The Exact Replica Threshold for Nonlinear Moments of Quantum States

TL;DR

This paper establishes the exact number of quantum state replicas needed for efficient nonlinear moment estimation, revealing a sharp threshold at loor t/2or fixed-order moments in the replica-limited measurement model.

Contribution

The authors prove that fewer than loor t/2ewer replicas require dimension-dependent sample complexity, identifying the precise replica threshold for nonlinear quantum state moments.

Findings

loor t/2ewer replicas lead to dimension-growing sample complexity.

loor t/2ewer replicas are insufficient for polynomial-sample estimation.

The threshold law extends to observable-weighted moments like Pauli observables.

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