Injective continuous frames and quantum detections

Abstract

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.