Continuous Injective Function With Compact Image

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.

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Acknowledgements

The authors would like to thank the referee for carefully reading the manuscript and providing many helpful comments and suggestions that help improve the quality of its presentation. Deguang Han is partially supported by NSF DMS-1712602, and Rui Liu is supported by the National Science Foundation of China (11671214, 11971348), "the Hundred Young Academia Leaders Program of Nankai University" (91923104, 91823003, 63174012), and "the Fundamental Research Funds for the Central Universities" (63191503, 63171225).

Author information

Authors and Affiliations

1. Department of Mathematics, University of Central Florida, Orlando, USA

   Deguang Han

2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China

   Qianfeng Hu & Rui Liu

Corresponding author

Correspondence to Rui Liu.

Additional information

Communicated by Joseph Ball.

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Cite this article

Han, D., Hu, Q. & Liu, R. Injective continuous frames and quantum detections. Banach J. Math. Anal. 15, 12 (2021). https://doi.org/10.1007/s43037-020-00086-7

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• Received: 27 February 2020

• Accepted: 23 July 2020

• Published: 14 October 2020

• DOI : https://doi.org/10.1007/s43037-020-00086-7

Keywords

• Quantum detection
• Quantum injectivity
• Continuous frame
• Positive operator-valued measure

Mathematics Subject Classification

• 42C15
• 46L10
• 47A05
• 42C99
• 46C10
• 47B38

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