  We provide a stochastic interpretation of non-commutative Dirichlet forms in
the context of quantum filtering. For stochastic processes motivated by quantum
optics experiments, we derive an optimal finite time deviation bound expressed
in terms of the non-commutative Dirichlet form. Introducing and developing new
non-commutative functional inequalities, we deduce concentration inequalities
for these processes. Examples satisfying our bounds include tensor products of
quantum Markov semigroups as well as Gibbs samplers above a threshold
temperature.
