## 数学代写|信息论作业代写information theory代考|COMP30690

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

To formulate von Neumann’s postulate 3, we now give a formal definition of a quantum observable below (see Chang [23]).

Definition 2.6.3. The triple $(\mathrm{X}, \mathcal{B}(\mathrm{X})$, a) is said to be a quantum observable if (i) $\mathrm{X}$ is a locally compact Hausdorff space; (ii) $\mathcal{B}(\mathrm{X})$ is the Borel $\sigma$-algebra of subsets of $\mathbb{X}$ and (iii) $\mathbf{a}$ is a positive self-adjoint operator valued measure $\mathbf{a}:(\mathbb{X}, \mathcal{B}(\mathbb{X})) \rightarrow \mathfrak{B}_{+}(\mathbb{H})$ such that $\mathbf{a}(E)$ is a positive self-adjoint operator on the complex Hilbert space $\mathbb{H}$ for every $E \in \mathcal{B}(\mathrm{X})$ that satisfies:

$\mathbf{0} \leq \mathbf{a}(E) \leq \mathbf{a}(\mathrm{X})$

$\mathbf{a}(\mathbb{X})=\tau$, where $\tau: \mathbb{H} \rightarrow \mathbb{C}$ is a bounded linear functional on $\mathbb{H}$ such that $\tau(\phi)=$ $|\phi|_{\mathrm{H}}$

$\mathbf{a}\left(\bigcup_{n=1}^{+\infty} E_n\right)=\sum_{n=1}^{+\infty} \mathbf{a}\left(E_n\right)$ for any sequence $\left{E_n, n=1,2, \ldots\right}$ of pairwise disjoint sets in $\mathcal{B}(\mathrm{X})$, where the summation in right-hand side is the $\sigma$-weakly convergence.
In this case, the measurable space $(X, \mathcal{B}(X))$ is said to be the value space of a.
The collection of bounded quantum observables will be denoted by $\mathcal{O}_{\mathbf{H}}(\mathbb{X}, \mathcal{B}(\mathbb{X}))$. As described in Chang [22], a “quantum observable” is the quantum physicist’s word for real random variable that describes a physical quantity (such as position, velocity, momentum, angular momentum, energy, etc.) of a quantum system that plays a central role in quantum mechanics. They are mathematical representations of physical quantities that can (in principle) be measured. However, arbitrary nonreal elements (or nonself-adjoint operators) do not represent in general complex random variables. Nonreal (i. e., complex) quantum random variables correspond to normal elements $\mathbf{a} \in \mathcal{A}$, which commutes with their adjoint, i. e., $\mathbf{a}(E) \mathbf{a}^(E)=\mathbf{a}^(E) \mathbf{a}(E)$ for all $E \in \mathcal{B}(\mathbb{X})$. To avoid unnecessarily confusion, we often take $\mathbb{X}$ to be $\mathbb{R}$ for simplicity. In this case, all quantum observables are assumed to be real-valued.
The definition of quantum probability space is given below.

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

Formulation of von Neumann’s postulate 4 is as follows. Recall that a quantum state $\rho$ is a positive trace-class operator such that $\operatorname{tr}(\rho)=1$ and a quantum observable $\mathbf{a}$ is a self-adjoint operator-valued map defined on the real measurable space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. By the von Neumann spectral theorem (Theorem 1.7.4), there exists a projection-valued measure $\mu_{\mathrm{a}}$ on $\left(\mathbb{R}, \mathcal{B}(\mathbb{R})\right.$ ) such that $\mathbf{a}(E)=\int_E \lambda \mu_{\mathrm{a}}(d \lambda)$ for all $E \in \mathcal{B}(\mathbb{R})$. The probability $P(\rho, \mathbf{a}, E)$ that in the quantum state $\rho$ the quantum observable a should take values in $E \in \mathcal{B}(\mathbb{R})$ is given by $P(\rho, \mathbf{a}, E)=\operatorname{tr}\left[\rho \mu_{\mathbf{a}}(E)\right]$.

