Reduced Density operator in matrix form

[munirah]

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

 

[anathema]

for sharing your understanding of reduced density operators and your attempt at solving this problem. To calculate the reduced density operator using matrix notation, you can follow these steps:

1. Write out the density operator $\rho$ in matrix form, as given in the question.
2. Write out the unitary transformation $U$ in matrix form, as given in the question.
3. Multiply the two matrices $\rho$ and $U$ together to get the new density operator $\rho_\text{new}$.
4. To find the partial trace of the first qubit, we need to sum over the elements of the matrix that correspond to the second qubit. In this case, the second qubit corresponds to the last two columns of the matrix. So, we can sum over the elements in these two columns to get the reduced density operator for the first qubit.
5. The elements in the first column of the reduced density operator correspond to the state $|0\rangle$, and the elements in the second column correspond to the state $|1\rangle$. So, the partial trace of the first qubit can be written as:

$$\rho_1 = \begin{pmatrix} a+1 & a+1 \\ 2 & 2\end{pmatrix}.$$

6. To measure the entanglement, you can calculate the von Neumann entropy of the reduced density operator $\rho_1$. This will give you a measure of the entanglement between the first and second qubits after the transformation.

I hope this helps! Let me know if you have any further questions.