Gradient flow techniques for quantum Markov semigroups

Quantum mechanics describes physical systems at microscopic scales. If such a system is open, that is, it interacts with its environment, then its time evolution can be modeled mathematically by quantum Markov semigroups. A general phenomenon for such open systems is the return to equilibrium: If the environment is in thermodynamic equilibrium, then the state of the open system that interacts with the environment will also tend to equilibrium. This is for example a challenge for the construction of quantum computers since this decoherence causes prepared quantum states to become more and more indistinguishable from the noise of the environment. Beyond this qualitative observation, it is of interest to make quantitative statements about this return to equilibrium. One general method for this comes from the theory of gradient flows. The time evolution of an open system is a gradient flow if the deviation from equilibrium does not only decrease, but decreases as fast as possible. To make this precise, one needs a measure for the deviation from equilibrium and an adapted metric on the state space, that is, a measure for the distance between to states of the system. One goal of this project is to construct such an adapted metric on the state space for a whole class of measures for the deviation from equilibrium. To do so, it is necessary to have a precise understanding of the structure of the quantum Markov semigroups that model the time evolution of open quantum systems. There is a fascinating correspondence between such quantum Markov semigroups and derivations operations that satisfy a product rule similar to the one of the ordinary derivation operator from calculus. However, so far this theory is limited to the (purely theoretical) case where the environment of the open system is at infinite temperature. The second goal of this project is to extend this correspondence to systems at finite temperature.

In this project, significant progress was made in the understanding of the mathematical structure of the master equations that describe the time evolution of open quantum systems coupled to an environment in equilibrium. These equations have been well-understood for systems with finitely many degrees of freedom, but the master equations of systems with infinitely many degrees of freedom had only been characterized in certain special cases. In particular, these equations were almost completely characterized for the GNS version of the detailed balance condition and connections with derivations were made for the KMS version of the detailed balance condition, which are new even for systems with finitely many degrees of freedom. In the study of the return to equilibrium of open quantum systems, it was established that gradient flow techniques can prove optimal decay rates of the relative entropy for qubit systems. The progress in the analysis of the long-time behavior of open quantum systems moreover lead to new insights, including the solution of a slightly relaxed version, into a conjecture of Montanaro and Osborne in quantum information theory with applications to quantum learning algorithms. On the purely mathematical side, this project has paved the way to the application of derivations in deformation / rigidity theory to type III von Neumann algebras. Furthermore, as part of the project, a new class of von Neumann algebras was introduced, which generalize central objects of operator-valued free probability and lead to new structural insights into these objects.