Doubts on Choi-Jamiolkowsky (CJ) matrix formulation

Question

We can represent a CP map $\mathcal{M}$ by a positive semidefinite matrix via Choi-Jamiolkowsky (CJ) isomorphism. The CJ matrix $M$ ∈ $L(H_1 \otimes H_2)$ corresponding to a linear map $\mathcal{M}$ : $L(H_1) \rightarrow L(H_2)$ is defined as $$M := I \otimes \mathcal{M} (|\psi\rangle\langle\psi|)$$ where $|\psi\rangle = \sum_{j=1}^{d}|jj\rangle ∈ H_1 ⊗ H_1$ is a maximally entangled state (not normalized), the set of states ${|j\rangle}_{j=1}^{d}$ is an orthonormal basis of $H_1$ and $I$ is the identity map.

In the appendix part of the paper 'Quantum correlations with no causal order', while calculating the probabilities for violation of the causal inequality the paper admits the following:

A CP map corresponding to the detection of a state $|\psi\rangle$ and repreparation of another state $|\phi\rangle$ has CJ matrix of the form: $|\psi\rangle\langle\psi| \ \otimes |\phi\rangle\langle\phi|$

I am sure this statement is general. But I am not able to see how do we get this form from the definition. Could someone provide some intuition?

Answer 1

Since you asked for intuition, let me give you my intuition: The first part of the Choi state tells you what happens to the input, the second part what will be the output.

While this might be a bit vague in the scenario of an arbitrary channel, in the case of this specific channel it is fully precise: The first component is $|\psi\rangle\langle\psi|$ -- it describes that the input is projected onto the state $|\psi\rangle$ (or possibly $|\psi^*\rangle$, this would require more careful analysis). The second component is $|\phi\rangle\langle\phi|$ -- it describes the output which is the state $|\phi\rangle$.

This corresponds to the channel $\mathcal M(\rho) = \langle\psi|\rho|\psi\rangle\, |\phi\rangle\langle\phi|$.