Certifying entangled measurements in known Hilbert spaces

We study under which conditions it is possible to assert that a joint measurement cannot be simulated by local operations and classical communication. More concretely, we consider a scenario where two parties, Alice and Bob, send each a state selected from a labeled set of unknown states to a third party, Charlie, who in turn interacts with the states in some undisclosed way and then announces an outcome. We show that, under the assumption that Alice and Bob know the dimensionality of their systems, there exist situations where the statistics of the outcomes reveals the nature of Charlie's measurement.