Conjectured upper bound relating classical mutual informations of Wigner marginals to quantum mutual information

Establish whether, for bipartite quantum systems with subsystems A and B, the inequality I(f^A : f^B) + I(g^A : g^B) ≤ I(ρ^A : ρ^B) holds, where f and g denote the Wigner marginal distributions of the canonically conjugate observables (φ, π) for the respective subsystems, I(· : ·) denotes the classical mutual information of these distributions, and I(ρ^A : ρ^B) denotes the quantum mutual information of the reduced density matrices.

Background

In the section connecting classical and quantum information, the authors note that for any bipartition A|B, the classical mutual information computed from measurement distributions obeys I(𝒪^A : 𝒪^B) ≤ I(ρ^A : ρ^B) due to entropic uncertainty relations. They then reference a stronger, conjectured inequality involving the sum of the classical mutual informations associated with the Wigner marginals f and g of conjugate observables.

Here, f^A and g^A are defined as the Wigner marginals obtained by integrating the local Wigner distribution over π and φ, respectively, and I(f^A : f^B) and I(g^A : g^B) are the corresponding classical mutual informations across the bipartition. Establishing this bound would strengthen the link between experimentally accessible classical correlations and underlying quantum correlations, potentially tightening entropic constraints beyond the Gaussian regime.