Optimal Quantum State Testing Even with Limited Entanglement

In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state $ρ$, test whether it matches some target state $σ$ or is $ε$-far from it. For certifying $d$-dimensional states, $Θ(d/ε^2)$ copies of $ρ$ are known to be necessary and sufficient.