# Reduced Density operator in matrix form

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1. Jun 18, 2016

### munirah

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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

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

3. The attempt at a solution

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

2. Jun 23, 2016