With public randomness and n_q-qubit channels, distributed quantum state certification needs O(d² / (2^{n_q} ε²)) samples, with a matching lower bound for mixedness-preserving channels and a higher lower bound without shared randomness.
Adversarially robust quan- tum state learning and testing
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Distributed Quantum Property Testing with Communication Constraints
With public randomness and n_q-qubit channels, distributed quantum state certification needs O(d² / (2^{n_q} ε²)) samples, with a matching lower bound for mixedness-preserving channels and a higher lower bound without shared randomness.