Reduced Density operator in matrix form

Question

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.

Answer 1

Your example matrix $A$ isn't unitary. The right-most column has a length of 2 instead of 1, and the left-most column is only perpendicular to the right-most column if $a=-1$.

You also seem to have computed $\rho_{\text{new}} = A \rho$ instead of $\rho_{\text{new}} = A \rho A^\dagger$. You can tell your $\rho_{\text{new}}$ must be wrong at a glance because it's not Hermitian; the bottom-left isn't the conjugate-transpose of the upper-right.

The textbook you mentioned reading covers all of this, as well as how to compute a partial trace, in Chapter 2 (specifically section 2.4 The density operator).

Once you have the correct density matrix, you can measure the entanglement in various ways (also there are various measures). Tracing out one of the qubits and computing the Von Neumann entropy of what's left is one way. Another is to just arrange the 4 amplitudes into a 2x2 grid, pretend its a matrix, compute the SVD, and return the entropy of the squared singular values as if they were probabilities.

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You can use simulation toys like Quirk to get a feel for how density matrices change in response to operations: