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 $t\ge 3$, existing protocols show that $\lceil t/2\rceil$ 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 $\lceil t/2\rceil-1$ 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 $\operatorname{tr}(Oρ^t)$, 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.

翻译：对量子态多个副本进行联合测量可提供非线性可观测量（如$\operatorname{tr}(ρ^t)$）的信息，但副本数是否构成清晰的信息论资源边界尚不明确。对于每个固定阶数$t\ge 3$，现有协议表明$\lceil t/2\rceil$个副本已足以实现$\operatorname{tr}(ρ^t)$的多项式样本估计，然而能否少用一个副本必然导致随维度增长的样本复杂度门槛仍悬而未决。我们证明在受限于副本数的联合测量样本/副本访问模型中，情况确实如此：任何限制在$\lceil t/2\rceil-1$个副本的协议需要随维度增长的样本复杂度，而$\lceil t/2\rceil$个副本则根据先前工作已足够。由此，固定阶纯态矩的精确副本数阈值为$\lceil t/2\rceil$。等价地，对于固定阶纯态矩，额外一个相干副本不仅有用，更标志着在副本受限模型中多项式样本估计与维度增长区域之间精确的分界阈值。我们进一步证明，该阈值规律可推广至广泛的加权可观测量矩$\operatorname{tr}(Oρ^t)$族，包括泡利可观测量及其他具有有界算子范数和宏观迹范数的可观测量。因此，相干副本数在非线性量子态估计中充当着真正离散的资源角色。