Exact Incompatibility-Breaking Criterion for Unital Qubit Channels

Quantum measurements are often incompatible, meaning they cannot be performed jointly. A central question in quantum information is whether noisy quantum channels that render projective measurements (PVMs) compatible also necessarily render all general measurements (POVMs) compatible. This paper investigates this "incompatibility-breaking" property for qubit channels and its implications for quantum steering.

The author employs a dual framework based on support functions to characterize the set of POVMs that can be simulated by a given parent POVM. By comparing the convex set of noisy POVMs with the set of POVMs obtainable via post-processing a parent POVM, the author reduces the problem of universal simulation to a geometric inequality on the Bloch sphere. This dual formulation allows for the construction of specific parent POVMs that certify the compatibility of noisy measurements.

For all unital qubit channels, the author derives an exact criterion for incompatibility breaking. The study proves that the boundary determined by projective measurements is sufficient to cover all POVMs, effectively closing the gap between PVM and POVM compatibility thresholds in this setting. Through the steering-joint-measurability correspondence, this result provides the exact POVM-steering boundary for two-qubit states with maximally mixed marginals. For nonunital channels, the author constructs an asymmetric parent POVM to derive a sufficient condition for incompatibility breaking, which translates to a sufficient unsteerability criterion for arbitrary two-qubit states.

This work provides a definitive answer to the PVM-POVM equivalence problem for the entire class of unital qubit channels, generalizing previous results that were limited to specific symmetric cases like the depolarizing channel. By establishing these exact boundaries, the paper clarifies the operational limits of nonclassicality under noise and provides a powerful dual-based methodology for addressing similar problems in higher dimensions or for nonunital channels.