A Note on Injective g-Frames in Quaternionic Hilbert Spaces

Abstract

Motivated by recent advancements in the quantum detection problem employing both discrete and continuous frames, this paper delves into a quantum detection problem utilizing g-frames within the context of quaternionic Hilbert spaces. We offer several equivalent representations of injective g-frames in separable quaternionic Hilbert spaces. By normalizing the trace, we establish a classification for the g-frame injectivity problem. Additionally, we propose a method to derive an injective g-frame by leveraging an injective frame within quaternionic Hilbert spaces. Furthermore, we demonstrate that the injectivity of a g-frame remains intact under a linear isomorphism, while injective g-frames exhibit instability in infinite-dimensional scenarios.

Keywords

• g-frame
• quantum injectivity
• quaternionic Hilbert space
• quantum detection