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For example, given a $$2$$-qubit system with density matrix $$$$\rho_{AB}=\begin{pmatrix} \rho_{11} &\rho_{12} &\rho_{13} &\rho_{14} \\ \rho_{21} &\rho_{22} &\rho_{23} &\rho_{24} \\ \rho_{31} &\rho_{32} &\rho_{33} &\rho_{34} \\ \rho_{41} &\rho_{42} &\rho_{43} &\rho_{44} \end{pmatrix},$$$$ one can compute the partial traces with respect to either system:
• if we "trace out" the second system, which we will label with $$B$$, we get the residual state: $$$$\newcommand{\Tr}{\operatorname{Tr}} \rho_{A}=\Tr_{B}(\rho_{AB})=\begin{pmatrix} \Tr\begin{pmatrix} \rho_{11} &\rho_{12} \\ \rho_{21} &\rho_{22} \end{pmatrix} &\Tr\begin{pmatrix} \rho_{13} &\rho_{14} \\ \rho_{23} &\rho_{24} \end{pmatrix}\\ \Tr\begin{pmatrix} \rho_{31} &\rho_{32} \\ \rho_{41} &\rho_{42} \end{pmatrix} &\Tr\begin{pmatrix} \rho_{33} &\rho_{34} \\ \rho_{43} &\rho_{44} \end{pmatrix} \end{pmatrix}= \begin{pmatrix} \rho_{11}+\rho_{22} & \rho_{13}+\rho_{24}\\ \rho_{31}+\rho_{24} & \rho_{33}+\rho_{44} \end{pmatrix}$$$$
• if we "trace out" the first system, labelled with $$A$$, the residual state is: $$$$\rho_{B}=\Tr_{A}(\rho_{AB})=\begin{pmatrix} \Tr\begin{pmatrix} \rho_{11} &\rho_{13} \\ \rho_{31} &\rho_{33} \end{pmatrix} &\Tr\begin{pmatrix} \rho_{12} &\rho_{14} \\ \rho_{32} &\rho_{34} \end{pmatrix}\\ \Tr\begin{pmatrix} \rho_{21} &\rho_{23} \\ \rho_{41} &\rho_{43} \end{pmatrix} &\Tr\begin{pmatrix} \rho_{22} &\rho_{24} \\ \rho_{42} &\rho_{44} \end{pmatrix} \end{pmatrix}= \begin{pmatrix} \rho_{11}+\rho_{33} & \rho_{12}+\rho_{34}\\ \rho_{21}+\rho_{43} & \rho_{22}+\rho_{44} \end{pmatrix}.$$$$