Checkpoint: Work with Density Matrices

This checkpoint chains together the whole module: build a joint density matrix, take a partial trace, and read off both a purity and an expectation value from the reduced state.

What you are computing

Start from the partially entangled pure state

|\psi\rangle = \cos t\,|00\rangle + \sin t\,|11\rangle, \qquad t = \frac{\pi}{8}.

Its joint density matrix is $\rho_{AB} = |\psi\rangle\langle\psi|$ . Tracing out the second qubit gives a diagonal reduced state — the squared Schmidt coefficients on the diagonal:

\rho_A = \operatorname{Tr}_B\,\rho_{AB} = \begin{pmatrix}\cos^2 t & 0\\ 0 & \sin^2 t\end{pmatrix}.

From $\rho_A$ you extract two numbers seen throughout the module:

\text{purity} = \operatorname{Tr}(\rho_A^2) = \cos^4 t + \sin^4 t, \qquad \langle Z\rangle_A = \operatorname{Tr}(\rho_A Z) = \cos^2 t - \sin^2 t = \cos 2t.

Checks to keep you honest

The reduced state is mixed because the original is entangled, so the purity must be strictly below 1 (and above $1/2$ ).
Because $\rho_A$ is diagonal, $\langle Z\rangle_A$ is simply the difference of its diagonal entries — here $\cos(\pi/4) = \sqrt2/2 \approx 0.707$ .
If you accidentally forget to trace out $B$ and instead compute the purity of the full $\rho_{AB}$ , you will get exactly $1$ (the joint state is pure) — a useful sanity check that you traced correctly.

Try it

Carry out all three steps — build $\rho_{AB}$ , trace out $B$ , then compute the purity and $\langle Z\rangle_A$ of $\rho_A$ — and return their sum $\operatorname{Tr}(\rho_A^2) + \langle Z\rangle_A$ .

Run your code to see the quantum state.