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

• URL: http://arxiv.org/abs/1005.2358v3
• Date: Fri, 5 Apr 2024 17:33:45 GMT
• Title: The $χ^2$-divergence and Mixing times of quantum Markov processes
• Authors: K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf, F. Verstraete,
• Abstract summary: We introduce quantum versions of the $chi2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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