A Hierarchy for Replica Quantum Advantage
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We prove that given the ability to make entangled measurements on at most k replicas of an n-qubit state ρ simultaneously, there is a property of ρ which requires at least order 2^n measurements to learn. However, the same property only requires one measurement to learn if we can make an entangled measurement over a number of replicas polynomial in k, n. Because the above holds for each positive integer k, we obtain a hierarchy of tasks necessitating progressively more replicas to be performed efficiently. We introduce a powerful proof technique to establish our results, and also use this to provide new bounds for testing the mixedness of a quantum state.
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