Part VI: Quantum Information
Information in the Quantum World
Quantum information theory extends classical information theory to systems governed by quantum mechanics. The fundamental unit is no longer the bit but the qubit— a two-state quantum system that can exist in superposition and become entangled with other qubits.
Quantum systems carry fundamentally different properties: entanglement enables correlations stronger than any classical channel allows (Bell inequalities), superposition enables quantum parallelism, and the no-cloning theorem prevents copying unknown quantum states.
This part covers the Von Neumann entropy — the quantum analog of Shannon entropy — quantum channels and their capacities, and quantum error correction, which makes fault-tolerant quantum computation possible.
Chapters in This Part
Chapter 16: Von Neumann Entropy
\(S(\rho) = -\text{Tr}(\rho \ln \rho)\), relation to Shannon entropy, subadditivity, strong subadditivity (Araki-Lieb), entanglement entropy, purification.
Chapter 17: Quantum Channels
Completely positive trace-preserving (CPTP) maps, Kraus representation, depolarizing and amplitude-damping channels, Holevo bound, quantum channel capacity.
Chapter 18: Quantum Error Correction
No-cloning theorem, Shor's 9-qubit code, stabilizer formalism, CSS codes, surface codes, fault-tolerant threshold theorem.
Key Quantum Information Inequalities
Von Neumann entropy: \( S(\rho) = -\text{Tr}(\rho \ln \rho) \geq 0 \)
Subadditivity: \( S(AB) \leq S(A) + S(B) \)
Araki-Lieb (triangle inequality): \( S(AB) \geq |S(A) - S(B)| \)
Holevo bound: \( I(X:B) \leq S(\rho_B) - \sum_x p_x S(\rho_x) \equiv \chi \)