We derive new concentration bounds for time averages of measurement outcomes
in quantum Markov processes. This generalizes well-known bounds for classical
Markov chains which provide constraints on finite time fluctuations of
time-additive quantities around their averages. We employ spectral,
perturbation and martingale techniques, together with noncommutative $L_2$
theory, to derive: (i) a Bernstein-type concentration bound for time averages
of the measurement outcomes of a quantum Markov chain, (ii) a Hoeffding-type
concentration bound for the same process, (iii) a generalization of the
Bernstein-type concentration bound for counting processes of continuous time
quantum Markov processes, (iv) new concentration bounds for empirical fluxes of
classical Markov chains which broaden the range of applicability of the
corresponding classical bounds beyond empirical averages. We also suggest
potential application of our results to parameter estimation and consider
extensions to reducible quantum channels, multi-time statistics and
time-dependent measurements, and comment on the connection to so-called
thermodynamic uncertainty relations.
