# Reduced Density operator in matrix form

## Homework Statement

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.

## Homework Equations

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My problem here I want to know the calculation/how to do reduced density operator if given In matrix form.
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

$$\rho_\text{new}=\begin{pmatrix} a+1 & 0 & 0 & a+1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2\end{pmatrix}.$$

## The Attempt at a Solution

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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 from given matrix.

Thank you