A General Class of Functionals for Certifying Quantum Incompatibility

TL;DR

This paper introduces a unified, optimization-free framework for certifying quantum incompatibility using convex functionals, applicable across dimensions and capable of outperforming linear inequalities in certain regimes.

Contribution

It develops a general, nonlinear incompatibility witness framework based on convex functionals, extending to measurement and instrument incompatibility, with practical examples.

Findings

Witnesses are nontrivial when the functional is non-affine on extremal points.

For pure bipartite states, witnesses provide lower bounds on entanglement measures.

The approach outperforms most linear steering inequalities for pure states.

Abstract

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 non-affine 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 $\ell_{2}$-type coherence functional.