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Stronger conjecture: collapse of both hierarchies at level 2 (E_{c,PPT}^{exact} = E_{χ,2} = E_{κ,2})

Prove that for all bipartite states ρ, E_{c,PPT}^{exact}(ρ) = E_{χ,2}(ρ) = E_{κ,2}(ρ), establishing that both χ- and κ-hierarchies collapse at the second level.

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Background

The authors note they could not find a gap between E_{χ,2} and E_{κ,2}, motivating a stronger conjecture that both hierarchies collapse at level 2. This would provide an even simpler single-letter expression for the entanglement cost.

Confirming this stronger conjecture would unify the two hierarchies and eliminate the need for any hierarchy limit in computing the cost.

References

In fact, we were not even able to find a gap between E_{χ,2} and E_{κ,2}, so one could even make the stronger conjecture that (ρ) = E_{χ,2}(ρ) = E_{κ,2}(ρ) holds for all states, which would entail that both hierarchies collapse at the second level.

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