Talk on
Spectral Convergence Bounds for Finite Classical and Quantum Markov Chains

Oleg Szehr (TUM)

In this talk I present a new framework that yields spectral
bounds on norms of functions of transition maps for finite, homogeneous
Markov chains. The techniques employed work for bounded semigroups, in
particular for classical as well as for quantum Markov chains and they
do not require additional assumptions like detailed balance,
irreducibility or aperiodicity. I use the method in order to derive
convergence bounds that improve significantly upon known spectral
bounds. The core technical observation is that power-boundedness of
transition maps of Markov chains enables a Wiener algebra functional
calculus in order to upper bound any norm of any holomorphic function of
the transition map.