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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?

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    $\begingroup$ You've written the answer in your question. To compute how your state updates due using the Choi matrix of the process you compute it with the formula that you wrote which involves the Choi matrix. The transpose is simply a feature of the way the Choi matrix is constructed. $\endgroup$
    – Rammus
    Nov 4, 2023 at 8:23

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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.

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