Choi–Jamiołkowski Isomorphism

In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels (described by completely positive maps) and quantum states (described by density matrices), this is introduced by Man-Duen Choi and Andrzej 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.

Definition

To study a quantum channel $\mathcal$ from system $S$ to $S'$, which is a trace-preserving completely positive map from operator spaces $\mathcal(\mathcal_S)$ to $\mathcal(\mathcal_)$, we introduce an auxiliary system $A$ with the same dimension as system $S$. Consider the maximally entangled state

: $, \Phi^+\rangle=\frac\sum_^, i\rangle\otimes, i\rangle=\frac(, 0\rangle\otimes, 0\rangle+\cdots+, d-1\rangle\otimes, d-1\rangle)$

in the space of $\mathcal_A\otimes\mathcal_S$. Since $\mathcal$ is completely positive, $(I_A\otimes \mathcal)(, \Phi^+\rangle\langle\Phi^+, )$ is a nonnegative operator. Conversely, for any nonnegative operator on $\mathcal_A\otimes\mathcal_$, we can associate a completely positive map from $\mathcal(\mathcal_S)$to $\mathcal(\mathcal_)$. This kind of correspondence is called Choi-Jamiołkowski isomorphism.

References

{{DEFAULTSORT:Choi-Jamiolkowski isomorphism
Quantum information theory