Incompatibility, cospark and support uncertainty for rank-one positive operator-valued measures

Measurement incompatibility is a fundamental feature of quantum mechanics and is closely connected with uncertainty relations and quantum information processing. This paper studies the exact support content of this phenomenon. We extend Xu’s s-order incompatibility, which was inspired by De Bièvre’s complete incompatibility for orthonormal bases, to rank-one positive operator-valued measures (POVMs) through their associated tight frames. For two orthonormal bases, the order is characterized by the spark of the concatenated basis matrix. However, for rank-one POVMs, the appropriate invariant is the cospark of the joint analysis operator, equivalently the complement of the largest indexed subfamily of measurement vectors that fails to span the Hilbert space. This distinction is essential for overcomplete POVMs. We use this frame-theoretic characterization to prove sharp support uncertainty relations, erasure-spanning thresholds, and multi-measurement extensions. This invariant is distinct from resource-theoretic joint measurability: it quantifies unavoidable nonzero outcome patterns and need not vanish for jointly measurable or even identical POVMs.