Executive Summary : | In today’s world, it is unimaginable to survive without information exchange. The underlying mathematical framework, the information theory, serves as the basis of all communications, networking, and data storage systems. Classical information theory deals with various information processing tasks using classical laws, whereas quantum information theory aims to accomplish such tasks using quantum mechanical means. The former has led us to modern telecommunication technology such as 5G, whereas the latter has already shown us glimpses of exotic equipments such as quantum computers, teleporters, etc. Probabilistic and statistical methods are indispensable to dealing with information theory, and Random matrix theory (RMT) appears as a natural candidate to analyze sophisticated and complex telecommunication protocols. Originally used by Wigner to study spectra of complex nuclei, RMT has gradually found its application in an ever increasing variety of fields, such as condensed matter, high energy physics, finance, neuroscience, etc., along with the information theory. In classical information theory, one of the key implementations of RMT is in modeling the communication channel in multiple-input-multiple-output systems. Depending on spatial and temporal correlations, signal fading conditions, number of users, signal multiplexing schemes, etc., various random matrix models are employed which can then be used to calculate quantities which assess the efficiency of the communication system, such as Shannon entropy, Bit Error Probability, etc. In quantum information theory, among other things, RMT finds its application in modeling random quantum states. These provide the most typical or generic description and therefore serve as reference states. They also play a crucial role in the subject of quantum chaos. Based on the random states under consideration, a suitable random matrix model is used and quantities of interest, such as von Neumann entropy and purity, are evaluated. Yet another important area, in which RMT techniques are implemented, concerns with the statistics of various distance metrics between quantum states. These results are pertinent to topics such as quantum communication protocols, quantification of quantum correlations, quantum algorithms in machine learning, quantum-state tomography. With the continuous advancement of sophisticated technologies in the area of information theory, it becomes essential to examine more complex random matrix models which can adequately take care of the system dependent constraints. The proposed project aims to address these issues by studying various structured random matrix models, their eigenvalue and eigenvector statistics, and eventually the desired observables. As an additional benefit, the analytical and computational techniques developed in the course of this project will be of relevance also to the areas of mathematical physics, statistical physics, quantum physics and computational physics. |