A measurement of the real quantum observable $(\mathbb{R}, \mathcal{B}(\mathbb{R})$, a) (or simply a) is a physical procedure or experiment that produces numerical results related to a. A process of measurement is the map $(\mathbf{a}, \rho) \mapsto \mu_{\mathrm{a}}$ from $\mathcal{A} \times \mathcal{S}(\mathcal{A})$ to $\mathcal{P}(\mathbb{R})$ (where $\mathcal{P}(\mathbb{R}$ ) is the space of probability measures on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, which to every observable $\mathbf{a} \in \mathcal{A}$ and state $\rho \in \mathcal{S}(\mathcal{A})$ assigns a probability measure $\mu$ on the Borel measurable space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. For every Borel subset $E \in \mathcal{B}(\mathbb{R})$, the quantity $0 \leq \mu_{\mathrm{a}}(E) \leq 1$ is the probability that for a quantum system in the state $\rho$ the result of a measurement of the observable $\mathbf{a}$ belongs to $E$. The expectation value (the mean-value) of the observable $\mathbf{a} \in \mathcal{A}$ is $\int_{\infty}^{\infty} \lambda d \mu_{\mathrm{a}}(\lambda)$, where $\mu_{\mathrm{a}}(\lambda)=\mu_{\mathrm{a}}(]-\infty, \lambda[)$ is a distribution function for the probability measure $\mu_{\mathrm{a}}$.
In any given measurement of the observable $\mathbf{a}$, the allowable results $a$ take values in $\sigma(\mathbf{a})$, the spectrum of $\mathbf{a}$. Given the state $\rho$, the value $a \in \sigma(\mathbf{a})$ is observed with probability $\operatorname{tr}\left(\rho \mathbf{P}{\psi(a)}\right)$, where $\mathbf{P}{\psi(a)}$ or simply $\mathbf{P}a$ is the one-dimensional vector space generated by the eigenvector $\psi(a)$ associated to the eigenvalue $a$ of $\mathbf{a}$. Consequently, the expectation of the observable $\mathbf{a}$ is given by $\mathbf{E}\rho(\mathbf{a})=\operatorname{tr}[\rho \mathbf{a}]$.

Suppose that a measurement of the observable a gives rise to the observation $a \in$ $\sigma(\mathbf{a})$. Then we must condition that state in order to predict the outcomes of subsequent measurements, by updating the state $\rho$ using
$$\rho \mapsto \rho^{\prime}[a]=\frac{\mathbf{P}_a \rho \mathbf{P}_a}{\operatorname{tr}\left(\rho \mathbf{P}_a\right)}$$
This is the so-called back-action of a quantum measurement.
Let $\mathcal{B}(\mathbb{R})$ be the $\sigma$-algebra of Borel subsets of $\mathbb{R}$ (see Wheeden and Zygmund [177] for a definition of Borel $\sigma$-algebra). Recall that $\mathfrak{L}_p(\mathbb{H})$ is the collection of projection operators on $\mathrm{H}$ (see Section 1.6 for the definition and properties of projection operators). In the following, we explore the concept of projection-valued measures.

# 信息论代写

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

$\mathbf{a}:(\mathbb{X}, \mathcal{B}(\mathbb{X})) \rightarrow \mathfrak{B}{+}(\mathbb{H})$ 这样 $\mathbf{a}(E)$ 是复 Hilbert 空 $$\mathbf{0} \leq \mathbf{a}(E) \leq \mathbf{a}(\mathrm{X})$$ $\mathbf{a}(\mathbb{X})=\tau$ ，在哪里 $\tau: \mathbb{H} \rightarrow \mathbb{C}$ 是上的有界线性泛函 $\mathbb{H}$ 这样 $\tau(\phi)=|\phi|{\mathrm{H}}$
$\mathbf{a}\left(\bigcup_{n=1}^{+\infty} E_n\right)=\sum_{n=1}^{+\infty} \mathbf{a}\left(E_n\right)$ 对于任何序列
Veft $\left{E_{_} n, n=1,2 ，\right.$ Vdots\right } } \text { 成对不相交的集合 } \mathcal { B } ( X ) \text { ， }

Chang [22] 所述，“量子可观察量”是量子物理学家对真 实随机变量的描述，它描述了一个量子系统的物理量
(如位置、速度、动量、角动量、能量等) 在量子力学 中的核心作用。它们是可以 (原则上) 测量的物理量的 数学表示。然而，任意非实数元素（或非自伴随算子)

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## MATLAB代写

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