# When should I use the Choi matrix and when should I use the $\chi$ matrix?

A quantum map on a $$d$$-dimensional space has the general representation: $$\mathcal{S}(\rho)=\sum_{\alpha,\beta}^{d^2}\chi_{\alpha\beta}\Gamma_{\alpha}\rho \Gamma_{\beta}^{\dagger},$$ where $$\chi$$ is the $$d^2\times d^2$$ process matrix, which is positive semidefinite and trace preserving.

On the other hand, 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. For a quantum process $$\mathcal{E}$$ acting only on the system $$S$$ of $$|\psi\rangle$$, the output state is given by $$\Upsilon_{\mathcal{E}}=(\mathcal{I} \otimes \mathcal{E})(|\psi\rangle\langle\psi|)=\sum_{k, l=1}^d|k\rangle\langle l| \otimes \mathcal{E}(|k\rangle\langle l|),$$ which is called the Choi matrix of the process $$\mathcal{E}$$.

Since they can all represent quantum process, so why this paper of process tomography uses $$\chi$$ matrix?

The two descriptions are entirely equivalent. It doesn't matter which you use when, it's just a case of using whichever description you personally find to be mathematically the most convenient.

• Is there any reason as to why someone would use the process matrix? Computing the Choi matrix seems much much easier. Commented Nov 3, 2023 at 13:46
• Well, let's take a silly extreme: the map is just a unitary. If I want to apply this unitary to a density matrix, I would much rather just calculate $U\rho U^\dagger$ than have the calculate $\text{Tr}_1(\rho^T\otimes U\cdot |\Omega\rangle\langle\Omega|\cdot I\otimes U^\dagger)$ Commented Nov 3, 2023 at 13:55

This is just a comment, but it's too long for a comment, so writing as an answer. As I haven't read the paper you are asking about, I cannot answer as to particularly why that paper is using the process matrix. Will update my answer later if I get time.

A map $$\mathcal{E}$$ is completely positive if and only if the Choi matrix $$\Upsilon_\mathcal{E}$$ is a positive matrix.

Equivalently, the super-operator $$\mathcal{E}$$, acting on any operator $$A$$ can be written as $$\mathcal{E}(A) = \sum_{\alpha,\beta}^{d^2}\chi_{\alpha\beta}\Gamma_{\alpha}A \Gamma_{\beta}^{\dagger} \,,$$ where $$\chi$$ is a positive matrix in some operator basis $$\{\Gamma_{\alpha}\}$$. However, if your $$\mathcal{E}$$ is not a CPTP map, then you will not be able to write the above equation with positive $$\chi$$.

Both the Choi matrix and the process matrix accomplish the same thing, which is to verify the complete positivity of your map/process $$\mathcal{E}$$, i.e., $$\mathcal{E}$$ correspond to some valid physical quantum process or not$$^1$$.

Generally, if you have $$\mathcal{E}$$ given, it's easier to calculate the Choi matrix to verify the complete positivity of $$\mathcal{E}$$.

Fun fact: $$\Upsilon_\mathcal{E}$$ and $$\chi$$ are related by just a unitary transformation.
1: If you need some intuition behind this statement, my this answer to another question would be helpful.