Concentration of measure on quantum states

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Quantum information processing is the science that explores the way information behaves in physical systems that can be controlled precisely enough for quantum mechanical effects to emerge. When these effects begin to appear, a large part of our basic intuition about how information behaves no longer holds; for instance, one cannot determine the state of an unknown quantum system without disturbing it. This has consequences on areas such as cryptography, which deals with information processing in the presence of malicious adversaries, and information theory, which studies the fundamental limits of tasks such as data compression and error correction. One of the main difficulties that arises in both of these fields is that many mathematical tools cannot be transposed directly from the classical world to the quantum world, such as the collection of results based on the concentration of measure phenomenon. This is a generalized version of the law of large numbers, which tells us that the average of the results of several experiments is almost always close to its expected value. These techniques have a very broad range of applications in computer science, physics, and far beyond, and have become a subfield of mathematics in their own right. However, they apply only when the experiment results are classical values; when they are themselves quantum systems, they often no longer work, and new methods are needed. The goal of this research program is to systematically develop a theory of concentration of measure for quantum states, together with its applications in quantum cryptography and quantum information theory. There is a vast array of open problems ranging from fundamental questions to more applied topics that I plan to investigate. Among the fundamental questions, even certain basic sampling problems involving quantum states remain open. On the cryptography side, I plan to develop new tools for "device-independent" protocols, in which devices can be certified from their input-output behaviour alone and can therefore be supplied by an untrusted manufacturer, and for continuous-variable protocols, which are far easier to implement experimentally but much more difficult to analyze mathematically. In information theory, I plan to explore the fundamental limits of quantum error correction and to gain a better understanding of the kinds of noise models that can arise in a quantum computer. This research program addresses both core questions about the nature of information in the physical world as well as current technological challenges, and has the potential for both near-future and long-term impact.