Informationally complete joint measurements on finite quantum systems

We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable the identification of all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If d is odd, then this observable is informationally complete. But if d is even, then the joint observable is not informationally complete, and one has to allow more noise in order to obtain informational completeness.