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  "path": "/abs/2604.22627v1",
  "publishedAt": "2026-04-27T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Shuai Zeng"
  ],
  "textContent": "**Authors:** Shuai Zeng\n\nJoint 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.",
  "title": "The Exact Replica Threshold for Nonlinear Moments of Quantum States"
}