Quantum state certification with entangled measurements has copy complexity Θ~(∥σ*∥_{1/2}/ε²), where σ* is the hypothesis state with a small tail of eigenvalues removed.
Distribution testing lower bounds via reductions from communication complexity
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Instance-Optimal Quantum State Certification with Entangled Measurements
Quantum state certification with entangled measurements has copy complexity Θ~(∥σ*∥_{1/2}/ε²), where σ* is the hypothesis state with a small tail of eigenvalues removed.