General Class of Functionals for Certifying Quantum Incompatibility

Quantum steering, measurement incompatibility, and instrument incompatibility have recently been recognized as unified manifestations of quantum incompatibility. Building on this perspective, we develop a general framework for constructing optimization-free, nonlinear incompatibility witnesses based on convex functionals, valid in arbitrary dimensions. We prove that these witnesses are nontrivial precisely when the underlying functional is nonaffine on extremal points (e.g., pure states for ensembles). For pure bipartite states, the witnesses yield lower bounds on entanglement measures, thereby outperforming most linear steering inequalities in the pure-state regime. Moreover, the construction extends in full generality to certify measurement and instrument incompatibility, where the witnesses act as genuine incompatibility monotones. We demonstrate the versatility of our approach with two operationally relevant functionals: the Wigner-Yanase skew information and an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mo>ℓ</a:mo> <a:mn>2</a:mn> </a:msub> </a:math> -type coherence functional.