Compatibility Complexity and the Compatibility Radius of Qubit Measurements

Measurement compatibility in quantum mechanics has been introduced and used as an extension of commutativity for positive operator-valued measurements (POVMs). While incompatible families of measurements are necessary to realize many fascinating quantum effects, in this work, we show that even compatible measurements can have a rich non-classical structure. In particular, we consider how large a 'paren' measurement must be to simulate an entire family of compatible measurements. For the case of spin-1/2 systems, we pursue a geometrical formulation of the problem and derive tight lower and upper bounds on the size of a parent POVM needed to simulate noisy spin measurements. Most notably, we prove that at critical noise thresholds, a parent POVM of unbounded size is needed to achieve the simulation. Part of our results is obtained by relating the compatibility of noisy spin measurements to the zonotope approximation problem studied in Banach space theory. Implications for local hidden state models of two-qubit Werner states are discussed.