  Quantum supermaps are a higher-order generalization of quantum maps, taking
quantum maps to quantum maps. It is known that any completely positive, trace
non-increasing (CPTNI) map can be performed as part of a quantum measurement.
By providing an explicit counterexample we show that, instead, not every
quantum supermap sending a quantum channel to a CPTNI map can be realized in a
measurement on quantum channels. We find that the supermaps that can be
implemented in this way are exactly those transforming quantum channels into
CPTNI maps even when tensored with the identity supermap. We link this result
to the fact that the principle of causality fails in the theory of quantum
supermaps.
