# 数学代写|信息论作业代写information theory代考|CO739

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## 数学代写|信息论作业代写information theory代考|Hilbert–Schmidt operator

Given an orthonormal basis $\left(\phi_n\right){n=1}^{+\infty}$ of the complex Hilbert space $\mathbb{H}$ and a bounded linear operator $\mathbf{T} \in \mathfrak{B}(\mathbb{H})$, we put the Hilbert-Schmidt norm $|\mathbf{T}|{\mathrm{HS}}$ as follows:
$$|\mathbf{T}|_{\mathrm{HS}}:=\left(\sum_{n=1}^{+\infty}\left|\mathbf{T} \phi_n\right|_{\mathbf{H}}^2\right)^{1 / 2} \leq+\infty .$$
It can be shown that $|\mathbf{T}|_{\mathrm{HS}}=\left|\mathbf{T}^*\right|_{\mathrm{HS}}$. An operator $\mathbf{T} \in \mathfrak{B}(\mathbb{H})$ is said to be a HilbertSchmidt operator if it has a finite Hilbert-Schmidt norm (i. e., $|\mathbf{T}|_{\mathrm{HS}}<+\infty$ ). The space of all Hilbert-Schmidt operators will be denoted by $\mathfrak{H} \mathfrak{S}(\mathbb{H})$. Note that $\mathfrak{H} \mathfrak{S}(\mathbb{H})$ is itself a Hilbert space under the Hilbert-Schmidt inner product $\langle\cdot, \cdot\rangle_{\mathrm{HS}}: \mathfrak{H} \mathfrak{S}(\mathbb{H}) \times \mathfrak{H} \mathfrak{S}(\mathbb{H}) \rightarrow \mathbb{C}$ defined by
$$\langle\mathbf{S}, \mathbf{T}\rangle_{\mathrm{HS}}=\sum_n\left\langle\mathbf{S} \phi_n, \mathbf{T} \phi_n\right\rangle_{\mathrm{H}}, \quad \mathbf{S}, \mathbf{T} \in \mathfrak{H} \subsetneq(\mathbb{H})$$
It appears that both the Hilbert-Schmidt norm $|\cdot|_{H S}$ and the trace defined above are expressed in terms of an orthonormal basis $\left(\phi_n\right)_{n=1}^{+\infty}$. However, a further analysis

(which we shall omit here) shows that they are both independent of the orthonormal basis chosen.

For a finite-dimensional Hilbert space $\mathbb{H}=\mathbb{C}^n$, the polar decomposition of an $n \times n$ real or complex matrix $\mathbf{A}$ is a factorization of the form $\mathbf{A}=\mathbf{U P}$, where $\mathbf{U}$ is a unitary matrix (a rotation matrix) and $\mathbf{P}$ is a positive-semidefinite Hermitian matrix (a scaling of the space along a set of $n$ orthogonal axes), both of size $n \times n$. The polar decomposition of a square matrix $\mathbf{A}$ always exists. In fact, if $\mathbf{A}$ is invertible, the decomposition is unique, and the factor $\mathbf{P}$ will be positive-definite. In that case, $\mathbf{A}$ can be written uniquely in the form $\mathbf{A}=\mathbf{U} e^{\mathbf{X}}$, where $\mathbf{U}$ is unitary and $\mathbf{X}$ is the unique self-adjoint logarithm of the matrix $\mathbf{P}$. The polar decomposition can also be defined as $\mathbf{A}=\mathbf{P U}$ where $\mathbf{P}$ and $\mathbf{U}$ have the same properties as above (but are different matrices, in general, for the same $\mathbf{A})$.

## 数学代写|信息论作业代写information theory代考|Formulation of quantum systems

In this chapter, we give a concise and yet rigorous mathematical formulation of a generic infinite-dimensional quantum system based on the following set of postulates originated from von Neumann [172] (see also Chang [22-24]). These postulates are commonly accepted by quantum probabilists and quantum physicists alike as the starting point for a systematic study of quantum systems.

