  We introduce a measure of average dimensionality (or coherence) for
high-dimensional quantum devices. This includes sets of quantum measurements,
steering assemblages, and quantum channels. For measurements and channels, our
measure corresponds to an average compression dimension, whereas for quantum
steering we get a semi-device independent quantifier for the average
entanglement dimensionality known as the Schmidt measure. We analyse the
measure in all three scenarios. First, we show that it can be decided via
semi-definite programming for channels and measurements in low-dimensional
systems. Second, we argue that the resulting steering measure is a
high-dimensional generalisation of the well-known steering weight. Finally, we
analyse the behaviour of the measure in the asymptotic setting. More precisely,
we show that the asymptotic Schmidt measure of bipartite quantum states is
equal to the entanglement cost and show how the recently introduced
entanglement of formation for steering assemblages can be related to our
measure in the asymptotic case.
