Characterize the operational set versus almost-iid state sequences

Characterize whether every power-law sublinear almost-iid state sequence along a single-copy state \(\rho\) belongs to the operational set \(\OP(\mathrm{tom}(\cdot);\rho)\) for tomography protocols satisfying the stated assumptions, thereby establishing or refuting the inclusion \(\mathcal{A}(\rho)\subseteq\OP(\mathrm{tom}(\cdot);\rho)\).

Background

The paper characterizes permutation-invariant sequences in the operational set after tracing out a sublinear number of systems: they become equivalent, in trace distance up to subexponentially decreasing error, to sequences in the class $\mathcal{A}(\rho)$ of almost-iid states with a power-law sublinear number of defects. The authors also prove that almost-iid sequences with sufficiently few defects belong to the efficient operational set when the tomography protocol is robust under noise.

The remaining issue is whether all sequences in $\mathcal{A}(\rho)$ can arise from the de Finetti-type characterization and therefore belong to the operational set. Resolving this would determine whether the structural almost-iid class exactly captures the operationally admissible resources, rather than merely providing a one-sided characterization.