The resource theory of incompatibility

We conceptualize quantum incompatibility as an intrinsically quantum relation between different ways of refining one's information about a system. This notion of incompatibility applies not only to sets of measurements (multi-meters), but to sets of sources of quantum states (multi-sources), to sets of quantum instruments (multi-instruments), and various generalizations of these (multi-devices). Specifically, we define the resource theory as follows: (i) a multi-device is deemed to be a free resource (having no incompatibility) if all of the devices in the set over which it ranges can be jointly simulated by a single device; (ii) one multi-device is above another in the partial order of all such resources---and therefore has more incompatibility---if the first can be processed to the second using only free multi-devices. We identify many monotones and conversion witnesses. In particular, we demonstrate how no-go conversion witnesses can be obtained as infeasibility certificates in a semi-definite programming feasibility problem. We also identify a low-level partial order over the resources based on the compatibility hypergraph of each mutli-device, where the hypergraph for a set of devices specifies which subsets are compatible.