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Conjecture: χ-hierarchy collapses at level 2 (E_{χ,3} = E_{χ,2})

Prove that for all bipartite states ρ, E_{χ,3}(ρ) = E_{χ,2}(ρ), thereby implying that the χ- and κ-hierarchies collapse and that the zero-error PPT entanglement cost satisfies E_{c,PPT}^{exact}(ρ) = E_{χ,2}(ρ) = E_{κ,3}(ρ).

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Background

Based on numerical evidence and structural observations, the authors hypothesize that the χ-hierarchy stabilizes at level 2. Such a collapse would yield a simple single-letter formula for the zero-error PPT entanglement cost via E_{χ,2}, and align the κ-hierarchy accordingly.

Proving this conjecture would remove the need for taking limits over hierarchy levels and provide a more direct analytic and computational characterization of the PPT entanglement cost.

References

This leads us to posit the following conjecture. For all states ρ = ρ{AB}, the χ-quantities defined by~chi_p_1_SM satisfy that E{χ,3} (ρ) = E_{χ,2}(ρ). As a consequence, the χ- and κ-hierarchies collapse and (ρ) = E_{χ,2} (ρ) = E_{κ,3}(ρ).

— Computable entanglement cost under positive partial transpose operations (2405.09613 - Lami et al., 15 May 2024) in Supplemental Material, Subsection “Open problem: hierarchy collapse”