Operator theory and matrix analysis methods in quantum information theory

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Quantum information theory (QIT) relates to the physics of interacting atoms and subatomic particles (quantum mechanics), while broadly concerning the notion of information in various ways (measuring information, transmitting it, keeping it secure, correcting errors, etc). It is important to understand the underlying mathematical nature of concepts in QIT so as to be able to take advantage of their properties and go beyond what is possible in (classical) information theory. Indeed, the advancement of QIT promises far-reaching implications on human activities both in terms of high level research as well as daily life activities. Research and development in the area is offering important advantages in the business sector. My research focuses on quantum state transfer, quantum resource theory (in particular, quantum coherence and entanglement), and positive operator-valued measures (in particular, quantum probability measures). Quantum state transfer concerns the ability to accurately transmit a quantum state from one location to another within a quantum computer, a process necessary for quantum computers to function to their full capacity. Coherence and entanglement are the two major resources that set QIT apart its classical counterpart. Quantum probability measures arise naturally in quantum mechanics, yet there are many open theoretical questions that must be answered in order to fully explain the phenomena observed in physical experiments. My work often involves majorization and generalizations thereof, which are used to compare objects and capture a sense of optimality. The major goal of my proposed research program is to fill in the gaps of mathematical knowledge with respect to these important concepts in order to make significant advances in both pure mathematics and QIT.  My research explores the mathematical foundations of quantum mechanics from the point of view of matrix analysis and operator theory; techniques from optimization theory, convex analysis, perturbation theory, linear and multilinear algebra, matrix theory, and approximation theory also play a role.