Quantum perspectives on transportation cost spaces and signal processing

Classical Optimal Transport searches for the most efficient way to move resources or information from one location to another, for example the most cost-effective way for a manufacturer to distribute its products from the factories where they are produced to the stores where they will be sold. This project will study the analogous question in Quantum Information Science pursues the best way to allocate quantum resources. Signal processing is the theory behind medical imaging. The investigator works on analogous versions of these tools in Quantum Tomography; for example, which conditions guarantee stability under small errors, a feature necessary for any mathematical theory to be applicable in the real-world. The project also includes the training of graduate students and early-career researchers working in quantum information science. The investigator will study a quantum version of optimal transport theory and specifically, developing and studying quantum versions of classical transportation cost spaces, associated Lipschitz functions, and applications in quantum information theory. Another theme of the project is on signal processing. The investigator studies various injectivity questions for quantum measurements and quantum channels related to Quantum Tomography. Dynamical Sampling aims to more economically recover a signal by reusing the same sensor multiple times (in the spirit of the well-known single-pixel camera), and the investigator works on developing both Banach-space and Quantum Information versions of this framework. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.