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

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• 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|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].

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