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. Probabilistic and statistical methods are indispensable to dealing with information theory, and Random matrix theory (RMT) appears as a natural candidate to analyze and model sophisticated and complex telecommunication protocols. This project aims to apply RMT to some problems in classical and quantum information theories by studying various structured random matrix models, their eigenvalue and eigenvector statistics, and eventually the desired observables. The key objectives in this project are,
- Investigation of various structured random matrix models and their eigenvalue statistics relevant to classical and quantum information theories, e.g. fixed trace product matrix ensembles, circulant random matrices and their variants, non-Hermitian matrices and their variants, etc.
- To derive analytical results for information entropies associated with the above matrix models, such as Shannon entropy in classical information theory, von Neumann in quantum information theory, etc.
- Exploration of higher order statistics of distance measures between quantum states and aim to derive analytical formulae, e.g., variance of Hilbert-Schmidt and Bures distances.
- To study the derived analytical results in context of quantum chaos by simulating model systems like coupled kicked tops, random spin chains, etc.
- To examine the universality aspects of eigenvalues and eigenvectors of the above matrix models using metric such as nearest-neighbor-spacing distribution, spacing-ratio distribution, complex spacing ratio distribution, generalized information entropies, multifractal dimensions, etc.
Dr. Santosh Kumar