Compatibility of Pauli Observables

Abstract

Given an arbitrary finite family of Pauli observables, how sharply can they be approximated by a single measurement? We derive an exact state-space correspondence that characterises the resulting sharpness tradeoffs, establishing a link between incompatibility and preparation uncertainty. In particular, suitable ensembles of quantum states determine the full compatibility region and generate explicit parent measurements through Pauli conjugation and classical randomisation. The compatibility region depends only on the anti-commutativity graph and determines all nonuniform tradeoffs under standard noise models, including depolarising, probabilistic and generalised noise. We show that optimal sharpness tradeoffs are determined by spectral problems for signed Pauli Hamiltonians, whose ground states supply the seeds for optimal parent measurements. The correspondence also shows that magic is necessary to attain the full compatibility region of every noncommuting Pauli family. We quantify the resulting sharpness advantage and relate it to the reduced robustness of magic. Finally, we apply the framework to several structured Pauli families, obtaining exact compatibility characterisations and explicit optimal parents.