AI 中文总结
该研究拓展量子信息中相位恢复的研究范围，将纯态推广到有界秩半正定算子、量子信道推广到一般超算子，推导了相关稳定性结果并刻画了块对角保厄米超算子的相位可恢复性条件。
AI 中文摘要
我们延续量子信息背景下的相位恢复研究，遵循Liu等人2023年的研究范式，在两个主要方面进行了更具一般性的拓展：(a) 除纯态外，还考虑了有界秩的态（及半正定算子），这与量子层析成像的做法一致；(b) 考虑了一般超算子，而非仅量子信道。我们证明，要实现针对有界秩半正定算子的相位恢复，只需区分完全可区分的对（即具有正交支撑的对）。由此我们证明，相位可恢复性关于迹范数始终是Lipschitz稳定的，随后利用Fuchs-van de Graaf不等式推导出关于Bures-Wasserstein距离的稳定性（从而推广了框架理论中相位恢复的已知稳定性结果）。对于取值为块对角矩阵的保厄米超算子，我们基于框架理论中经典互补性质的启发式条件刻画了其相位可恢复性。
英文摘要
We continue the study of Phase Retrieval in the Quantum Information setting in the style of \cite{liu2023phase}, in a manner which is more general in two primary ways: (a) Instead of only pure states we consider also states (and positive semidefinite operators) of bounded rank, as is done in Quantum Tomography; (b) We consider general super operators instead of only quantum channels. We show that in order to have phase retrieval with respect to positive semidefinite operators of bounded rank, it suffices to discriminate between perfectly distinguishable pairs (i.e. those with orthogonal supports). From this we prove that phase retrievability is always Lipschitz stable with respect to the trace norm, and then we use the Fuchs-van de Graaf inequalities to deduce stability with respect to the Bures--Wasserstein distance (thus generalizing the known stability results for phase retrieval in Frame Theory). For Hermitian-preserving super operators taking values in block-diagonal matrices, we characterize phase retrievability in terms of conditions inspired by the classical complement property from Frame Theory.
Comments31 pages
