I was reading TQI-notes by Watrous where they introduce different representations for quantum channels and wondering how to go from one to the other. I have:
\begin{align} &|\Phi(\rho)\rangle\!\rangle \tag{1} \\ &= K(\Phi)|\rho\rangle\!\rangle \tag{2} \\ &=|\text{tr}_2[(I\otimes \rho^T) J(\Phi)]\rangle\!\rangle \tag{3} \end{align}
where $\Phi$ is the channel, $\rho$ is the density operator of the quantum state input into the channel, $|\rangle\!\rangle$ represents vectorization, $K(\Phi)$ is the normal representation of the channel ($= \sum_i A_i \otimes \bar{A}_i$ where $A_i$ are the Kraus operators), and $J(\Phi)$ is the Choi representation ($=\sum_i |A_i\rangle\!\rangle \langle\!\langle A_i|$) of the channel.
Questions:
- Are the equations correct?
- Is there a more straightforward way to rewrite Eq. (3) in terms of $J(\Phi)$? Choi representation seems very convoluted so where can this be useful?
- I have omitted the Strinespring representation as I don't have a clear intuition for it but how can it be related to Eq.(1)?
