Self-averaging of the filtered stabilizer Rényi entropy

Prove self-averaging for the filtered stabilizer Rényi entropy of states sampled from the thermal Scrooge ensemble, rather than only for the related linearized filtered stabilizer Rényi entropy.

Background

The paper uses annealed Scrooge averages as approximations to quenched averages of the filtered stabilizer Rényi entropy and argues numerically that fSRE values concentrate around their ensemble average as system size increases. A rigorous concentration result is established only for the linearized SRE, using a Lipschitz bound and Lévy’s lemma for GAP measures.

Consequently, whether the actual nonlinear fSRE is self-averaging remains unresolved. Establishing this would justify replacing typical-state fSRE values by the corresponding Scrooge-ensemble average in the thermodynamic analysis.