最优低秩量子态层析成像：有界样本联合测量
Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements
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中文总结 AI 辅助
本研究确定了低秩量子态层析成像在每次测量最多作用于t个样本时的最优样本复杂度，并证明联合测量至多带来√t倍提升，且r²阶联合测量即可达到集体速率。
中文摘要 AI 辅助
我们确定了当每次测量最多可联合作用于 $t$ 个样本时，低秩量子态层析成像的最优样本复杂度。对于足够小的 $\varepsilon$，在 $\mathbb{C}^d$ 上估计秩至多为 $r$ 的未知态至迹范数误差 $\varepsilon$ 并以恒定成功概率完成，需要且可通过 $$ \Theta\left( \frac{dr}{\varepsilon^2} \max\left\{1,\frac r{\sqrt t}\right\} \right)$$ 个样本实现。下界允许协议利用所有先前的经典结果自适应地选择每次联合测量；匹配的上界则是非自适应的。因此，对至多 $t$ 个样本的联合测量至多能将基于单样本测量的算法的复杂度提升 $\sqrt t$ 倍。此外，联合测量 $r^2$ 阶样本对于达到无限制的集体速率是必要且充分的。对于下界，我们改变具有固定均匀谱的态的支持集，并限制每次对 $t$ 个样本的联合测量的Fisher信息迹。自适应Fisher链式法则和van Trees不等式随后给出迹范数下界。对于上界，我们构造并分析了一个基于高斯联合测量的非自适应层析成像协议。一个显式的二阶矩恒等式和态支持集外的条件高斯分布给出了依赖于秩的误差分析，从而得到匹配的速率。
英文摘要
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, $$Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)$$ 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.
发表机构
- University of Waterloo（滑铁卢大学）
- University of British Columbia（不列颠哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。
