  For the model of probabilistic labelled transition systems that allow for the
co-existence of nondeterminism and probabilities, we present two notions of
bisimulation metrics: one is state-based and the other is distribution-based.
We provide a sound and complete modal characterisation for each of them, using
real-valued modal logics based on the Hennessy-Milner logic. The logic for
characterising the state-based metric is much simpler than an earlier logic by
Desharnais et al. as it uses only two non-expansive operators rather than the
general class of non-expansive operators.
