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Journal of Mathematical Physics / Volume 51 / Issue 12 / ARTICLES / Quantum Information and Computation

J. Math. Phys. 51, 122201 (2010); http://dx.doi.org/10.1063/1.3511335 (19 pages)

The χ2-divergence and mixing times of quantum Markov processes

K. Temme1, M. J. Kastoryano2,3, M. B. Ruskai4, M. M. Wolf2, and F. Verstraete1

1Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
2Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
3Niels Bohr International Academy, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
4Department of Mathematics, Tufts University, Medford, Massachusetts 02155, USA

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(Received 20 June 2010; accepted 13 October 2010; published online 14 December 2010)

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.

© 2010 American Institute of Physics

Article Outline

  1. INTRODUCTION
    1. Formal setting and notation
  2. THE QUANTUM χ2 -DIVERGENCE
    1. Monotone Riemannian metrics and generalized relative entropies
    2. Properties of the quantum χ2 -divergence
  3. MIXING TIME BOUNDS AND CONTRACTION OF THE χ2 -DIVERGENCE UNDER CPT MAPS
    1. Mixing time Bounds
    2. Contraction Coefficients
  4. QUANTUM DETAILED BALANCE
  5. QUANTUM CHEEGER’S INEQUALITY
    1. Example: Conductance bound for unital qubit channels
  6. CONCLUSION AND DISCUSSION

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KEYWORDS and PACS

Keywords

convergence of numerical methods, Markov processes, quantum theory

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3511335

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    D. Petz and Cs. Sudar, “Geometries of quantum states,” J. Math. Phys. 37, 2662 (1996)JMAPAQ000037000006002662000001.

    A. Lesniewski and M. B. Ruskai, “Monotone Riemannian metrics and relative entropy on noncommutative probability spaces,” J. Math. Phys. 40, 5702 (1999)JMAPAQ000040000011005702000001.

    M.B. Hastings, “Random untaries give quantum expanders,” Phys. Rev. A 76, 032315 (2007).

    D. Perez-Garcia, M. M. Wolf, D. Petz, and M. B. Ruskai, J. Math. Phys. 47, 083506 (2006)JMAPAQ000047000008083506000001.

    W. A. Majewski, “The detailed balance condition in quantum statistical mechanics,” J. Math. Phys. 25, 614 (1984)JMAPAQ000025000003000614000001.



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