数学代写|信息论代写information theory代考|COMP2610

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我们提供的信息论information theory及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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
数学代写|信息论代写information theory代考|Information Theory Re-founded and Re-envisioned

数学代写|信息论代写information theory代考|Quantum Information Theory Re-envisioned

This “classical” (i.e., non-quantum) logical information theory is developed with the data of two or more partitions (or random variables) on a set with point probabilities. The transition to the quantum case was guided by the method of linearizing set concepts to obtain the corresponding vector space concepts, which are then specialized to finite-dimensional Hilbert spaces. The underlying set is replaced by a basis set of simultaneous orthonormal eigenvectors of two or more commuting self-adjoint linear operators (observables), the partitions are replaced by the direct-sum decompositions of the eigenspaces of the operators, Cartesian products are replaced by tensor products, and the point probabilities are provided by the state to be measured. The first fundamental theorem for quantum logical entropy and measurement established a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that

are distinguished by the measurement (decohered in the post-measurement mixed state). This theorem establishes quantum logical entropy as a natural notion for a quantum information theory focusing on distinguishing states.

The classical theory may also be developed just using two probability distributions, $p=\left(p_{1}, \ldots, p_{n}\right)$ and $q=\left(q_{1}, \ldots, q_{n}\right)$, indexed by the same set, and this is generalized to the quantum case where we are just given two density matrices $\rho$ and $\tau$ representing two states in a Hilbert space. Then the Hamming distance can be carried over from the classical to the quantum case where it is equal to the Hilbert-Schmidt norm (square of the trace distance and quantum analogue of square of Euclidean distance). A second theorem relating measurement and logical entropy is that the Hilbert-Schmidt norm (= quantum logical Hamming distance) between any pre-measurement state $\rho$ and the state $\hat{\rho}$ resulting from a projective measurement of the state is just the difference in their logical entropies, $h(\hat{\rho})-h(\rho)$. There are other benefits of quantum logical entropy (as opposed to von Neumann entropy) emphasized by Manfredi and Feix [5] as well as Tamir and Cohen $[12,13]$ but our points focused on the natural connections with projective quantum measurement (described by the Lüders mixture operation) and notions of distance between quantum states, i.e., Hilbert-Schmidt norm $=$ Square of trace distance $=$ Quantum logical Hamming distance.

数学代写|信息论代写information theory代考|What Is to Be Done

This re-envisioning of information theory is probably controversial due in part to the role of logical entropy to directly measure information-as-distinctions. We have presented only bits and pieces of the intriguing connections between logical entropy and Shannon entropy, e.g., the dit-bit transform. There is work to be done to uncover the underlying mathematical theory that will fully account for those connections.
The results on Boltzmannian entropy in statistical mechanics and Shannon entropy in information theory were largely cautionary. Conceptual conclusions have been drawn from the role of Shannon entropy as a numerical approximation to Boltzmannian entropy, and that has engendered many confusions and overstatements in the literature.

We gave a few basic theorems relating quantum logical entropy to projective quantum measurement and to the Hilbert-Schmidt norm. Quantum information theory has seen enormous growth in the last half-century, so much work might be done to see how logical entropy relates to those results (e.g., [12] and [13]) and what new results can be obtained following the linearization methodology to relate classical results to quantum results. And there has been extensive and exhausting work following the seminal 1936 paper of Birkhoff and von Neumann [2] that first linearized the Boolean logic of subsets to the quantum logic of subspaces. But we have noted that the same linearization methodology linearizes the logic of partitions on sets (see the Appendix) to the logic of direct-sum decompositions of vector spaces which provides the dual form of quantum logic when specialized to Hilbert spaces-but that logic has barely been investigated [4].

数学代写|信息论代写information theory代考|Subset Logic and Partition Logic

Since logical entropy arises as the quantitative version of the logic of partitions just as logical or Laplacian probability arises as the quantitative version of the logic of subsets, this Appendix will give a few basics the logic of partitions $[5,6]$.

In classical propositional logic, the atomic variables and compound formulas are usually interpreted as representing propositions (see any textbook in logic). But in terms of mathematical entities, the variables and formulas may be taken as representing subsets of some fixed universe set $U(|U| \geq 1)$ with the propositional interpretation being identified with the special case of subsets $0=\emptyset$ and 1 of a one element set 1. Alonzo Church noted that George Boole and Augustus De Morgan originally interpreted logic as a logic of subsets or classes.
The algebra of logic has its beginning in 1847 , in the publications of Boole and De Morgan. This concerned itself at first with an algebra or calculus of classes, to which a similar algebra of relations was later added.[3, pp. 155-156]
Today, largely due to the efforts of F. William Lawvere, subsets were generalized to subobjects or “parts” (equivalence classes of monomorphisms) so that logic has become the logic of subobjects or parts in a topos (such as the category of sets). In the basic case of the category of sets, it is again the logic of subsets.

