Problem 1. Consider the density matrix:

$$\rho =\left(\begin{array}{cc}3/4& \frac{\sqrt{2}}{4}{e}^{-\mathit{i}\varphi }\\ \frac{\sqrt{2}}{4}{e}^{-\mathit{i}\varphi }& 1/4\end{array}\right)$$

go through the conditions to ensure that this matrix is indeed a density matrix. Furthermore is the matrix a pure or mixed state, i.e. $\rho =|\psi ⟩⟨\psi |$ or $\rho =\sum _i{\mathrm{p}}_{\mathrm{i}}|{\psi }_i⟩⟨{\psi }_i|$.

Also calculate the $\mathrm{t}\mathrm{r}({\sigma }_1\rho )$.

Solution 1. The criteria that a matrix be a density matrix is the following:

1. Positive semidefinite operator, therefore is Hermitian with eigenvalues $\ge 0$.

2. Has $\mathrm{t}\mathrm{r}(\rho )=1$

So the easiest condition to check is the second one by taking the trace of the ensity matrix, as below:

It satisfies this criteria. Now we need to check if its positive semidefinite. Since this requires that its Hermitian, we can check that $\rho ={\rho }^{†}$.

ρ == ρ.adjoint()

$\rho$ is Hermitian.

Now to see if the is postive valued, we can get the eigenspectrum[1]:

$e_1=\frac{1}{2}-\frac{\sqrt{3}}{4}=0.067$

$e_2=\frac{\sqrt{3}}{4}+\frac{1}{2}=0.933$

as we can see $\rho$ is indeed positive and semidefinite and therefore a density matrix.

The next part is to determine if the density matrix corresponds to a pure or mixed state.

For a pure state we can check that ${\rho }^2=|\psi ⟩⟨\psi |\psi ⟩⟨\psi |=\rho$:

so its not a pure state but a mixed state. Finally we want to evaluate the expectation of the Pauli operator ${\sigma }_x$ using $\mathrm{t}\mathrm{r}({\sigma }_x\rho )$.