最优紧框架与量子测量
Optimal Tight Frames and Quantum Measurement
摘要：
紧框架和秩一量子测量被证明是密切相关的。事实上，量子力学系统所处空间的规范化紧框架族正是该空间上的秩一广义量子测量（POVM）族。利用这种关系，各种量子力学的概念和结果的框架理论的类似物开发。最小二乘量子测量的类似物是在最小二乘意义上最接近给定向量集的紧框架。最小二乘紧帧被发现的情况下，其中指定的帧的缩放（约束最小二乘帧（CLSF））和缩放是自由的情况下（无约束最小二乘帧（ULSF））。著名的规范框架被证明是成比例的ULSF和符合CLSF与一定的缩放。最后，对应于一个几何均匀向量集的规范框架向量被证明是几何均匀的，并具有相同的对称性作为原始向量集。
英文摘要：
Tight frames and rank-one quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank-one generalized quantum measurements (POVMs) on that space. Using this relationship, frame-theoretical analogues of various quantum-mechanical concepts and results are developed. The analogue of a least-squares quantum measurement is a tight frame that is closest in a least-squares sense to a given set of vectors. The least-squares tight frame is found for both the case in which the scaling of the frame is specified (constrained least-squares frame (CLSF)) and the case in which the scaling is free (unconstrained least-squares frame (ULSF)). The well-known canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling. Finally, the canonical frame vectors corresponding to a geometrically uniform vector set are shown to be geometrically uniform and to have the same symmetries as the original vector set.
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