The existence of incompatible observables is a cornerstone of quantum mechanics and a valuable resource in quantum technologies. Here we introduce a measure of incompatibility, called the mutual eigenspace disturbance (MED), which quantifies the amount of disturbance induced by the measurement of a sharp observable on the eigenspaces of another. The MED provides a metric on the space of von Neumann measurements, and can be efficiently estimated by letting the measurement processes act in an indefinite order, using a setup known as the quantum switch, which also allows one to quantify the noncommutativity of arbitrary quantum processes. Thanks to these features, the MED can be used in quantum machine learning tasks. We demonstrate this application by providing an unsupervised algorithm that clusters unknown von Neumann measurements. Our algorithm is robust to noise can be used to identify groups of observers that share approximately the same measurement context.

Thermalization processes degrade the states of any working medium, turning any initial state into a passive state from which no work can be extracted. Recently, it has been shown that this degradation can be avoided if two identical thermalization processes take place in coherently controlled order, in a scenario known as the quantum SWITCH. In some situations, control over the order even enables work extraction when the medium was initially in a passive state. This activation phenomenon, however, is subject to a limitation: to extract non-zero work, the initial temperature of the medium should be less than half of the temperature of the reservoirs. Here we analyze this limitation, showing that it still holds true even when the medium interacts with $N\ge 2$ reservoirs in a coherently-controlled order. Then, we show that the limitation can be lifted when the medium and the control systems are initially correlated. In particular, when the medium and control are entangled, work extraction becomes possible for every initial value of the local temperature of the medium.