QCQI – Chapter 12 Detailed Summary: Quantum Cryptography & Communication - Quantum Computing & Quantum Information

Course Notes (MSU)

Learning Objectives¶

Explain why no-cloning and measurement disturbance enable quantum key distribution (QKD).
Implement and analyze BB84: basis sifting, QBER estimation, information reconciliation, privacy amplification.
Understand B92 (two-state) and E91 (entanglement-based) variants and their security intuition.
Recognize practical attacks (intercept-resend, photon-number splitting) and countermeasures (decoy states).
Know quantum no-go results: bit commitment impossibility; basics of coin flipping and secret sharing.

Foundations¶

No-cloning theorem. No unitary can copy two nonorthogonal states: if $U|\psi\rangle|0\rangle=|\psi\rangle|\psi\rangle$ and $U|\phi\rangle|0\rangle=|\phi\rangle|\phi\rangle$ then inner products force $\langle \psi|\phi\rangle=\langle \psi|\phi\rangle^2$ , hence $\langle \psi|\phi\rangle\in\{0,1\}$ .

Measurement disturbance and complementarity (e.g., $Z$ vs. $X$ bases) ensure eavesdropping leaves detectable errors.

Eureka! Security is physical: copying unknown quantum states is impossible, and incompatible measurements disturb the signal.

BB84 Protocol¶

Bases: $Z=\{|0\rangle,|1\rangle\}$ and $X=\{|+\rangle,|-\rangle\}$ with $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$ .

Pseudo-code: BB84 (honest, noiseless skeleton)

Alice picks random bits $a_i$ and random bases $b_i\in\{Z,X\}$ ; sends $|\psi_i\rangle$ accordingly Bob measures each qubit in a random basis $b'_i$ to get outcomes $x_i$ Publicly compare bases; keep indices with $b_i=b'_i$ (sifted key) Reveal a random test subset to estimate QBER; if low enough, proceed Run information reconciliation (IR) to correct mismatches Apply privacy amplification (PA) to compress and remove Eve’s information

Intercept-resend attack. Eve randomly measures in $Z$ or $X$ and resends. When $b_i=b'_i$ , Eve introduces errors with probability $1/4$ on average, yielding $\text{QBER}\approx 25\%$ (detectable).

Eureka! QBER is a thermometer for eavesdropping: higher disturbance $\Rightarrow$ lower extractable secret key.

Information Reconciliation (IR)¶

Classically remove discrepancies without revealing too much. Practical protocols (e.g., Cascade, LDPC) use parities of blocks with limited leakage.

Privacy Amplification (PA)¶

Apply a universal hash (e.g., Toeplitz) to the reconciled key to shrink Eve’s information to negligible levels. Asymptotic key rate (very roughly) is $R \approx q\left[1 - f\,h(e_b) - h(e_b)\right],$ where $q$ is sifting efficiency, $f\ge 1$ is IR inefficiency, $e_b$ is QBER, and $h$ is the binary entropy (exact rates come from rigorous finite-key analyses).

B92 (Two-State QKD)¶

Alice uses two nonorthogonal states (e.g., $|0\rangle$ , $|+\rangle$ ). Bob performs a measurement with an inconclusive outcome. Security arises from the impossibility of perfectly distinguishing nonorthogonal states.

Eureka! Inconclusive outcomes are a feature: they prevent perfect eavesdropper discrimination.

E91 (Entanglement-Based)¶

Share EPR pairs; measure in randomly chosen complementary bases. A Bell-inequality violation bounds Eve’s information; the entanglement-based picture is equivalent in security to prepare-and-measure BB84.

Eureka! Security can be certified by nonlocality: a CHSH violation implies limited eavesdropper knowledge.

Practical Attacks & Countermeasures¶

Photon-number splitting (PNS). Weak coherent pulses sometimes have multiple photons; Eve can split off one and wait. Decoy states with different intensities foil PNS by revealing rate anomalies.

Other vectors: Trojan-horse (probe back-reflections), detector blinding/time-shift; countered by monitoring, randomization, and device-independent approaches (beyond scope here).

No-Go: Bit Commitment & More¶

Unconditionally secure quantum bit commitment is impossible: with entanglement, a cheating committer can delay a local measurement (EPR attack) to open either bit.

Coin flipping admits quantum protocols with reduced cheating probability but not zero.

Secret sharing encodes a secret into multipartite entanglement (e.g., GHZ) so only authorized sets can recover it.

Eureka! Quantum advantages exist, but some cryptographic dreams remain impossible without extra assumptions.

Pseudo-code Aids¶

Pseudo-code: Intercept-resend simulator (concept)

For each signal: Alice picks $(a,b)\in\{0,1\}\times\{Z,X\}$ Eve picks random basis $e$ ; measures, resends eigenstate Bob picks $b'$ ; measures and records bit After sifting ( $b=b'$ ), compute empirical QBER

Pseudo-code: Simple IR (block parity) + PA (Toeplitz hash)

Partition sifted bits into blocks; publish parities; correct by binary search Count parity bits leaked $\ell_{\text{IR}}$ Pick random Toeplitz hash matrix $H$ ; output key $k=H \cdot s$ with length $|s|-\ell_{\text{IR}}-\text{margin}$

Schematics¶

BB84 Flow (Concept)¶

E91 (CHSH Sketch)¶

Hands-on Notebook (Multi-Backend)¶

Run: L16_QCQI_Ch12_QKD_and_Quantum_Cryptography

In the first cell set backend to one of: cirq, pennylane, braket, pyquil, or qiskit.

Mini-Lab Ideas¶

Simulate BB84 with/without intercept-resend; plot QBER vs. eavesdropping fraction.
Implement a toy IR (block parities) and PA (Toeplitz hashing); estimate net key length vs. QBER.
Build an E91 toy (Bell pairs, random bases) and compute a CHSH statistic.
Explore B92 by varying the angle between the two states and measuring the conclusive rate.