# Spectral gap implies rapid mixing for commuting Hamiltonians

### APA

Capel Cuevas, A. (2023). Spectral gap implies rapid mixing for commuting Hamiltonians. Perimeter Institute. https://pirsa.org/23100096

### MLA

Capel Cuevas, Angela. Spectral gap implies rapid mixing for commuting Hamiltonians. Perimeter Institute, Oct. 16, 2023, https://pirsa.org/23100096

### BibTex

@misc{ pirsa_PIRSA:23100096, doi = {10.48660/23100096}, url = {https://pirsa.org/23100096}, author = {Capel Cuevas, Angela}, keywords = {Quantum Information}, language = {en}, title = {Spectral gap implies rapid mixing for commuting Hamiltonians}, publisher = {Perimeter Institute}, year = {2023}, month = {oct}, note = {PIRSA:23100096 see, \url{https://pirsa.org}} }

Angela Capel Cuevas University of Tübingen

## Abstract

Quantum systems typically reach thermal equilibrium rather quickly when coupled to an external thermal environment. The usual way of bounding the speed of this process is by estimating the spectral gap of the dissipative generator. However, the gap, by itself, does not always yield a reasonable estimate for the thermalization time in many-body systems: without further structure, a uniform lower bound on it only constraints the thermalization time to be polynomially growing with system size. In this talk, we will discuss that for all 2-local models with commuting Hamiltonians, the thermalization time that one can estimate from the gap is in fact much smaller than direct estimates suggest: at most logarithmic in the system size. This yields the so-called rapid mixing of dissipative dynamics. We will show this result by proving that a finite gap directly implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for the class of models we consider. The result is particularly relevant for 1D systems, for which we can prove rapid thermalization with a constant decay rate, giving a qualitative improvement over all previous results. It also applies to hypercubic lattices, graphs with exponential growth rate, and trees with sufficiently fast decaying correlations in the Gibbs state. This has consequences for the rate of thermalization towards Gibbs states, and also for their relevant Wasserstein distances and transportation cost inequalities.

---

Zoom link https://pitp.zoom.us/j/91315419731?pwd=TGpFTjlHWEJZVWZkdTh6bDFKMjhQZz09