In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels (described by complete positive maps) and quantum states (described by density matrices), this is introduced by M. D. Choi and A. Jamiołkowski . It is also called channel-state duality by some authors in the quantum information area, but mathematically, this is a more general correspondence between positive operators and the complete positive superoperators.
To study a quantum channel from system to , which is a trace-preserving complete positive map from operator spaces to , we introduce an auxiliary system with the same dimension as system . Consider the Greenberger–Horne–Zeilinger state
in the space of , since is complete positive, is a nonnegative operator. Conversely, for any nonnegative operator on , we can associate a complete positive map from to , this kind of correspondece is called Choi-Jamiolkowski isomorphism.
- Haapasalo, Erkka (2019-06-27). "The Choi-Jamiolkowski isomorphism and covariant quantum channels".
- Choi, M. D. (1975). Completely positive linear maps on complex matrices. Linear algebra and its applications, 10(3), 285-290.
- Jamiołkowski, A. (1972). Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3(4), 275-278.
- Jiang, Min; Luo, Shunlong; Fu, Shuangshuang (2013-02-13). "Channel-state duality". Physical Review A. 87 (2). doi:10.1103/PhysRevA.87.022310. ISSN 1050-2947.
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