Existence of divergences separating completely positive and positive data processing

Determine whether there exists a divergence that satisfies the data processing inequality for quantum channels, meaning completely positive maps, but fails to satisfy it for merely positive maps.

Background

The paper discusses the extension of data processing inequalities from completely positive maps to merely positive maps. Although the data processing inequality is known for positive maps for most divergences, the authors state that no counterexample is known in which a divergence is monotone under quantum channels but not under all positive maps. Determining whether such a divergence exists would clarify the distinction between complete positivity and positivity in the monotonicity theory of quantum divergences.

References

For most divergences the data processing inequality is now known to hold for positive maps (see, e.g., Appendix E for a discussion), and there is no known example of a divergence that satisfies the data processing inequality for quantum channels (completely positive maps), but not for merely positive maps.

— Sufficient positive maps between von Neumann algebras: Rényi divergences  (2608.28510 - Jenčová et al., 28 Aug 2026) in Section 1, Introduction