Amount of quantum coherence needed for measurement incompatibility
Jukka Kiukas, Daniel McNulty, Juha-Pekka Pellonpää · Physical Review A · 2022
A pair of quantum observables diagonal in the same ``incoherent'' basis can be measured jointly, so some coherence is obviously required for measurement incompatibility. Here we first observe that coherence in a single observable is linked to the diagonal elements of any observable jointly measurable with it, leading to a general criterion for the coherence needed for incompatibility. Specializing to the case where the second observable is incoherent (diagonal), we develop a concrete method for solving incompatibility problems, tractable even in large systems by analytical bounds, without resorting to numerical optimization. We verify the consistency of our method by a quick proof of the known noise bound for mutually unbiased bases, and apply it to study emergent classicality in the spin-boson model of an $N$-qubit open quantum system. Finally, we formulate our theory in an operational resource-theoretic setting involving ``genuinely incoherent operations'' used previously in the literature, and show that if the coherence is insufficient to sustain incompatibility, the associated joint measurements have sequential implementations via incoherent instruments.