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Volume 12
Issue 14
10.3390/math12142174
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Article

Quantum Injectivity of Frames in Quaternionic Hilbert Spaces

by
Zhenheng Xu
1,
Guoqing Hong
2,*,
Zuhua Guo
3 and
Jianxia Zhang
4
1
School of Vehicle and Traffic Engineering, Henan Institute of Technology, Xinxiang 453003, China
2
School of Science, Henan Institute of Technology, Xinxiang 453003, China
3
School of Computer Science and Technology, Henan Institute of Technology, Xinxiang 453003, China
4
School of Intelligent Engineering, Henan Institute of Technology, Xinxiang 453003, China
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(14), 2174; https://doi.org/10.3390/math12142174
Submission received: 1 June 2024 / Revised: 7 July 2024 / Accepted: 9 July 2024 / Published: 11 July 2024
(This article belongs to the Section A: Algebra and Logic)
Versions Notes

Abstract

A quantum injective frame is a frame capable of differentiating states based on their respective frame measurements, whereas the quantum-detection problem associated with frames endeavors to delineate all such frames. In the present paper, the concept of injective frames in infinite dimensional quaternionic Hilbert spaces is introduced. Further, some properties of injective frames such as the invariance of injective frames under invertible operators are discussed and several solutions to the frame quantum-detection problem are given. Finally, by employing operator theory and frames theory in quaternionic Hilbert spaces, some characterizations and classifications of frames for solving the injectivity problem are given.
Keywords: frame; quaternionic Hilbert space; quantum detection; quantum injectivity frame; quaternionic Hilbert space; quantum detection; quantum injectivity

Share and Cite

MDPI and ACS Style

Xu, Z.; Hong, G.; Guo, Z.; Zhang, J. Quantum Injectivity of Frames in Quaternionic Hilbert Spaces. Mathematics 2024, 12, 2174. https://doi.org/10.3390/math12142174

AMA Style

Xu Z, Hong G, Guo Z, Zhang J. Quantum Injectivity of Frames in Quaternionic Hilbert Spaces. Mathematics. 2024; 12(14):2174. https://doi.org/10.3390/math12142174

Chicago/Turabian Style

Xu, Zhenheng, Guoqing Hong, Zuhua Guo, and Jianxia Zhang. 2024. "Quantum Injectivity of Frames in Quaternionic Hilbert Spaces" Mathematics 12, no. 14: 2174. https://doi.org/10.3390/math12142174

APA Style

Xu, Z., Hong, G., Guo, Z., & Zhang, J. (2024). Quantum Injectivity of Frames in Quaternionic Hilbert Spaces. Mathematics, 12(14), 2174. https://doi.org/10.3390/math12142174

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