Topic area: Mathematical Sciences, Modeling of Complex Systems. Project Title: Von Neumann Subfactors, Noncommutativity and Quantum Computing.
Abstract
The discovery of quantum mechanics, and quantum field theory at the beginning of the 20th century revolutionized our understanding of the world around us. Recent advances in information processing using quantum physical systems place us at the brink of another intellectual and technological revolution that will be a true game changer. The impact of quantum computing and quantum information technologies on science, privacy and national security will be profound. The PI proposes to investigate the concept of noncommutativity, which is the mathematical reason for the ``quantumness of a physical system and implies fundamental laws of nature such as Heisenberg s uncertainty principle. In recent work, he has shown that quantum operations can have different degrees of noncommutativity. Understanding this phenomenon qualitatively and quantitatively would lead to a deeper understanding of the quantum world. Entanglement of quantum systems is a key resource for quantum information processing. It has been experimentally verified and owes its existence to the noncommutativity of physical systems. Quantum technology has to deal with the fact that quantum states are short-lived and unstable. This is a potential problem for the usage of quantum devices in applications that require a large number of quantum bits. Freedman and Kitaev proposed to search for physical systems that feature certain topological properties and use them to carry out quantum computations. Such a topological quantum computer would then be more robust against noise. As is the case for qubit based quantum computers, it is important to understand what entanglement should mean for topological quantum states, and how to quantify it. To this end, the PI proposes to study a notion of entropy for algebras introduced by Connes and Stormer. It is in spirit a von Neumann entropy that has a relative version defined for pairs of algebras. This fact makes it a promising candidate to detect and measure entanglement of topological quantum states. The proposal has several additional goals. It is planned to employ ideas and techniques from von Neumann algebra theory to advance the understanding of quantum transforms, explore new avenues of constructing quantum algorithms and error correcting codes, and investigate the mathematical structures that underly exotic states of matter. The latter may have important implications for the discovery of new materials. The PI has a successful research record in von Neumann algebra theory and has been the director of the noncommutative geometry and operator algebras research group at Vanderbilt University since 2007. He will mentor PhD students and postdoctoral fellows who will work on aspects of the suggested research projects. One key strength of this proposal are the explicit questions and examples presented in the proposal that can be investigated by junior researchers. Many of the mathematical concepts that go into the proposed research can be traced back to John von Neumann (von Neumann factors), Alain Connes (noncommutative geometry and von Neumann factors) and Vaughan Jones (theory of subfactors and planar algebras). This project will expand the mathematical insights in these areas with a focus on new mathematical structures that could profoundly impact our understanding of quantum information.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Dec 22, 2022
- Source ID
- W911NF2310026
Entities
People
- Dietmar Bisch
Organizations
- Army Contracting Command
- United States Army
- Vanderbilt University