[论文] Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements (arXiv:2609.10514)

论文概要

研究领域: ML
作者: Ashwin Nayak, Xingyu Zhou
发布时间: 2026-09-09
arXiv: 2609.10514

中文摘要

本文确定了低秩量子态层析的最优样本复杂度，其中每次测量最多联合作用于t个样本。对于足够小的ε，以常数成功概率将C^d上秩不超过r的未知状态估计到迹范数误差ε，需要且可达到Θ(dr/ε^2 · max{1, r/√t})个样本。下界允许协议自适应选择每次联合测量；匹配的上界是非自适应的。因此，最多t个样本的联合测量相比单样本测量最多改善√t倍的复杂度。此外，联合测量约r^2个样本是达到无限制集体速率所必需且充分的。

原文摘要

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most \(t\) samples. For sufficiently small \(\varepsilon\), estimating an unknown state on \(\mathbb{C}^d\) of rank at most \(r\) to trace norm error \(\varepsilon\) with constant success probability requires, and is achievable with,

samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most \(t\) samples improve the complexity of algorithms making single-sample measurements by at most a factor \(\sqrt t\). Further, measuring order \(r^2\) samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on \(t\) samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

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自动采集于 2026-09-11

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