Choi–Jamiołkowski isomorphism

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.