Establish equality of operational and standard entanglement cost

Determine whether the operational entanglement cost \(\tilde{E}_C(A:B)_{\rho}\) equals the standard iid entanglement cost \(E_C(A:B)_{\rho}\) for every bipartite density operator \(\rho_{AB}\).

Background

For LOCC transformations, the paper proves that operational entanglement distillation coincides with its iid counterpart, E~D(A:B)ρ=ED(A:B)ρ\tilde{E}_D(A:B)_\rho=E_D(A:B)_\rho. For entanglement cost, it establishes only the inequality E~C(A:B)ρ≤EC(A:B)ρ\tilde{E}_C(A:B)_\rho\leq E_C(A:B)_\rho, while the reverse inequality is not obtained.

The unresolved converse inequality is a concrete instance of the general equality problem for operational and iid transformation rates. Proving equality would show that the usual entanglement cost remains valid even when the available resource is only operationally certified as many copies of ρ\rho, rather than known to have an exact tensor-power structure.

References

However, it remains open whether the equality between operational asymptotic transformation rates $\tilde{R}\F$ and the traditional iid rates $R\F$ holds in general. In particular, for the case of entanglement cost,~\cref{intro:thm_almost-iid-iid-rates-relation} implies that the operational version is smaller than the standard one, $\tilde{E}C(A:B){\rho} \leq E_C(A:B)_{\rho}$, while the converse inequality remains unclear.

— Operational transformation rates of quantum states  (2610.02025 - Mazzola et al., 1 Oct 2026) in Question immediately following the Outlook discussion of entanglement cost

Thus this ancillary relative-entropy construction cannot improve the regularised Rains converse. Whether the first inequality in~extended_rains_sandwich is always an equality before regularisation is not established here.

— On PPT entanglement distillation  (2610.12454 - Lami, 8 Oct 2026) in Section 5, subsection “Retrieving and extending the Rains bound”