arXiv:2604.22627v1 Announce Type: cross
Abstract: Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as $operatorname{tr}(rho^t)$, 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 $operatorname{tr}(rho^t)$, 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 $operatorname{tr}(Orho^t)$, 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.