Quantum injectivity of multi-window Gabor frames in finite dimensions
Quantum injectivity of multi-window Gabor frames in finite dimensions
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DOI：
10.1007/s43034-022-00208-2
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发表时间：
2022-08
影响因子：
1
通讯作者：
D. Han;Qianfeng Hu;Rui Liu;Heying Wang
中科院分区：
文献类型：
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作者：
D. Han;Qianfeng Hu;Rui Liu;Heying Wang
近年来，有限Gabor框架在信号处理，特别是相位恢复中得到了广泛的研究。相位恢复问题要求用正算符值测度（POVM）区分纯态和它们的量子测量，而量子检测问题要求区分所有态和它们的测量。受离散帧和连续帧的量子检测问题研究的启发，本文研究了多窗口Gabor帧的量子检测问题。首先给出了多窗Gabor框架是量子内射的一个充分必要条件。这推广了单窗口情况下的已知结果。作为这个特征的结果，所有的多窗口生成器的集合为内射的Gabor框架是Zerkki稠密的，因此，每个一般的多窗口的Gabor框架是内射的，所有的注入窗口的Gabor框架的集合是稳定的扰动。特别是，我们提出了一个定量的稳定性结果的指标之一。最后给出了一些例子来说明多窗Gabor帧的这种特征化的必要性。
Finite Gabor frames forhave been extensively studied in the context of signal processing, in particular, phase-retrieval in recent years. While the phase-retrieval problem asks to distinguish the pure states from their quantum measurements with a positive operator valued measure (POVM), the quantum detection problem asks to distinguish all the states from their measurements. Inspired by some recent work on the quantum detection problem by (discrete) frames and continuous frames, in this paper we examine the quantum detection problem with multi-window Gabor frames. We firstly obtain a necessary and sufficient condition in terms of the window vectors for a multi-window Gabor frame to be quantum injective. This generalizes the known result for the single-window case. As a consequence of this characterization, the set of all the multi-window generatorsfor injective Gabor frames is Zariski dense inand consequently every generic multi-window Gabor frame is injective and the set of all the injectives-window Gabor frames is stable under perturbation. In particular, we present a quantitative stability result for one of the metrics. Some examples are also provided to demonstrate the necessity of such a characterization for multi-window Gabor frames.