标题： 混合量子态的特征值确定使用重叠统计 显示英文标题

标题： Eigenvalue Determination for Mixed Quantum States using Overlap Statistics

摘要： 我们研究了混合态与其在随机酉变换下的像之间的重叠统计量。从酉群中选择变换，并赋予其不变（哈尔）测度时，重叠的分布仅依赖于混合态的本征值。这使得我们可以从重叠统计数据来估计这些本征值。在这项工作的第一部分，我们给出了三能级系统（qutrits）的明确结果，包括对本征值估计中预期不确定性的讨论。在第二部分，我们假设可用的酉变换集被限制为$SO(3)$，表示为维格纳$D$-矩阵。在这种情况下，重叠统计数据不仅依赖于本征值，还依赖于所研究混合态的本征态。此时，重叠分布显示出一个复杂的模式，可以将其视为混合态的指纹。当使用酉群中的随机变换时，本征值可以从重叠统计数据的下限和上限简单确定。在$SO(3)$情况下，这仍然是可能的，但代价是存在有限的系统性不确定性。 显示英文摘要

摘要： We consider the statistics of overlaps between a mixed state and its image under random unitary transformations. Choosing the transformations from the unitary group with its invariant (Haar) measure, the distribution of overlaps depends only on the eigenvalues of the mixed state. This allows one to estimate these eigenvalues from the overlap statistics. In the first part of this work, we present explicit results for qutrits, including a discussion of the expected uncertainties in the eigenvalue estimation. In the second part, we assume that the set of available unitary transformations is restricted to $SO(3)$, realized as Wigner $D$-matrices. In that case, the overlap statistics does not depend only on the eigenvalues, but also on the eigenstates of the mixed state under scrutiny. The overlap distribution then shows a complicated pattern, which may be considered as a fingerprint of the mixed state. When using random transformations from the unitary group, the eigenvalues can be determined quite simply from the lower and the upper limit of the overlap statistics. This may still be possible in the $SO(3)$ case, but only at the expense of a finite systematic uncertainty.