Positive semidefiniteness versus complete positivity in the range of Ψ

Establish whether every checkerboard copula A that can be represented as A=Ψ(C) for some bivariate copula C has completely positive mass matrix M_A and density matrix G_A.

Background

The paper studies the transformation Ψ(C)=CT*C and characterizes several properties of checkerboard copulas in its range. For a checkerboard copula A on a symmetric mesh, the authors prove that A=Ψ(C) implies that its value, mass, and density matrices are positive semidefinite. They also prove that when the preimage is itself a checkerboard copula, the mass and density matrices are completely positive.

The unresolved implication (P3) asks whether complete positivity of the mass and density matrices necessarily follows even when the representing copula C is an arbitrary bivariate copula rather than a checkerboard copula. The question is nontrivial because complete positivity is strictly stronger than entrywise nonnegativity and positive semidefiniteness in sufficiently large dimensions.

References

We collect the findings of the last three theorems in a schematic summary, along with two implications that are still open, see Figure~\ref{fig:results}.

— On diagonal sections of Markov products of copulas  (2610.13846 - Drnovšek et al., 9 Oct 2026) in Section 3, Figure 1 (implication (P3)); concluding remarks, Section 7