The ${\chi }^2$-divergence and Mixing times of quantum Markov processes

Abstract

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.

Proposition 6 of the older arxiv version has been removed in the published version as it is in error

References in corpus (5)

• Quantum Metropolis Sampling
• Metric adjusted skew information
• Introduction to quantum Fisher information
• A quantum version of Wielandt's inequality
• Random Unitaries Give Quantum Expanders

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