AI 中文总结
研究连续变量系统中高斯态层析成像样本复杂度问题，证明高斯测量协议有\(\log \log E\)依赖，给出自适应与能量依赖权衡协议，还表明非高斯测量及特定简单协议可降低样本复杂度，阐明能量作用及相关相互作用。
AI 中文摘要
与有限维情况不同，连续变量系统中的量子信息有一特殊性质：在不施加物理约束时，态层析成像的样本复杂度可能无界。值得注意的是，对于学习高斯态的现有最先进协议（其具有有限维描述）也是如此，已知最佳速率与\(\log \log E\)成比例，其中\(E\)是系统能量。我们证明这并非现有分析的人为产物，而是所用测量的基本限制。我们表明：（1）任何使用高斯测量的协议，即使是纠缠或自适应选择的协议，都必然有\(\log \log E\)的依赖关系，这回答了先前许多工作提出的一个开放性问题。（2）在自适应轮数和能量依赖之间存在平滑权衡，我们给出了一个实现此插值速率的匹配协议。（3）使用高度纠缠的非高斯测量，可以用\(O(n^2 / \epsilon^2)\)个样本学习\(n\)模纯高斯态，与\(E\)无关，这回答了Chen等人提出的一个开放性问题。（4）基于Holevo和Helstrom的单拷贝规范相位POVM的简单协议可以用\(O(1/\epsilon^2)\)个样本学习单模纯高斯态，同样与\(E\)无关。我们的结果阐明了能量在玻色子态层析成像中的作用，并为量子学习中自适应、纠缠和神奇性之间有趣的相互作用提供了新的见解。
英文摘要
Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with $\log \log E$, where $E$ is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a $\log \log E$ dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn $n$-mode pure Gaussian states with $O(n^2 / ε^2)$ samples, independent of $E$. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with $O(1/ε^2)$ samples, again independent of $E$. Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.
Comments70 pages, 2 figures, comments welcome
