Structural classification of tightness for the sample-complexity sandwich bound

Classify the structural conditions under which the sandwiched bounds relating the minimum-error discrimination sample complexity to the quantum mixing time are tight, equivalently, under which the minimum-error sample complexities at error thresholds ε and ε²/16 satisfy k_min(ε)=Θ_ε(k_min(ε²/16)).

Background

For arbitrary uniform mixed-state ensembles, the paper establishes a sandwich between the fixed-initial-state quantum weak mixing time and the minimum-error discrimination sample complexity. The authors then show that a generalized, stabilized Dobrushin-type coefficient gives a tight characterization for worst-case (minimax) discrimination, but not necessarily for minimum-error discrimination.

The paper identifies a sufficient tightness criterion in terms of the stability of the minimum-error sample complexity under the relevant change in error threshold: the sandwich is tight when k_min(ε)=Θ_ε(k_min(ε²/16)). It does not classify which ensembles or structural properties satisfy this condition, and explicitly leaves that classification unresolved.

References

Eq. eq: sandwiched bound for sample complexity is tight when $k_{\min}(\epsilon)=\Theta_\epsilon(k_{\min}(\frac{\epsilon2}{16}))$. We leave the structural classification of this condition to future research.

— A mixing time method for estimating the sample complexity of quantum state discrimination  (2609.40182 - Zhou et al., 30 Sep 2026) in Section 6, subsection “Tight estimate for worst-case discrimination sample complexity by Dobrushin-type coefficient”