A quantum injective frame is a frame that can be used to distinguish density operators (states) from their frame measurements, and the frame quantum detection problem asks to characterize all such frames. This problem was recently settled in Botelho-Andrade et al. (Springer Proc Math Stat 255:337–352, 2017) and Botelho-Andrade et al. (J Fourier Anal Appl 25:2268–2323, 2019) mainly for finite or infinite but discrete frames. In this paper, we consider the continuous frame version of the quantum detection problem. Instead of using the frame element itself, we use discrete representations of continuous frames to obtain several versions of characterizations for quantum injective continuous frames. With the help of these characterizations, we also examine the issues involving constructions and stability of continuous quantum injective frames. In particular, we show that injective continuous frames are stable for finite-dimensional Hilbert spaces but unstable for infinite-dimensional cases.

中文翻译：

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注入连续帧和量子检测

量子注入框架是一种框架，可用于从它们的框架测量中区分密度算子（状态），框架量子检测问题要求表征所有此类框架。这个问题最近在 Botelho-Andrade 等人中得到解决。(Springer Proc Math Stat 255:337–352, 2017) 和 Botelho-Andrade 等人。(J Fourier Anal Appl 25:2268–2323, 2019) 主要用于有限或无限但离散的框架。在本文中，我们考虑了量子检测问题的连续帧版本。我们不使用框架元素本身，而是使用连续框架的离散表示来获得量子注入连续框架的多个版本的表征。在这些表征的帮助下，我们还研究了涉及连续量子注入框架的构造和稳定性的问题。