Amount of quantum coherence needed for measurement incompatibility

TL;DR

This paper investigates the minimal quantum coherence required for measurement incompatibility, providing a new criterion, analytical bounds, and applications to quantum systems and resource theory.

Contribution

It introduces a general criterion linking coherence to measurement incompatibility and develops a tractable method for solving incompatibility problems in large systems.

Findings

Established a coherence-incompatibility criterion

Derived analytical bounds for measurement incompatibility

Applied the method to spin-boson models and resource theory

Abstract

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. Specialising 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 optimisation. 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.