关于基于特征谱的量子协方差矩阵分类的一个结果
A Result About the Classification of Quantum Covariance Matrices Based on Their Eigenspectra
AI总结：
针对连续变量量子系统协方差矩阵受不确定性原理约束的问题，本文找到了一类非平凡特征谱，证明其对应的量子协方差矩阵均可通过正交辛变换关联，且与识别高斯态热参数和压缩参数的非简并特征谱类重合。
摘要：
由于海森堡不确定性原理，具有有限自由度的连续变量量子系统的协方差矩阵集合是实正定矩阵集合的一个严格子集。这意味着，通常情况下，量子协方差矩阵的每一个正交变换并不都会产生服从不确定性原理的正定矩阵。因此，一个自然的问题出现了，即寻找与给定特征谱一致的量子协方差矩阵集合。对于纯高斯态这一特殊类别，具有给定特征谱的量子协方差矩阵集合由正交辛群作用的单条轨道组成。该类中某个态的协方差矩阵的特征谱由乘积均为一的数对组成。我们的主要贡献是找到了一类非平凡的特征谱，其性质是：与该类中任意特征谱对应的量子协方差矩阵集合均可通过正交辛变换相互关联。我们证明了具有此性质的所有非简并特征谱必定属于该类，并且此类特征谱集合与识别高斯态物理相关热参数和压缩参数的非简并特征谱类相重合。
英文摘要：
The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices due to Heisenberg's uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix produces a positive-definite matrix that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a covariance matrix of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
