  In studies of entanglement, finding out if a state is entangled and
quantifying the amount of entanglement contained in a state are related but
different questions. Similarly in studies of causality, finding out the causal
structures compatible with a model and quantifying the causal strengths are
related but different questions. Recently much research have been directed
towards the first question but considerably less attention is paid to the
second one. In this paper we propose axioms for all reasonable quantitative
measures of causality to obey. The axioms apply to a broad family of
operational probabilistic theories with and without definite causal structure.
For quantum models, we study causality measures based on one-shot quantum
capacities in detail. These measures are used to define the notions of quantum
signalling and quantum causality measures in order to quantify quantum
causality.