In view of the general subset interpretation, “Boolean logic” or “subset logic” would be a better name for what is usually called “propositional logic.” Using this subset interpretation of the connectives such as join, meet, and implication, then a tautology, herein Subset tautology, is any formula such that regardless of what subsets of $U$ are assigned to the atomic variables, the whole formula will evaluate to the universe set $U$. Remarkably, to define subset tautologies, it is sufficient (a fact known to Boole) to restrict attention to the two subsets of a singleton $U=1$ (or, equivalently, only the subsets $\emptyset$ and $U$ of a general $U$ ) which is done, in effect, in the usual propositional interpretation where tautologies are defined as truth-table tautologies. The truth-table notion of a tautology should be a theorem, not a definition; indeed it is a theorem that extends to valid probability formulas [12].

数学代写|信息论代写information theory代考|Information Theory Re-founded and Re-envisioned


数学代写|信息论代写information theory代考|Quantum Information Theory Re-envisioned



经典理论也可以仅使用两个概率分布来发展,p=(p1,…,pn)和q=(q1,…,qn), 由同一个集合索引,这被推广到量子情况,我们只得到两个密度矩阵ρ和τ表示希尔伯特空间中的两个状态。然后,汉明距离可以从经典情况转移到等于希尔伯特-施密特范数(轨迹距离的平方和欧几里得距离平方的量子模拟)的量子情况。与测量和逻辑熵相关的第二个定理是任何预测量状态之间的希尔伯特-施密特范数(=量子逻辑汉明距离)ρ和国家ρ^由状态的投影测量产生的只是它们逻辑熵的差异,H(ρ^)−H(ρ). Manfredi 和 Feix [5] 以及 Tamir 和 Cohen 强调了量子逻辑熵(与 von Neumann 熵相反)的其他好处[12,13]但我们的观点集中在与射影量子测量的自然联系(由 Lüders 混合运算描述)和量子态之间的距离概念,即 Hilbert-Schmidt 范数=跟踪距离的平方=量子逻辑汉明距离。

数学代写|信息论代写information theory代考|What Is to Be Done

这种对信息论的重新构想可能是有争议的,部分原因是逻辑熵在直接测量信息作为区分方面的作用。我们只介绍了逻辑熵和香农熵之间有趣的联系的零碎部分,例如,dit-bit 变换。需要做一些工作来揭示能够充分解释这些联系的基本数学理论。

我们给出了一些将量子逻辑熵与射影量子测量和希尔伯特-施密特范数相关的基本定理。量子信息论在过去的半个世纪中取得了巨大的发展,可能需要做很多工作来了解逻辑熵与这些结果的关系(例如,[12] 和 [13]),以及在线性化之后可以获得哪些新结果将经典结果与量子结果联系起来的方法。在 Birkhoff 和 von Neumann [2] 的开创性论文 1936 年首次将子集的布尔逻辑线性化为子空间的量子逻辑之后,进行了广泛而艰巨的工作。

数学代写|信息论代写information theory代考|Subset Logic and Partition Logic


在经典命题逻辑中,原子变量和复合公式通常被解释为代表命题(参见任何逻辑教科书)。但就数学实体而言,变量和公式可以看作是某个固定宇宙集的子集在(|在|≥1)命题解释与子集的特殊情况相同0=∅和一个元素集 1 中的 1 个。Alonzo Church 指出,George Boole 和 Augustus De Morgan 最初将逻辑解释为子集或类的逻辑。
逻辑代数始于 1847 年,在 Boole 和 De Morgan 的出版物中。这首先与类的代数或微积分有关,后来添加了类似的关系代数。 [3, pp. 155-156]
今天,很大程度上由于 F. William Lawvere 的努力,子集被推广到子对象或“部分”(单态的等价类),因此逻辑已成为拓扑中的子对象或部分的逻辑(例如集合的范畴)。在集合范畴的基本情况下,它又是子集的逻辑。

鉴于一般的子集解释,“布尔逻辑”或“子集逻辑”将是通常称为“命题逻辑”的更好名称。使用连接词的这个子集解释,例如连接、相遇和蕴涵,然后重言式,这里的子集重言式,是任何公式,无论是什么子集在分配给原子变量,整个公式将评估为宇宙集在. 值得注意的是,要定义子集重言式,将注意力限制在单例的两个子集上就足够了(布尔已知的事实)在=1(或者,等效地,只有子集∅和在一般的在) 这实际上是在通常的命题解释中完成的,其中重言式被定义为真值表重言式。重言式的真值表概念应该是一个定理,而不是一个定义;事实上,它是一个扩展到有效概率公式的定理 [12]。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





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多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。



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