Complete Resource Theory of Quantum Incompatibility as Quantum Programmability

Measurement incompatibility describes two or more quantum measurements whose expected joint outcome on a given system cannot be defined. This purely nonclassical phenomenon provides a necessary ingredient in many quantum information tasks such as violating a Bell inequality or nonlocally steering part of an entangled state. In this Letter, we characterize incompatibility in terms of programmable measurement devices and the general notion of quantum programmability. This refers to the temporal freedom a user has in issuing programs to a quantum device. For devices with a classical control and classical output, measurement incompatibility emerges as the essential quantum resource embodied in their functioning. Based on the processing of programmable measurement devices, we construct a quantum resource theory of incompatibility. A complete set of convertibility conditions for programmable devices is derived based on quantum state discrimination with postmeasurement information.

Figure 1

PMDs processing, according to Definition 3. Time flows from left to right. The program $y$ (i.e., the postinformation) arrives after the preprocessing has been performed. Since the only quantum memory resides in the PMD, the quantum input must be committed to the PMD until the program arrives. On the other hand, the classical output $i$ of the preprocessing instrument can be stored in a classical memory and interact with the program before it reaches the PMD. Notice that, even though it is not explicitly depicted in the picture, classical randomness can be shared between all processing boxes (orange on-line), so that the set of possible processings is convex.

Figure 2

The spatial model of PMD processing. The quantum and program inputs are separated between Alice and Bob, and the free operations depicted in Fig. 1 translate into one-way LOCC maps from Alice to Bob.