Checkpoint: Analyze a Channel

This checkpoint ties the module together. You will take a concrete channel, apply it to a coherent superposition through its Kraus operators, and read off a quantitative property of the output — the kind of end-to-end calculation that underlies every noise analysis.

What to recall

• A channel acts as $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ in the operator-sum form.
• Amplitude damping with rate $\gamma$ has $K_0=\operatorname{diag}(1,\sqrt{1-\gamma})$ and $K_1=\sqrt{\gamma}\,|0\rangle\langle1|$. It damps the excited population, transfers it to the ground state, and shrinks coherences by $\sqrt{1-\gamma}$.
• The purity $\operatorname{tr}(\rho^2)$ measures how mixed a state is: $1$ for a pure state, $1/d$ for the maximally mixed state. For a qubit, $\operatorname{tr}(\rho^2)=\tfrac12(1+|\vec r\,|^2)$.

The problem

Prepare the coherent superposition $|+\rangle = \tfrac{1}{\sqrt2}(|0\rangle+|1\rangle)$, so $\rho = |+\rangle\langle+| = \tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix}$. Send it through the amplitude-damping channel with $\gamma = \tfrac12$. Then compute the purity of the output state, $\operatorname{tr}\big(\mathcal{E}(\rho)^2\big)$.

Working it out by hand

Applying the channel to $\rho$:

The purity is the sum of the squared magnitudes of all entries (since the matrix is Hermitian, $\operatorname{tr}(\rho^2)=\sum_{ij}|\rho_{ij}|^2$):

The pure input ($\text{purity }1$) has been driven to a partially mixed output — damping has injected entropy. Confirm $0.875$ by building $\mathcal{E}(\rho)$ from the Kraus operators and squaring.

Your task

Build $\mathcal{E}(\rho)$ for $\rho=|+\rangle\langle+|$ and $\gamma=\tfrac12$ using $\mathcal{E}(\rho)=K_0\rho K_0^\dagger + K_1\rho K_1^\dagger$, then return $\operatorname{tr}\big(\mathcal{E}(\rho)^2\big)$.

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