Complexity of self-consistent entanglement certification

Abstract

In recent work Phys. Rev. X 16, 031057 (2026), we proposed a self-consistent approach to entanglement certification based on generalized noncontextuality. It requires no prior characterization of the measurement devices and, given access to all local measurements, can certify every entangled state in a Bell circuit. Here, we study the complexity of this protocol in a finite experiment. At fixed local dimension $d$, we show that the optimal number of distinct local effects needed to reach a trace-distance \textit{entanglement resolution} $\gamma$ is $\Theta_d(\gamma^{-(d-1)})$. This scaling is necessary even when the measurements are tailored to the target state. Independent Haar-random projective measurements instead require $\Theta_d(\gamma^{-(d-1)}\log(1/\gamma))$ effects. Finally, we show that, once the exact operational identities are known, $\Theta_d(\gamma^{-2}\log(1/\delta))$ copies of the target state are necessary and sufficient for certification with error probability at most $\delta$. Determining the optimal sample complexity when those identities must instead be inferred from the same finite target-state data remains open.