Exercises: Chapter 9 — Quantum Teleportation: Transmitting Quantum Information Using Entanglement (It's Not Sci-Fi — It's a Protocol)

Exercise 9.1: Derive the Teleportation Algebra

Starting from the initial state $|\Psi_0\rangle = (\alpha|0\rangle_A + \beta|1\rangle_A) \otimes |\Phi^+\rangle_{BC}$, derive the state after each step of the protocol. Verify that Bob's final state (after correction) is exactly $\alpha|0\rangle + \beta|1\rangle$ for all four measurement outcomes. Show your work for each outcome.

Exercise 9.2: Teleport an Entangled Qubit

Modify the teleportation circuit to teleport one qubit of an entangled pair. Start with a Bell state on qubits 0 and 1, then teleport qubit 0 to qubit 3 (using qubit 2 as Alice's half of the EPR pair). Verify that qubits 1 and 3 are now entangled.

Hint: The circuit should have 4 qubits. Qubits 0 and 1 start in a Bell state. Qubits 2 and 3 form the shared EPR pair. Teleport qubit 0 using the EPR pair (qubits 2 and 3), and then check the entanglement between qubits 1 and 3.

Exercise 9.3: Fidelity Under Noise

Implement the teleportation protocol with varying levels of depolarizing noise (0.1%, 0.5%, 1%, 2%, 5%, 10%). For each noise level, compute the fidelity $F = \langle \psi | \rho_{\text{out}} | \psi \rangle$ between the input state and Bob's output state. Plot fidelity vs. noise level. At what noise level does the fidelity drop below the classical limit of 2/3?

Exercise 9.4: Entanglement Swapping

Implement a three-node entanglement swapping chain: A ↔ R1 ↔ R2 ↔ B. Verify that after both swapping operations, A and B share a Bell state. Compute the fidelity as a function of the number of repeater nodes, assuming each link has fidelity $F = 0.95$.

Exercise 9.5: The No-Signaling Proof

Prove mathematically that without the classical communication, Bob's reduced density matrix is $I/2$ regardless of Alice's input state. This proves that teleportation cannot be used for faster-than-light signaling. Verify numerically using Qiskit.

Exercise 9.6: Teleportation with Werner State Entanglement

(a) Simulate teleportation where the shared entanglement is a Werner state $\rho_W = F|\Phi^+\rangle\langle\Phi^+| + \frac{1-F}{3}(I - |\Phi^+\rangle\langle\Phi^+|)$ for $F \in \{0.5, 0.6, 0.7, 0.8, 0.9, 1.0\}$.

(b) Plot the teleportation fidelity vs. $F$. Verify the theoretical formula $\mathcal{F} = (2F+1)/3$.

(c) At what value of $F$ does the teleportation fidelity drop below the classical limit of 2/3?

Exercise 9.7: Teleportation Circuit Optimization

(a) The standard teleportation circuit uses 2 CNOTs and 1 Hadamard for the Bell measurement, plus up to 2 single-qubit corrections. Can you find a more efficient implementation (fewer total gates)?

(b) Transpile the teleportation circuit for a heavy-hex topology (like ibm_brisbane) and report the depth, gate count, and number of SWAPs. How much overhead does the transpilation add?

Exercise 9.8: State Tomography After Teleportation

(a) Implement full quantum state tomography on Bob's qubit after teleportation. Measure in the X, Y, and Z bases to reconstruct the density matrix.

(b) Compare the reconstructed density matrix with the ideal state using fidelity. What is the fidelity on a noiseless simulator?

(c) Add depolarizing noise and plot how the tomographic fidelity degrades.

Exercise 9.9: Superdense Teleportation

(a) If Alice and Bob share a 3-qubit GHZ state instead of a Bell pair, can they teleport a 2-qubit state? Design the protocol.

(b) How many classical bits need to be transmitted?

(c) Implement this in Qiskit and verify.

Exercise 9.10: Teleportation of a Mixed State

(a) Can teleportation transmit a mixed state? If Alice has a qubit in state $\rho = p|0\rangle\langle 0| + (1-p)|1\rangle\langle 1|$, what does Bob receive?

(b) Verify your answer by simulating teleportation of a mixed state in Qiskit.

(c) Prove that the teleportation channel is a completely positive trace-preserving (CPTP) map.