The $\chi ^2$χ2-divergence and mixing times of quantum Markov processes

Abstract

We introduce quantum versions of the \documentclass[12pt]{minimal}\begin{document}$\chi ^2$\end{document}χ2-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 the literature 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 \documentclass[12pt]{minimal}\begin{document}$\chi ^2$\end{document}χ2-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyze different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.