The $\chi^2$-divergence and Mixing times of quantum Markov processes
Abstract
We introduce quantum versions of the -divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3] for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore the contractive behavior of the -divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyse different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.
Keywords
Cite
@article{arxiv.1005.2358,
title = {The $\chi^2$-divergence and Mixing times of quantum Markov processes},
author = {K. Temme and M. J. Kastoryano and M. B. Ruskai and M. M. Wolf and F. Verstraete},
journal= {arXiv preprint arXiv:1005.2358},
year = {2024}
}
Comments
Proposition 6 of the older arxiv version has been removed in the published version as it is in error