Operator theory with applications to quantum information theory

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Quantum information theory is the study of quantum properties that can be used to store, transmit, and process information in an efficient, accurate, and secure way. My approach is to build up the mathematical foundations for physical realizations in quantum mechanics through operator theory and matrix algebra techniques with the end goal of advancing the mathematics behind quantum information theory. Quantum error correction is used to recover information from errors introduced by noise that occurs when sending quantum information through a quantum channel. Quantum cryptography, on the other hand, is used to hide information when sending quantum information through a quantum channel so that the original message cannot be recovered by a third party. Intriguingly, there is an algebraic bridge linking quantum error correction with quantum cryptography, and the two fields can be thought of as two sides of a coin. At the same time, my research has shown that this connection breaks down in a certain general setting; I am presently exploring this in further detail. Being able to protect information against possibly malicious eavesdroppers has clear real-world applications, and it is therefore important to develop a clear mathematical framework for how to do so. Given a particular quantum channel, it would be desirable to have a straightforward procedure for determining which states (if any) can be sent privately over the channel. My work aims to address this issue through the development of an overarching private quantum code theory. Entanglement is a key to quantum information theory; being able to manipulate entanglement makes quantum information a powerful tool. A major area of research in quantum information theory is the problem of entanglement transformations: that is, can one manipulate a pure state of a composite system via local operations and classical communication and have it transform into another particular state? Recently, this question has been answered using majorization theory, thus giving majorization an important role in quantum information theory. The entropy of entanglement of an infinite-dimensional pure state can be infinite, meaning that assigning a value of entanglement to an infinite-dimensional state is not a straightforward generalization of the finite-dimensional setting. I plan to focus on the infinite-dimensional setting, since quantum mechanics is inherently infinite-dimensional. Understanding entanglement transformations, and in particular the partial orders of majorization and trumping on quantum states, informs our understanding of entanglement, which will ultimately lead to the full use of entanglement as a resource. If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Preprocessing by a quantum channel leads to the partial order “cleaner than” on quantum probability measures and the notion of a quantum probability measure having the desirable (optimal) quality of being “clean”. Some work has been done to formalize the very general issue of optimality in quantum measurements; I plan to pursue this topic through a novel use of measurement spaces. I hope to provide new insights into the subject of optimality via structural aspects of the measurement spaces. Many of the structural questions I wish to answer are of independent interest in operator theory.