Completeness of correlated-catalytic PPT monotones

Determine whether the monotones defined by the regularized PPT_k relative entropies for all finite k together with the regularized PPT relative entropy form a complete family of monotones for correlated catalytic transformations under completely PPT-preserving maps.

Background

The paper constructs the family of monotones D_{PPT_k}^{\infty} for finite k and establishes the correlated-catalytic monotonicity of D_{PPT}^{\infty}. These quantities constrain transformations assisted by catalysts that may become correlated with the output while being returned unchanged marginally.

The authors leave unresolved whether these monotones are sufficient to characterize all correlated-catalytic transformations under completely PPT-preserving operations, rather than merely providing necessary constraints. A positive answer would establish that the constructed family completely captures the conversion preorder in this operational setting.