Can any expert help me in explaining how this example below get the reduced density matrix from the density matrix in bipartite system.(adsbygoogle = window.adsbygoogle || []).push({});

$$\rho =\frac{1}{4}\begin{pmatrix} 1 & 1 & cos(\frac{\alpha}{2})-sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{2})+sin(\frac{\alpha}{2}) \\ 1 & 1 & cos(\frac{\alpha}{2})-sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{2})+sin(\frac{\alpha}{2}) \\ cos(\frac{\alpha}{2})-sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{2})-sin(\frac{\alpha}{2}) & 1-sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{4}) \\ cos(\frac{\alpha}{2})+sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{2})+sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{4}) & 1+sin(\alpha)\end{pmatrix}$$

For the subsystems, this yields, as below.

$$\rho_\text{A}=Tr_\text{B}(\rho)=\frac{1}{2}\begin{pmatrix}1 & cos(\frac{\alpha}{2}) \\ cos(\frac{\alpha}{2}) & 1\end{pmatrix}$$

and

$$\rho_\text{B}=Tr_\text{A}(\rho)=\frac{1}{2}\begin{pmatrix}1-\frac{1}{2}sin(\alpha) & cos^2(\frac{\alpha}{2}) \\ cos^2(\frac{\alpha}{2}) & 1+\frac{1}{2}sin(\alpha)\end{pmatrix}$$

How the reduced to each system A and B obtained?

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# I How reduced density matrix obtained from the matrix.

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