Abstract
Finite Gabor frames for \({\mathbb {C}}^N\) have 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 generators \((\varphi _1, \ldots , \varphi _s)\) for injective Gabor frames is Zariski dense in \({\mathbb {C}}^{N}\oplus \cdots \oplus {\mathbb {C}}^N,\) and consequently every generic multi-window Gabor frame is injective and the set of all the injective s-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.
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Acknowledgements
The authors would like to express their great appreciation to the referees for valuable suggestions which led to the current version of the paper. Deguang Han acknowledges the support from NSF under the Grant DMS-2105038. Qianfeng Hu, Rui Liu and Heying Wang acknowledge partial support by National Natural Science Foundation of China (11671214, 11971348, 12071230), Hundred Young Academia Leaders Program of Nankai University (63223027, ZB22000105), Undergraduate Education and Teaching Project of Nankai University (NKJG2022053), and National College Students’ Innovation and Entrepreneurship Training Program of Nankai University (202210055048).
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Han, D., Hu, Q., Liu, R. et al. Quantum injectivity of multi-window Gabor frames in finite dimensions. Ann. Funct. Anal. 13, 59 (2022). https://doi.org/10.1007/s43034-022-00208-2
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DOI: https://doi.org/10.1007/s43034-022-00208-2