I already read book of Quantum Computation and Quantum Information by Nielsen and Chuang according to reduced density operator and I already understand how to do the reduced density using Dirac notation, Ket Bra.
My problem here I want to know the calculation/how to do reduced density operator if given In matrix form. Sorry because this is first time I write the formula.
As example, given density operator in matrix, [in Dirac notation is represented by $\frac{|00\rangle+|11\rangle}{\sqrt{2}}\;$]
$$\rho =\frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1\end{pmatrix}$$
has been transform by local unitary, U as example
$$A=\begin{pmatrix} a & b & c & 1 \\ 0 & c & d & 0 \\ 0 & e & f & 0 \\ 1 & g & h & 1\end{pmatrix}.$$
After the transformation, I will get new density operator
$$\rho_\text{new}=\begin{pmatrix} a+1 & 0 & 0 & a+1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2\end{pmatrix}.$$
My problem here, I want to measure the entanglement. In Dirac notation, I already not to trace the second qubit if I want to find partial trace of first qubit. But how to know in matrix which one represent the 1st or second qubit and how to did the partial trace.