Generalized quantum Jensen inequality for deriving data processing

Determine whether an appropriate generalized quantum Jensen inequality can derive the quantum data-processing inequality from the Kirkwood–Dirac quasiprobability fluctuation theorem even when Horváth’s criterion for the Jensen–Steffensen inequality is violated.

Background

For real signed trajectory distributions, Horváth’s criterion characterizes when the Jensen–Steffensen inequality holds for every continuous convex function. The paper demonstrates that amplitude-damping examples can violate this criterion while retaining the specific exponential relation implied by the integral quasiprobability fluctuation theorem and the quantum data-processing inequality.

The unresolved issue is whether a suitably generalized quantum Jensen inequality could replace the ordinary Jensen–Steffensen framework and provide a derivation of the quantum data-processing inequality directly from the Kirkwood–Dirac quasiprobability fluctuation theorem. Resolving this would connect trajectory-level quantum information dynamics with the average-level data-processing principle despite quasiprobability negativity.