, let us present in a similar manner the remaining summands in the sum:
\begin{aligned} &{[[Q, R], P]=Q R P-R Q P-P Q R+P R Q} \ &{[[R, P], Q]=R P Q-P R Q-Q R P+Q P R .} \end{aligned}
The sum of right-hand values (2.7), (2.8), and (2.9), as can be easily seen after the conversion of such summands, is zero. In this way, the Jacobi identity is proved.
Sometimes, in linear algebra and its applications, one has to use matrices split into rectangular parts or blocks [3,4]. Consider the rectangular matrix $A=\left(a_{i j}\right)$, where $1 \leqslant i \leqslant m, 1 \leqslant j \leqslant n$. Let $m=m_{1}+m_{2}$ and $n=n_{1}+n_{2}$.

Let us draw horizontal and vertical lines and split the matrix $A$ into four rectangular blocks:Thus, the matrix $A$ is presented in the form of a block matrix, consisting of the blocks $B_{11}, B_{12}, B_{21}$, and $B_{22}$ of size $m_{1} \times n_{1}, m_{1} \times n_{2}, m_{2} \times n_{1}$, and $m_{2} \times n_{2}$, respectively.

## 物理代写|量子计算代写Quantum computer代考|Dirac matrices

As an example of a block matrix setting, we provide the definition of the Dirac matrices. Four Dirac matrices $\alpha_{1}, \alpha_{2}, \alpha_{3}$, and $\beta$ are part of the equation named after him for a half-integer spin relativistic particle, and are expressed in terms of the Pauli matrices $\sigma_{k}, k=1,2,3$, as follows [5]:
$$\alpha_{k}=\left[\begin{array}{cc} O & \sigma_{k} \ \sigma_{k} & O \end{array}\right], \quad \beta=\left[\begin{array}{cc} I & O \ O & -I \end{array}\right]$$
(Relativistic particles are the particles whose velocity is close to the velocity of light.)
Each of the Dirac matrices has a Hermitian property and a property of being unitary. Moreover, for all $l, m \in{1,2,3}$ the equalities
\begin{aligned} \alpha_{l} \alpha_{m}+\alpha_{m} \alpha_{l} &=2 \delta_{l m} I, \ \alpha_{l} \beta+\beta \alpha_{l} &=O \end{aligned}
are valid. Note that the size of matrices $I$ and $O$ in formulas (2.12) and (2.13) is equal to $4 \times 4$.

Using the concept of a block matrix, it is easy to write down the definition of the tensor product of square matrices $A \otimes B$, defined as $A=\left(a_{i j}\right), i, j=1,2, \ldots, n_{A}$, and $B=\left(b_{i j}\right), i, j=1,2, \ldots, n_{B}$ :
$$A \otimes B=\left[\begin{array}{cccc} a_{11} B & a_{12} B & \ldots & a_{1 n_{A}} B \ a_{21} B & a_{22} B & \ldots & a_{2 n_{A}} B \ \vdots & \vdots & \ddots & \vdots \ a_{n_{A} 1} B & a_{n_{A} 2} B & \ldots & a_{n_{A} n_{A}} B \end{array}\right]$$
where $a_{11} B$ is a block with size $n_{B} \times n_{B}$, consisting of elements of the form $a_{11} b_{i j}$, block $a_{12} B$ consists of elements of the form $a_{12} b_{i j}$, etc.

Note that for the tensor product operation, the sizes of matrices $A$ and $B$ do not have to be the same, i.e. $n_{A} \neq n_{B}$.

## 物理代写|量子计算代写Quantum computer代考|Pauli Matrices and Dirac Matrices

σ1=[01 10],σ2=[0−一世 一世0],σ3=[10 0−1],

(1) Pauli 矩阵是 Hermitian 矩阵1和单一的（参见第 103 页的附录 B）：

∀ķ∈1,2,3σķ=σķ†=σķ−1.
(2) ∀ķ∈1,2,3，泡利矩阵的平方等于单位矩阵：

σ一世2=[10 01]
(3) ∀一世,j∈1,2,3, 等式

σ一世σj+σjσ一世=2d一世j[10 01]

\left{\sigma_{i}, \sigma_{j}\right}=2 \delta_{i j} I 。\left{\sigma_{i}, \sigma_{j}\right}=2 \delta_{i j} I 。

## 物理代写|量子计算代写Quantum computer代考|Let us prove the Jacobi2 identity

$$[[P, Q], R]+[[Q, R], P]+[[R, P], Q] \equiv O$$

$$[[P, Q], R]=[P Q-Q P, R]=(P Q-Q P) R-R(P Q-Q P) \quad=P Q R-Q P R-R P Q+R Q P$$

$$[[Q, R], P]=Q R P-R Q P-P Q R+P R Q \quad[[R, P], Q]=R P Q-P R Q-Q R P+Q P R$$

## 物理代写|量子计算代写Quantum computer代考|Dirac matrices

(相对论粒子是速度接近光速的粒子。)

$$\alpha_{l} \alpha_{m}+\alpha_{m} \alpha_{l}=2 \delta_{l m} I, \alpha_{l} \beta+\beta \alpha_{l} \quad=O$$

$A \otimes B=\left[\begin{array}{lllllllllll}a_{11} B & a_{12} B & \ldots & a_{1 n_{A}} B & a_{21} B & a_{22} B & \ldots & a_{2 n_{A}} B & \vdots & \ddots & \vdots a_{n_{A} 1} B\end{array}\right.$

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