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.

Definition

To study a quantum channel ${\displaystyle {\mathcal {E}}}$ from system ${\displaystyle S}$ to ${\displaystyle S'}$, which is a trace-preserving complete positive map from operator spaces ${\displaystyle {\mathcal {L}}({\mathcal {H}}_{S})}$ to ${\displaystyle {\mathcal {L}}({\mathcal {H}}_{S'})}$, we introduce an auxiliary system ${\displaystyle A}$ with the same dimension as system ${\displaystyle S}$. Consider the maximally entangled state

${\displaystyle |\Phi ^{+}\rangle ={\frac {1}{\sqrt {d}}}\sum _{i=0}^{d-1}|i\rangle \otimes |i\rangle ={\frac {1}{\sqrt {d}}}(|0\rangle \otimes |0\rangle +\cdots +|d-1\rangle \otimes |d-1\rangle )}$

in the space of ${\displaystyle {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{S}}$, since ${\displaystyle {\mathcal {E}}}$is complete positive, ${\displaystyle I\otimes {\mathcal {E}}(|\Phi ^{+}\rangle \langle \Phi ^{+}|)}$ is a nonnegative operator. Conversely, for any nonnegative operator on ${\displaystyle {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{S'}}$, we can associate a complete positive map from ${\displaystyle {\mathcal {L}}({\mathcal {H}}_{S})}$to ${\displaystyle {\mathcal {L}}({\mathcal {H}}_{S'})}$, this kind of correspondece is called Choi-Jamiolkowski isomorphism.

References

1. Haapasalo, Erkka (2019-06-27). "The Choi-Jamiolkowski isomorphism and covariant quantum channels". arXiv:1906.11442v1. Bibcode:2019arXiv190611442H. Cite journal requires |journal= (help)
2. Choi, M. D. (1975). Completely positive linear maps on complex matrices. Linear algebra and its applications, 10(3), 285-290.
3. Jamiołkowski, A. (1972). Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3(4), 275-278.
4. Jiang, Min; Luo, Shunlong; Fu, Shuangshuang (2013-02-13). "Channel-state duality". Physical Review A. 87 (2): 022310. Bibcode:2013PhRvA..87b2310J. doi:10.1103/PhysRevA.87.022310. ISSN 1050-2947.