Abstract
A generalization of the Choi–Jamiołkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels, the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in three dimensions.
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Acknowledgements
The author would like to thank Dr. Juha-Pekka Pellonpää, Dr. Jukka Kiukas, and Dr. Roope Uola for reading earlier versions of the manuscript and for providing their constructive feedback. Also, Dr. Yui Kuramochi is thanked for pointing out an error in an earlier version of this manuscript. This research has received funding from the National Natural Science Foundation of China (Grant No. 11875110).
Funding
This research has received funding from the National Natural Science Foundation of China (Grant No. 11875110).
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On behalf of all authors, Erkka Haapasalo states that there is no conflict of interest.
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Haapasalo, E. The Choi–Jamiołkowski isomorphism and covariant quantum channels. Quantum Stud.: Math. Found. 8, 351–373 (2021). https://doi.org/10.1007/s40509-021-00249-7
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DOI: https://doi.org/10.1007/s40509-021-00249-7