# The Exact Replica Threshold for Nonlinear Moments of Quantum States

arXiv: https://arxiv.org/abs/2604.22627 · HTML: https://mlsanity.com/paper/2604.22627/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 8 references that are in the map.
- First posted on arXiv 24 Apr 2026.

## Builds on (most cited first, 1 of 1)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)

## Abstract (arXiv)

Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as operatornametr(ρt), but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order t≥ 3, existing protocols show that lceil t/2rceil replicas already suffice for polynomial-sample estimation of operatornametr(ρt), 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 operatornametr(Oρt), 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.