B7 Markovian Dynamics in Many-Body Quantum Systems
The interplay between static and dynamical properties of classical lattice spin systems is by now largely understood. For example, the equivalence between decay of correlations of a Gibbs state and the rapid convergence of its corresponding Glauber dynamics has been established in high generality on mathematical grounds. However, the quantum counterpart of the theory remains largely unexplored. The long-term goal of the proposed project is to gain a better mathematical understanding of intrinsically quantum properties of local dissipative evolutions over generic spin many-body quantum systems. We will focus on four main properties of such systems, namely locality of interactions, decay of correlations, rapid mixing and Markovianity.
We will divide this goal into three related tasks: First, we will study how various forms of correlations decay on Gibbs states of (non-necessarily commuting) Hamiltonians between spatially separated regions. Second, we will derive sufficient static conditions for the entropic convergence of local Markovian evolutions for systems with finite-dimensional degrees of freedom. Finally, we will investigate the possibility of efficiently predicting the Markovianity of an evolution exhibiting a given propagation bound.
Given a Markovian evolution modeled by a Davies map, the main long-term goal of this project is to characterize the conditions on the decay of correlations in the equilibrium Gibbs states under which the associated Davies semigroup mixes rapidly. The outcomes of this project will have implications of both theoretical and practical value, ranging from the robust and efficient preparation of topologically ordered phases of matter via dissipation, the classification of dissipative phases of matter, to the design of more efficient quantum error-correcting codes based optimized for correlated Markovian noise models.