The landscape of causal relations that can hold among a set of systems in
quantum theory is richer than in classical physics. In particular, a pair of
time-ordered systems can be related as cause and effect or as the effects of a
common cause, and each of these causal mechanisms can be coherent or not.
Furthermore, one can combine these mechanisms in different ways: by
probabilistically realizing either one or the other or by having both act
simultaneously (termed a physical mixture). In the latter case, it is possible
for the two mechanisms to be combined quantum-coherently. Previous work has
shown how to experimentally realize one example of each class of possible
causal relations. Here, we make a theoretical and experimental study of the
transitions between these classes. In particular, for each of the two distinct
types of coherence that can exist in mixtures of common-cause and cause-effect
relations--coherence in the individual causal pathways and coherence in the way
the causal relations are combined--we determine how it degrades under noise and
we confirm these expectations in a quantum-optical experiment.Comment: 9 pages, 6 figure