Exercises: Chapter 5 — Multiple Qubits — Tensor Products, Entanglement, Bell States, and the Resource That Makes Quantum Computing Powerful

Exercise 5.1 — Separability Test. Determine whether each of the following two-qubit states is separable or entangled. For separable states, find the single-qubit factors.

(a) $|\psi_a\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$

(b) $|\psi_b\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |01\rangle)$

(c) $|\psi_c\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle - |11\rangle)$

(d) $|\psi_d\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$

Exercise 5.2 — Bell Basis Transformation. Show that the four Bell states form an orthonormal basis for $\mathbb{C}^4$. Verify that $\langle \Phi^+ | \Psi^+ \rangle = 0$ and compute all other inner products. Then show that the Bell measurement circuit (CNOT followed by H) maps each Bell state to a unique computational basis state.

Exercise 5.3 — Partial Measurement. The state $|\Psi\rangle = \frac{1}{2}|00\rangle + \frac{\sqrt{3}}{2}|11\rangle$ is measured: qubit 1 is measured in the computational basis. What are the possible outcomes, their probabilities, and the post-measurement states? Compute the reduced density matrix $\rho_2$ of the second qubit before measurement. Compare the purity $\text{Tr}(\rho_2^2)$ of $\rho_2$ with that of a maximally mixed state.

Exercise 5.4 — CHSH Simulation. Write a Qiskit program that computes the CHSH correlator $S$ for the singlet state $|\Psi^-\rangle$ using the optimal measurement angles. Verify that $S > 2$. Then try non-optimal angles (e.g., all measurements at angles 0, π/6, π/3, π/2) and observe how $S$ changes. At what angles does $S$ reach its maximum? Its minimum?

Exercise 5.5 — Entanglement Entropy. Compute the entanglement entropy $E(|\Psi\rangle)$ for the state $|\Psi\rangle = \sqrt{p}|00\rangle + \sqrt{1-p}|11\rangle$ as a function of $p \in [0, 1]$. Plot $E(p)$ and identify the maximum. What happens at $p = 0$ and $p = 1$? Verify that $E(p) = H(p) = -p\log_2 p - (1-p)\log_2(1-p)$.

Exercise 5.6 — Teleportation Verification. Implement the quantum teleportation protocol in Qiskit for an arbitrary state $|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$ with $\theta = \pi/3, \phi = \pi/4$. Verify that the teleported state matches the original by computing the fidelity. What happens if Alice's measurement results are lost?

Exercise 5.7 — Schmidt Decomposition. Find the Schmidt decomposition of the following states and compute their entanglement entropies: (a) $\frac{3}{5}|00\rangle + \frac{4}{5}|11\rangle$, (b) $\frac{1}{\sqrt{6}}|00\rangle + \frac{1}{\sqrt{6}}|01\rangle + \frac{1}{\sqrt{6}}|10\rangle + \sqrt{\frac{3}{6}}|11\rangle$, (c) $\frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$.

Exercise 5.8 — Superdense Coding. Implement the superdense coding protocol in Qiskit. Verify that Alice can encode all four possible two-bit messages by applying $\{I, X, Z, XZ\}$ to her half of a Bell pair, and that Bob can decode the message by performing a Bell measurement.

Exercise 5.9 — No-Cloning vs CNOT. Show explicitly that applying CNOT to $|\psi\rangle \otimes |0\rangle$ for $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ produces the entangled state $\alpha|00\rangle + \beta|11\rangle$, not the product state $|\psi\rangle \otimes |\psi\rangle$. Compute the reduced density matrix of each qubit in the entangled state and show they are both mixed.

Exercise 5.10 — GHZ Mermin Game. Consider the three-qubit GHZ state $|GHZ\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$. Alice, Bob, and Charlie each receive one qubit and choose to measure in the X or Y basis. Show that if they all measure X, the product of outcomes equals +1. If two measure Y and one measures X, the product equals -1. This is a three-party Bell inequality violation.