Instance-Optimal Quantum State Certification with Entangled Measurements

We consider the task of quantum state certification: given a description of a hypothesis state σ and multiple copies of an unknown state ρ, a tester aims to determine whether the two states are equal or є-far in trace distance. It is known that Θ(d/є2) copies of ρ are necessary and sufficient for this task, assuming the tester can make entangled measurements over all copies. However, these bounds are for a worst-case σ, and it is not known what the optimal copy complexity is for this problem on an instance-by-instance basis. While such instance-optimal bounds have previously been shown for quantum state certification when the tester is limited to measurements unentangled across copies, they remained open when testers are unrestricted in the kind of measurements they can perform.