How to get the quantum process back from its Choi matrix?

Question

The unnormalized maximally entangled bipartite state between a quantum system $S$ and an ancilla system $A$ is $|\psi\rangle=\sum_{k=1}^d|k\rangle_A|k\rangle_S$ , where $\{|k\rangle\}_{k=1}^d$ represents an orthonormal basis. A quantum process $\mathcal{E}$ acting only on the system $S$ of $|\psi\rangle$, the Choi matrix of the process $\mathcal{E}$ is given by $$ \Upsilon_{\mathcal{E}}=(\mathcal{I} \otimes \mathcal{E})(|\psi\rangle\langle\psi|)=\sum_{k, l=1}^d|k\rangle\langle l| \otimes \mathcal{E}(|k\rangle\langle l|), $$ Given the Choi matrix $\Upsilon_{\mathcal{E}}$, how can I get the process $\mathcal{E}$ by using the Choi matrix? In what situation will we use the transpose operation?

Answer 1

Given a map $\Phi$, define its Choi representation as $J(\Phi)=\sum_{ij}\Phi(E_{ij})\otimes E_{ij}$ with $E_{ij}\equiv |i\rangle\!\langle j|$. Then you can express the map from the Choi via $$\Phi(X) = \operatorname{tr}_2[J(\Phi)(I\otimes\rho^T)].$$ You can also work out directly the (a set of) Kraus operators. See e.g. Deduce the Kraus operators of the dephasing channel using the Choi for an example of this.