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arXiv · 1005.2358

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

K. Temme·M. J. Kastoryano·M. B. Ruskai·M. M. Wolf·F. Verstraete
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Abstract

We introduce quantum versions of the $\chi^2$-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 $\chi^2$-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.

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K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf, F. Verstraete. 2010-05-13. The $\chi^2$-divergence and Mixing times of quantum Markov processes. https://doi.org/10.1063/1.3511335

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