The Exact Replica Threshold for Nonlinear Moments of Quantum States

Abstract

Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as $\mathrm{tr}({\rho }^t)$, but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order $t\ge 3$, existing protocols show that $\lceil t/2\rceil$ replicas already suffice for polynomial-sample estimation of $\mathrm{tr}({\rho }^t)$, 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 $\lceil t/2\rceil -1$ replicas requires dimension-growing sample complexity, while $\lceil t/2\rceil$ replicas suffice by prior work. Thus the exact replica threshold for fixed-order pure moments is $\lceil t/2\rceil$. 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 $\mathrm{tr}(O{\rho }^t)$, 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.

Withdrawn because the main exact-threshold claim is not valid as stated. Stronger pre-existing replica-limited lower bounds, including arXiv:2509.24000, show that dimension-dependent sample complexity persists beyond the threshold claimed in this version