# Choi–Jamiołkowski isomorphism

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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.[citation needed]

## Definition

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

$|\Phi ^{+}\rangle ={\frac {1}{\sqrt {d}}}\sum _{i=0}^{d-1}|i\rangle \otimes |i\rangle ={\frac {1}{\sqrt {d}}}(|0\rangle \otimes |0\rangle +\cdots +|d-1\rangle \otimes |d-1\rangle )$ in the space of ${\mathcal {H}}_{A}\otimes {\mathcal {H}}_{S}$ , since ${\mathcal {E}}$ is complete positive, $I\otimes {\mathcal {E}}(|\Phi ^{+}\rangle \langle \Phi ^{+}|)$ is a nonnegative operator. Conversely, for any nonnegative operator on ${\mathcal {H}}_{A}\otimes {\mathcal {H}}_{S'}$ , we can associate a complete positive map from ${\mathcal {L}}({\mathcal {H}}_{S})$ to ${\mathcal {L}}({\mathcal {H}}_{S'})$ , this kind of correspondece is called Choi-Jamiolkowski isomorphism.