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.

  • $\begingroup$ 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? $\endgroup$
    – glS
    Aug 1 '20 at 11:39
  • $\begingroup$ hope it's okay now! $\endgroup$ Aug 5 '20 at 10:15

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