Exercises: Chapter 10 — Superdense Coding, Quantum Key Distribution, and the Communication Applications of Entanglement

Exercise 10.1: Superdense Coding with Imperfect Entanglement

Modify the superdense coding circuit to use a partially entangled state $|\psi\rangle = \sqrt{p}|00\rangle + \sqrt{1-p}|11\rangle$ instead of a perfect Bell state. Compute the success probability for each of the four messages as a function of $p$. At what value of $p$ does superdense coding cease to offer any advantage over classical communication?

Exercise 10.2: BB84 with a Realistic Channel

Extend the BB84 simulation to include: (a) Depolarizing noise (modeled as random Pauli errors) (b) Photon loss (some qubits never arrive at Bob) (c) Dark counts (Bob's detectors click even when no photon arrives) Compute the secret key rate as a function of distance for fiber with 0.2 dB/km loss.

Exercise 10.3: Detect an Eavesdropper

Using the BB84 simulation, implement an intercept-resend eavesdropper. Show that: (a) The QBER increases to ~25% when Eve intercepts all qubits (b) Alice and Bob can detect Eve by checking a subset of the sifted key (c) If Eve intercepts only a fraction $f$ of qubits, the QBER is approximately $0.25f$ (d) Plot the detection probability as a function of Eve's interception fraction for different sample sizes.

Exercise 10.4: Privacy Amplification

Implement a simple privacy amplification scheme using a Toeplitz matrix. Given a sifted key of length $n$ with known QBER, compute the length of the final secret key after privacy amplification. Verify that the output is statistically random using the NIST statistical test suite (or a simplified version).

Exercise 10.5: Compare BB84 and E91

Discuss the advantages and disadvantages of prepare-and-measure (BB84) vs. entanglement-based (E91) QKD. Consider: implementation complexity, security assumptions, loss tolerance, and compatibility with quantum repeaters. Implement both protocols and compare their error tolerance.

Exercise 10.6: Superdense Coding with More Than 2 Bits

(a) Show that if Alice and Bob share an $n$-qubit GHZ state, Alice can encode $2^n$ messages by applying $n$ local operations, and Bob can decode them using an $n$-qubit measurement.

(b) Is this "superdense coding with exponential capacity"? What is the catch?

(c) Implement 3-qubit superdense coding in Qiskit.

Exercise 10.7: Error Correction in BB84

(a) Implement the Cascade error correction protocol for BB84. Given a sifted key with QBER = 5%, how many passes are needed to correct all errors?

(b) What fraction of the key bits are revealed during error correction? How does this affect the final key length?

Exercise 10.8: Quantum Bit Error Rate Analysis

(a) Derive the theoretical QBER for an intercept-resend attack in BB84. Show that it is exactly 25%.

(b) For a more general attack where Eve applies a unitary $U$ to each qubit and an ancilla, compute the QBER as a function of $U$. What attack minimizes the QBER while maximizing Eve's information?

Exercise 10.9: Entanglement-Based Superdense Coding

(a) In superdense coding, the entanglement is consumed after one use. Design a protocol where Alice and Bob can reuse entanglement for multiple rounds of communication. What are the limitations?

(b) Implement a protocol where Alice and Bob share 3 Bell pairs and use superdense coding to transmit 6 classical bits by sending 3 qubits.

Exercise 10.10: Quantum Internet Protocol Stack

(a) Design a simple link-layer protocol for entanglement generation between two nodes connected by a fiber with 0.2 dB/km loss and 0.1% depolarizing error per km.

(b) Compute the entanglement generation rate as a function of distance, assuming photons are generated at 10 MHz and the detector efficiency is 10%.

(c) At what distance does the entanglement generation rate drop below 1 Hz?