NWO-DFG Quantum Markov semigroups: from geometry to classification

• Applicant: Dr. Melchior Wirth
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 565405322

Quantum Markov semigroups are quantum mechanical analogues of classical Markov processes. They describe the time propagation of a quantum mechanical system that is open and in which energy dissipates into an ambient environment. Quantum Markov semigroups are pervasive in mathematics and mathematical physics, where they have been thoroughly studied at least since the work of Lindblad and Gorini-Kossakowski-Sudarshan in the 1970s. Much more recently, several tools in modern probability and Markov theory have found suitable quantum analogues, yielding groundbreaking consequences for our understanding of quantum entropy, Fisher information, and structure of von Neumann algebras (partly by the PI's). This proposal builds on this successful line of research and adds a new chapter to the theory of quantum probability. Firstly, we aim to construct the quantum counterpart of optimal transport theory and information theory. Secondly, we target concrete classification problems for von Neumann algebras, one of the models for quantum mechanics. While doing this, we will use quantum Markov semigroups in our approach and show that they yield a rich geometry that has massive applications for optimal transport and classification problems at the same time. This key idea hinges on the collective expertise of the two PI's. If carried out successfully, this proposal will have a lasting impact on the structure and geometry of von Neumann algebras and will tie these seemingly different aspects of the theory together.

• DFG Programme: Research Grants
• International Connection: Netherlands
• Cooperation Partner: Professor Dr. Martijn Caspers