The Exact Replica Threshold for Nonlinear Moments of Quantum States

Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as operatorname{tr}\(ρ^t\), but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order tge 3, existing protocols show that lceil t/2rceil replicas already suffice for polynomial-sample estimation of operatorname{tr}\(ρ^t\), yet it has remained open whether one fewer replica must necessarily incur a sample-complexity barrier growing with the dimension. We prove that this is indeed the case in the sample/copy-access model with replica-limited joint measurements: any protocol restricted to lceil t/2rceil-1 replicas requires dimension-growing sample complexity, while lceil t/2rceil replicas suffice by prior work. Thus the exact replica threshold for fixed-order pure moments is lceil t/2rceil. Equivalently, for fixed-order pure moments, one additional coherent replica is not merely useful but marks the exact threshold between polynomial-sample estimation and a dimension-growing regime in the replica-limited model. We further show that the same threshold law extends to a broad family of observable-weighted moments operatorname{tr}\(Oρ^t\), including Pauli observables and other observables with bounded operator norm and macroscopic trace norm. Coherent replica number therefore acts as a genuinely discrete resource for nonlinear quantum-state estimation.