No Arabic abstract
This article serves as a brief introduction to the Shannon information theory. Concepts of information, Shannon entropy and channel capacity are mainly covered. All these concepts are developed in a totally combinatorial flavor. Some issues usually not addressed in the literature are discussed here as well. In particular, we show that it seems we can define channel capacity differently which allows us to potentially transmit more messages in a fixed sufficient long time duration. However, for a channel carrying a finite number of letters, the channel capacity unfortunately remains the same as the Shannon limit.
A concise introduction to quantum entanglement in multipartite systems is presented. We review entanglement of pure quantum states of three--partite systems analyzing the classes of GHZ and W states and discussing the monogamy relations. Special attention is paid to equivalence with respect to local unitaries and stochastic local operations, invariants along these orbits, momentum map and spectra of partial traces. We discuss absolutely maximally entangled states and their relation to quantum error correction codes. An important case of a large number of parties is also analysed and entanglement in spin systems is briefly reviewed.
We introduce and motivate generative modeling as a central task for machine learning and provide a critical view of the algorithms which have been proposed for solving this task. We overview how generative modeling can be defined mathematically as trying to make an estimating distribution the same as an unknown ground truth distribution. This can then be quantified in terms of the value of a statistical divergence between the two distributions. We outline the maximum likelihood approach and how it can be interpreted as minimizing KL-divergence. We explore a number of approaches in the maximum likelihood family, while discussing their limitations. Finally, we explore the alternative adversarial approach which involves studying the differences between an estimating distribution and a real data distribution. We discuss how this approach can give rise to new divergences and methods that are necessary to make adversarial learning successful. We also discuss new evaluation metrics which are required by the adversarial approach.
These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the CMSA Math Science Literature Lecture Series in May 2020. We attempt to give a birds-eye view of basic aspects of the theory of quantum groups.
In the past several years, observational entropy has been developed as both a (time-dependent) quantum generalization of Boltzmann entropy, and as a rather general framework to encompass classical and quantum equilibrium and non-equilibrium coarse-grained entropy. In this paper we review the construction, interpretation, most important properties, and some applications of this framework. The treatment is self-contained and relatively pedagogical, aimed at a broad class of researchers.
We summarize recent contributions in the broad area of age of information (AoI). In particular, we describe the current state of the art in the design and optimization of low-latency cyberphysical systems and applications in which sources send time-stamped status updates to interested recipients. These applications desire status updates at the recipients to be as timely as possible; however, this is typically constrained by limited system resources. We describe AoI timeliness metrics and present general methods of AoI evaluation analysis that are applicable to a wide variety of sources and systems. Starting from elementary single-server queues, we apply these AoI methods to a range of increasingly complex systems, including energy harvesting sensors transmitting over noisy channels, parallel server systems, queueing networks, and various single-hop and multi-hop wireless networks. We also explore how update age is related to MMSE methods of sampling, estimation and control of stochastic processes. The paper concludes with a review of efforts to employ age optimization in cyberphysical applications.