The ¿2-divergence and mixing times of quantum Markov processes

K. Temme; Michael James Kastoryano; M.B. Ruskai; Michael Marc Wolf

doi:10.1063/1.3511335

The ¿²-divergence and mixing times of quantum Markov processes

K. Temme, Michael James Kastoryano, M.B. Ruskai, Michael Marc Wolf

53 Citations (Scopus)

Abstract

We introduce quantum versions of the 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 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.

Original language	English
Journal	Journal of Mathematical Physics
Volume	51
Issue number	12
Pages (from-to)	122201
Number of pages	19
ISSN	0022-2488
DOIs	https://doi.org/10.1063/1.3511335
Publication status	Published - 1 Dec 2010

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title = "The ¿2-divergence and mixing times of quantum Markov processes",

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AU - Kastoryano, Michael James

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AU - Wolf, Michael Marc

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Y1 - 2010/12/1

N2 - We introduce quantum versions of the 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 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.

AB - We introduce quantum versions of the 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 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.

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JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 12

ER -

The ¿2-divergence and mixing times of quantum Markov processes

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The ¿²-divergence and mixing times of quantum Markov processes