Trajectory-level formulation of Rényi data processing

Determine whether the monotonicity of quantum Rényi divergences has a trajectory-level counterpart for the Kirkwood–Dirac quasiprobability description, specifically whether the corresponding Rényi data-processing inequality holds precisely when the Kirkwood–Dirac quasiprobability distribution satisfies a generalized Jensen–Steffensen condition, and whether failure of the inequality can be attributed to loss of this convexity compatibility.

Background

The paper studies fluctuation-theorem-like relations for quantum mutual-information dissipation using Kirkwood–Dirac quasiprobability trajectories. These trajectories may contain negative or complex weights, so standard probability-based convexity arguments do not automatically apply. The authors show that the exponential Jensen-like relation associated with the integral fluctuation theorem and the quantum data-processing inequality can remain valid even when Horváth’s necessary and sufficient criterion for the Jensen–Steffensen inequality fails.

The proposed open problem asks whether an analogous relationship exists for quantum Rényi divergences. More specifically, it asks whether Rényi data-processing monotonicity can be characterized by a generalized Jensen–Steffensen condition on the associated quasiprobability trajectories, and whether violations of Rényi data processing are linked to incompatibility with the relevant convexity structure.