Commutativity in quantum mechanics is historically identified with the possibility of two (or more) measurements originating from a single setup; it is a sufficient and necessary condition for compatibility of observables (represented by self-adjoint operators). The development of the theory, however, led to the mathematical framework of POVMs (positive operator valued measures), which act as generalisations of the notion of observables as self-adjoint operators.
In the framework of POVMs, commutativity still plays a role, but it is demoted to simply being a sufficient condition for compatibility. There exist measurements that do not commute, yet they can originate from a single observable. None the less, commutativity is still considered an important measure of incompatibility.
In our recent paper we show, quite surprisingly, that when considering the incompatibility of quantum testers, commutativity becomes completely irrelevant. Before we continue, let me explain what these testers are. Whilst we usually measure a state given to us, testers (as shown in their simplest form in the picture below) are designed to measure the devices responsible for changes to states: the channels. To this end, one does not only measure what is on the output, but one can also choose the initial state that is injected into the channel. Simply put, a tester could tell us what the channel does to states.
Let us consider a simple example of a measurement where the channel preserves the polarisation of light. We can prepare a horizontally polarised photon and measure whether, after leaving the channel, it is still horizontally polarised. We can also measure how the vertical polarisation is preserved in a similar way. But can we measure both of these features at the same time? As it turns out, one can not.
One can immediately see that in this case the incompatibility will not lie in the final measurement, as the measurement of vertical/horizontal polarisation is the same. The incompatibility in this case is contained solely in the preparation of the state – one cannot prepare a state whose polarisation is both horizontal and vertical. Intuitively one might be tempted to ask whether a superposition of the two polarisations would work, but unfortunately by preparing the superposition one loses the information about the action of the channel on the individual states that are superposed. In fact, the two testers of the polarisation-preserving channels are maximally incompatible, even though (in their mathematical form) they commute – it takes the most amount of work to make them compatible in some sense.