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We address the problem of characterising the compatible tuples ofmeasurements that admit a unique joint measurement. We derive a uniquenesscriterion based on the method of perturbations and apply it to show thatextremal points of the set of compatible tuples admit a unique jointmeasurement, while all tuples that admit a unique joint measurement lie in theboundary of such a set. We also provide counter-examples showing that none ofthese properties are both necessary and sufficient, thus completely describingthe relation between joint measurement uniqueness and the structure of thecompatible set. As a by-product of our investigations, we completelycharacterise the extremal and boundary points of the set of general tuples ofmeasurements and of the subset of compatible tuples.

Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements