Joint measurements are pivotal in quantum theory, featuring prominently in key quantum information primitives such as entanglement swapping, teleportation, and dense coding. A historically important question has been whether joint measurements can be performed at a distance, without bringing together the particles to be measured. The original motivation for asking this was understanding whether the measurement process in quantum mechanics is compatible with relativity, which was already put into question by Landau and Peierls in 1933. However, its interest is not only fundamental, since being able to perform joint measurements locally will have a strong impact in the development of protocols for quantum communication and distributed quantum computing.

In this talk I will describe a protocol for localizing any joint quantum measurement, given sufficient pre-shared entanglement. Because the protocol consists on multiple rounds of blind teleportation, the amount of rounds needed for localizing a given measurement can be understood as a measure of its complexity. Thus, we give the first complete classification of joint quantum measurements. I will show that many of the joint measurements that are now routinely studied theoretically and implemented experimentally have low complexity according to this measure. This indicates that there are still many interesting phenomena to discover when it comes to joint quantum measurements.