Is it possible to partial trace the $\chi$-matrix of $4$ qubits $q_0,q_1,q_2,q_3$ to obtain a description of what happens to $q_1$?

Considering a $$\chi$$-matrix of a circuit with, say, 4 qubits, is it possible to trace out 3 of them from $$\chi$$ - for example qubits $$q_0$$, $$q_2$$ and $$q_3$$ - thus gaining the process matrix describing what happens to $$q_1$$? If yes, since it seems to me different from partial trace for a density matrix, could you please give me the definition?

The $$\chi$$-matrix of an $$n$$-qubit quantum channel $$\mathcal{E}$$ is a matrix $$\chi$$ such that the evolution of a density matrix $$\rho$$ is given by $$\begin{gather} \mathcal{E}(\rho) = \sum_{i,j}\chi_{i,j}P_i \rho P_j \end{gather}$$

where $$\{P_0, P_1,\dots ,P_{d^{2}−1}\}$$ is the $$n$$-qubit Pauli basis containing $$d^{2} = 4^{n}$$ elements.

An estimation method to compute the $$\chi$$-matrix is available at 8.4.2 of Nielsen & Chuang's Quantum Computation and Quantum Information.

• can you add the definition of $\chi$-matrix you are going by? Or if you are just asking what is the definition of $\chi$-matrix, why doesn't quantumcomputing.stackexchange.com/a/11814/55 answer it?
– glS
Aug 1 '20 at 11:39
• hope it's okay now! Aug 5 '20 at 10:15