k-fold unbiased measurements and maximal incompatibility

Quantum measurements that cannot be performed jointly are a cornerstone of quantum theory, underlying phenomena like Bell non-locality and steering. While mutually unbiased bases (MUBs) are well-understood as a canonical example of incompatible measurements, extending this concept to higher-order tuples (beyond pairs) has proven difficult. This paper introduces k-fold unbiased measurements (k-UMs), a generalization that moves from rank-one basis measurements to arbitrary-rank projective measurements. The authors define both algebraic and spectral versions of k-UMs and investigate their existence, structural properties, and operational significance in terms of measurement incompatibility.

The authors establish that for rank-one measurements and triples (3-UMs), the algebraic and spectral definitions of k-UMs coincide. They prove significant no-go results, showing that rank-one k-UMs for k >= 3 are rare and that no three-outcome 3-UM exists at any rank. However, by utilizing real Hadamard matrices and complex Clifford algebras, they construct an infinite family of higher-rank 3-UMs.

Crucially, the authors determine the generalized incompatibility robustness for these 3-UMs exactly. They show that for any 3-UM with n outcomes, the robustness is given by the largest root of a universal polynomial divided by three. Using a symmetry-reduced sum-of-squares hierarchy, they provide high-precision numerical evidence that these constructed 3-UMs are among the most incompatible triples possible, suggesting that the noise thresholds derived from their k-UM analysis characterize the asymptotic behavior of the hierarchy.

This work bridges the gap between abstract algebraic constructions and the operational quantification of quantum resources. By providing an exact, dimension-independent benchmark for measurement incompatibility, the authors offer a powerful tool for analyzing multisetting steering and other device-independent quantum protocols. The Hadamard-Clifford construction provides concrete, implementable examples of highly incompatible measurements, which are essential for applications in quantum cryptography and state discrimination.