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

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

2010-05-13

arxiv.org/abs/1005.2358v2

J. Math. Phys. 51, 122201 (2010)

Physics

Quantum Physics

Scientific paper

10.1063/1.3511335

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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