Most incompatible measurements and sum-of-squares optimisation

TL;DR

This work tackles the quantitative assessment of measurement incompatibility in finite-dimensional quantum systems by introducing a sum-of-squares hierarchy that yields universal parent measurements. The authors establish analytical degree-two results showing that sets of anticommuting dichotomic observables are maximally incompatible for several robustness measures, and they develop a scalable SOS framework to push bounds to higher orders and more general settings. Degree-three constructions provide explicit bounds for fixed outcome counts and fixed dimension, including a dimension witness for genuine high-dimensional steering. Beyond degree three, the paper outlines a hierarchy for degree-four and higher, including normalisable and pinched-marginalisable SOS approaches, with preliminary numerical evidence suggesting practical improvements and a path toward proving extremality for complete sets of mutually unbiased bases. Overall, the results advance the certification of incompatibility in larger measurement assemblies, with direct implications for high-dimensional steering and one-sided device-independent dimensionality witnessing.

Abstract

Measurement incompatibility, or joint measurability, is a cornerstone of quantum theory and a useful resource. For finite-dimensional systems, quantifying this resource and establishing universal bounds valid for all measurements is a long-standing problem. In this work, we exhibit analytical universal parent measurements giving access to bounds that beat the state of the art. In particular, we can show that, for relevant robustnesses, sets of anticommuting observables give rise to the most incompatible dichotomic measurements. We also formalise the construction of such universal parent measurements in the framework of sum-of-squares optimisation and obtain preliminary numerical results demonstrating the power of the method by improving on our own analytical values. All results find direct application for demonstrating genuine high-dimensional steering, that is, certifying the dimensionality of a quantum system in a one-sided device-independent manner.