Postulate 1. With every quantum system, there is associated a separable complex Hilbert space $\mathbb{H}$ on which a $C^*$ or von Neumann algebras of linear operators $\mathcal{A}$ is defined. This complex Hilbert space $\mathbb{H}$ is called in physics terminology the space of states. The Hilbert space of a composite quantum system can be represented as a tensor product of Hilbert spaces of the component systems involved.

Postulate 2. Given a $C^*$ or von Neumann algebra of operators $\mathcal{A}$ on $\mathbb{H}$ for the quantum system, the space of quantum states $\mathcal{S}(\mathcal{A})$ of the quantum system then consists of all positive trace-class operators $\rho \in \mathcal{A}$ with unit trace, $\operatorname{tr}[\rho]=1$. The pure states are projection operators onto one-dimensional subspaces of $\mathrm{H}$. A state $\rho$ will be called the density operator or density matrix if $\operatorname{tr}[\rho \mathbf{a}]=\operatorname{tr}[\mathbf{a}]$ for all $\mathbf{a} \in \mathcal{A}$.

Postulate 3. An observable of the quantum system is a positive operator-valued measure a defined on a certain Borel measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. Specifically, for each Borel set $B \in \mathcal{B}(\mathbb{R}), \mathbf{a}(B)$ is a self-adjoint linear (but not necessarily bounded) operator on the Hilbert space $\mathbb{H}$.

Postulate 4. A process of measurement in a quantum system is the correspondence between the observable-state pair $(\mathbf{a}, \rho)$ and the probability measure $\mu_{\mathrm{a}}$ on the real Borel measurable space. For every Borel subset $E \in \mathcal{B}(\mathbb{R})$, the quantity $0 \leq \mu_{\mathrm{a}}(E) \leq$ 1 is the probability that when a quantum system is in the state $\rho$, the result of the measurement of the observable a belongs to $E$. The expectation value (the mean value) of the observable $\mathbf{a}$ in the state $\rho$ is
$$\langle\mathbf{a} \mid \rho\rangle=\int_{-\infty}^{\infty} \lambda d \mu_{\mathbf{a}}(\lambda),$$
where $\mu_{\mathrm{a}}(\lambda)=\mu_{\mathrm{a}}((-\infty, \lambda))$ is the distribution function for the probability measure $\mu_{\mathrm{a}}$.

# 信息论代写

## 数学代写|信息论作业代写information theory代考|Hilbert–Schmidt operator

$$|\mathbf{T}|{\mathrm{HS}}:=\left(\sum{n=1}^{+\infty}\left|\mathbf{T} \phi_n\right|{\mathbf{H}}^2\right)^{1 / 2} \leq+\infty$$ 可以证明 $|\mathbf{T}|{\mathrm{HS}}=\left|\mathbf{T}^*\right|{\mathrm{HS}}$. 操作员 $\mathbf{T} \in \mathfrak{B}(\mathbb{H})$ 如果它 具有有限的 Hilbert-Schmidt 范数（即， $|\mathbf{T}|{\mathrm{HS}}<+\infty$ ). 所有 Hilbert-Schmidt 算子的空间将表示为 $\mathfrak{H} \mathfrak{S}(\mathbb{H})$. 注意 $\mathfrak{S} \mathfrak{S}(\mathbb{H})$ 本身就是 Hilbert-Schmidt 内积下的 Hilbert 空间 $\langle\cdot, \cdot\rangle_{\text {HS }}: \mathfrak{H S}(\mathbb{H}) \times \mathfrak{H S}(\mathbb{H}) \rightarrow \mathbb{C}$ 被定 以为
$$\langle\mathbf{S}, \mathbf{T}\rangle_{\mathrm{HS}}=\sum_n\left\langle\mathbf{S} \phi_n, \mathbf{T} \phi_n\right\rangle_{\mathrm{H}}, \quad \mathbf{S}, \mathbf{T} \in \mathfrak{H} \subsetneq(\mathbb{H})$$

(我们将在这里省略) 表明它们都独立于所选择的正交基。

## 数学代写|信息论作业代写information theory代考|Formulation of quantum systems

$$\langle\mathbf{a} \mid \rho\rangle=\int_{-\infty}^{\infty} \lambda d \mu_{\mathbf{a}}(\lambda)$$

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