Optimal robustness of self-testing

Real experiments have a nonzero Bell-value deficit. Robust self-testing quantifies the worst-case reference-state fidelity obtainable by local extraction maps from statistics with that deficit. Bell normalization, fidelity convention and the allowed extraction maps must be fixed before comparing bounds.

Operator sum-of-squares techniques, swap isometries and SDPs provide many bounds. A square-root estimate in state-vector norm and a linear estimate in infidelity can describe compatible behavior, not different optimal exponents. Kaniewski established a tight three-qubit Mermin/GHZ benchmark and nearly optimal CHSH bounds; the catalogue should not present all standard tests as wholly unresolved.

Preprint progress (31 August 2026): Cao–Zhang–Shi–Zhao claim analytic $n$-qubit GHZ bounds with size-independent robustness, linear violation-error dependence and a factor-two comparison to an upper bound. Numerical checks through $n=100$ support a conjectured optimum but are not a general exact proof.

The incremental target is matching upper/lower robustness curves for named tests, with the same metric and normalization, and transferable methods for additional targets.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: Size-independent analytic GHZ robustness within a factor of two; numerical tests up to 100 qubits support, but do not prove, the general optimum. Evidence