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

K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf, F. Verstraete

Abstract

We introduce quantum versions of the $χ^2$-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 $χ^2$-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.

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

Abstract

We introduce quantum versions of the -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 -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.

Paper Structure

This paper contains 15 sections, 17 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Formal setting and notation
  3. The quantum $\chi^2$-divergence
  4. Monotone Riemannian metrics and generalized relative entropies.
  5. Properties of the quantum $\chi^2$-divergence
  6. Mixing time bounds and Contraction of the $\chi^2$-divergence under CPT maps
  7. Mixing time Bounds
  8. Contraction Coefficients
  9. Quantum Detailed Balance
  10. Quantum Cheeger's Inequality
  11. Example: Conductance bound for unital qubit channels
  12. Conclusion and Discussion
  13. Acknowledgements
  14. Families of divergences
  15. Proof of a key inequality

Key Result

Theorem 3

For every $k \in \mathcal{K}$, there is a $g \in \mathcal{G}$ such that for a given $\sigma \in {\cal S}_d$, and $A,B$ hermitian traceless, we get: and, $k$ is related to $g$ by

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Theorem 9
  • Lemma 10
  • ...and 14 more