# Choi–Jamiołkowski isomorphism

In quantum information theory and operator theory, the **Choi–Jamiołkowski isomorphism**^{[1]} 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^{[2]} and A. Jamiołkowski ^{[3]}. It is also called channel-state duality by some authors in the quantum information area,^{[4]} but mathematically, this is a more general correspondence between positive operators and the complete positive superoperators.^{[citation needed]}

## Definition[edit]

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.

## References[edit]

**^**Haapasalo, Erkka (2019-06-27). "The Choi-Jamiolkowski isomorphism and covariant quantum channels". Cite journal requires`|journal=`

(help)**^**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.