  We study quantum causal inference in a set-up proposed by Ried et al. [Nat.
Phys. 11, 414 (2015)] in which a common-cause scenario can be mixed with a
cause-effect scenario, and for which it was found that quantum mechanics can
bring an advantage in distinguishing the two scenarios: Whereas in classical
statistics, interventions such as randomized trials are needed, a quantum
observational scheme can be enough to detect the causal structure if the common
cause results from a maximally entangled state. We analyze this setup in terms
of the geometry of unital positive but not completely positive qubit-maps,
arising from the mixture of qubit-channels and steering maps. We find the range
of mixing parameters that can generate given correlations, and prove a quantum
advantage in a more general setup, allowing arbitrary unital channels and
initial states with fully mixed reduced states. This is achieved by
establishing new bounds on signed singular values of sums of matrices. Based on
the geometry, we quantify and identify the origin of the quantum advantage
depending on the observed correlations, and discuss how additional constraints
can lead to a unique solution of the problem.
