Quantum Information Meets Mathematics: the Blessing of High Dimension

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This project involves research on the interface of quantum information science and mathematics. Over the last few years it has become clear that quantum information theory enjoys very close links to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability) a.k.a. asymptotic geometric analysis, which belongs to the expertise of Dr. Szarek. In a nutshell, asymptotic geometric analysis studies quantitative properties of convex sets (or other geometric structures) and their "approximate" symmetries as the dimension goes to infinity. This makes it ideally suited to the study of quantum systems, where the setting is inherently high-dimensional. While classically analyzing high-dimensional phenomena often suffers from the curse of dimensionality (the complexity of the problem exploding with the increase in dimension, so that the question quickly ceases to be tractable), we may say that asymptotic geometric analysis exploits the blessing of dimensionality, with the symmetries mentioned above becoming apparent only when the dimension is large. Dr. Szarek's goals for this project are centered on acquiring a more thorough understanding of relevant aspects of computer science and of quantum physics, which will permit exploring in greater depth the links between asymptotic geometric analysis and the quantum theory. Subsequent objectives include identifying mathematical techniques that may be helpful in solving known problems and, more generally, in developing mathematical framework pertinent to processes associated with quantum information tasks. Specific problems and issues to be studied are related to distillability, the PPT property, probabilistic models for quantum objects/processes, derandomization and the role of duality.The quest to build a quantum computer is one of the major scientific and technological challenges of the 21st century, and quantum information theory provides the theoretical framework for that quest. A major event in the development of that field will be a semester-long program Mathematical Challenges in Quantum Information, to be held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England in the fall of 2013. The program will bring together theoretical physicists, computer scientists and mathematicians with interests in the subject and will supply an ideal environment for an interdisciplinary interaction between experts with different backgrounds. NSF funding will support Dr. Szarek's residency at the Newton Institute for the duration of the program. While Dr. Szarek is a theoretical mathematician, the research is expected to contribute to the understanding of the capabilities and limitations of quantum information systems and thus may have far reaching direct and indirect impact. Additionally, the project will ultimately result in involvement of graduate and undergraduate students in intensive research, thus contributing to the development of scientific base and infrastructure.