# calculate the reduced density matrix of a 2 qubit state and compare the eigenvalues

So I have the exercise to apply a Cz gate to the following 2 Qubit state

$$|a\rangle \otimes |b\rangle = (a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)\\\\$$

Afterwards, I should find the reduced density matrix for both the qubits and show that they are the same for both. So i got the following reduced density matrix.

$$\begin{pmatrix} \mid b_0 \mid^2 & b_0\overline{b_1}(\mid a_0\mid^2 - \mid a_1\mid^2) \\ b_1\overline{b_0}(\mid a_0\mid^2 - \mid a_1\mid^2)& \mid b_1 \mid^2 \end{pmatrix}$$

for the Subsystem of my second qubit and for the subsystem for the first

$$\begin{pmatrix} \mid a_0 \mid^2 & a_1\overline{a_0}(\mid b_0\mid^2 - \mid b_1\mid^2) \\ a_0\overline{a_1}(\mid b_0\mid^2 - \mid b_1\mid^2)& \mid a_1 \mid^2 \end{pmatrix}$$

But now I cant figure out how the eigenvalues of these matrices are the same

• The density matrices look fine to me, and they both have a determinant $4|a_0a_1b_0b_1|^2$. Oct 20, 2023 at 6:22