泡利概率谱携带纯态量子费希尔度量
The Pauli Probability Spectrum Carries the Pure-State Quantum Fisher Metric
- Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）
- University of Augsburg（奥格斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究发现N个量子比特纯态的经典费希尔信息矩阵是量子费希尔信息矩阵的2倍，可通过固定的成对贝尔测量获取，还推导了混合态下贝尔采样不可见的缺失费希尔信息，揭示了相关统计、几何与读出的联系。
AI中文摘要：
稳定子Rényi熵将纯量子态的平方泡利期望值分布压缩为非稳定子性的标量度量。本文表明，在这种压缩前，该分布具有极为简洁的信息几何结构。对于任意N个量子比特的光滑纯态族，其经典费希尔信息矩阵在所有正则点上恰好是该态量子费希尔信息矩阵的2倍。该信息可直接获取：通过制备该态及其复共轭，并在对应量子比特间执行成对贝尔测量，可采样得到相同的泡利分布，且达到共轭对的量子费希尔信息矩阵。该读出方式固定，不依赖于待估计参数数量或特定纯态传感模型。我们证明该结果的起源比泡利代数更具普遍性：共轭将加倍后的纯态与向量化密度算符等同，后者在任意厄米希尔伯特-施密特算子基下的坐标均为实数。混合态的情况则不同，此时贝尔概率不再与归一化平方泡利期望值一致，且相同的固定读出通常不再最优。我们推导了缺失费希尔信息的精确半正定表达式，明确了贝尔采样不可见的态变化分量。研究结果揭示了稳定子Rényi统计、量子态几何与固定多参数读出之间的直接联系。
英文摘要：
How a quantum state responds to small parameter changes underlies state distinguishability, many-body response, and metrological sensitivity. The Pauli probability spectrum provides a natural description of many-body quantum states, but it is not evident how much of this response survives in the corresponding classical distribution. Here we show that, for pure states, the complete labeled Pauli distribution retains the full local metric of the pure-state manifold, reproducing the quantum Fisher metric up to a fixed normalization. This correspondence has a direct measurement realization: preparing a state together with its complex conjugate and performing pairwise Bell measurements between corresponding degrees of freedom converts the quantum response into an ordinary classical distribution without losing any of the sensitivity available in the two inputs. The readout is fixed, factorizes into local pairwise measurements even for strongly entangled many-body states, and preserves the response along all parameter directions simultaneously. We trace this correspondence to the reality of the underlying operator amplitudes and show that the same mechanism extends beyond qubits. When the correspondence fails, we identify explicitly where the missing information resides: part of the response can remain hidden in phases for identical-copy and complex-coordinate readouts, while mixed states develop transverse operator-space motion that is invisible to the Bell probabilities. These results provide a direct link between Pauli/Bell statistics, many-body response, nonstabilizerness, and quantum geometry.
