Existence of states not certifiable with few single-qubit measurements

Identify explicit families of n-qubit target states for which no certification protocol limited to a polynomial number of single-qubit measurements can verify high fidelity with the target state, thereby delineating fundamental limits of single-qubit-measurement-based certification.

Background

The paper demonstrates efficient certification for almost all states under a relaxation-time condition and provides examples of structured families (e.g., phase states, GHZ-like states with suitable adaptations) where certification is feasible. However, it remains to determine whether there exist explicit states for which any protocol confined to polynomially many single-qubit measurements fails.

Clarifying such limitations would help characterize the boundary between certifiable and non-certifiable targets under single-qubit measurement constraints, guiding both theoretical development and experimental practice.