Determine equality of operational and iid transformation rates

Determine whether the operational almost-iid asymptotic transformation rate \(\tilde{R}_{\F}(\rho\to\sigma)\) equals the iid asymptotic transformation rate \(R_{\F}(\rho\to\sigma)\) for every pair of density operators \(\rho\) and \(\sigma\) and every fixed set of free operations \(\F\) satisfying the paper’s quantum-resource-theory assumptions.

Background

The paper defines operational transformation rates by requiring a single sequence of free operations to transform every state sequence accepted by a tomography-based operational criterion. After characterizing these rates through almost-iid sequences, it proves the lower bound $R_{\F}(\rho\to\sigma)\leq\tilde{R}_{\F}(\rho\to\sigma)$ and derives an upper bound involving the regularized relative entropy of resource.

Whether the broader operational class can improve the achievable rate over the usual iid class remains unresolved in general. Equality would show that iid rates are robust against correlations and defects that cannot be excluded operationally; strict inequality would demonstrate a genuine separation between experimentally certifiable resources and ideal iid resources.