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 Citationer (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.

Originalsprog	Engelsk
Tidsskrift	Journal of Mathematical Physics
Vol/bind	51
Udgave nummer	12
Sider (fra-til)	122201
Antal sider	19
ISSN	0022-2488
DOI	https://doi.org/10.1063/1.3511335
Status	Udgivet - 1 dec. 2010

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10.1063/1.3511335

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

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.",

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AU - Temme, K.

AU - Kastoryano, Michael James

AU - Ruskai, M.B.

AU - Wolf, Michael Marc

PY - 2010/12/1

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.

U2 - 10.1063/1.3511335

DO - 10.1063/1.3511335

M3 - Journal article

SN - 0022-2488

VL - 51

SP - 122201

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