A thorough initiation into the fundamental aspects of quantum communications. The course is divided into four blocks. It begins with an introduction to data compression and communication over noisy channels in classical information theory. The second part discusses quantum entropies, distance measures between quantum states and quantum data compression. The third part of the course introduces quantum channels, and derives the fundamental properties for communications over quantum channels. The course ends with an in-depth introduction to quantum key distribution including its security proof.
1. Introduction to information theory
The communications model
Formalization of data compression and noisy channel coding
2. Refresher on probability theory
Random variables
Jensen’s, Markov, Chebyshev’s inequalities
Weak law of large numbers
3. Information measures
Entropy, conditional entropy, joint entropy
Mutual information
4. Data compression
Symbol codes
Typical sequences
Source coding theorem
5. The noisy channel coding theorem
Joint typicality
Random coding
Fano’s inequality
Converse theorem
Communication with feedback
6. Linear codes
Codes as subspaces
Generator and parity check matrices
Hamming weight
Minimum distance
Hamming bound
Singleton bound
Dual code
7. Good codes exist: low density parity check codes
Random linear codes achieve capacity
Random LDPC ensembles
The sum product decoder
Density evolution
8. Refresh axioms of quantum mechanics
9. Refresher on Hilbert spaces for quantum information
Euclidean norm, Cauchy-Schwarz, Dual vector, Tensor product
Linear operators
Trace, partial trace and properties of trace
Purification
Transpose, partial transpose
Hilbert-Schmidt inner product and norm
Singular value, polar decomposition
Entanglement and purification
10. Quantum entropies
Von Neumann entropy
Coherent information
Mutual information
Relative entropy
11. Distance measures
Fidelity
Trace distance
12. Quantum data compression
One-shot quantum data compression
Quantum data compression theorem
13. Quantum channels
Stinespring dilation theorem
Kraus representation
Choi representation
Completely positive and trace preserving maps
Measurements as quantum instruments
14. Hypothesis testing
Type I and type II errors
Symmetric and asymmetric hypothesis testing
15. Data processing inequalities
Data processing of relative entropy
Strong subadditivity
16. Entanglement theory and LOCC protocols
Teleportation
LOCC protocols
Entanglement measures
Entanglement distillation
Hashing bound
17. Entanglement-assisted classical communication
Communications scheme
Position-based coding
Sequential decoding
Entanglement assisted capacity
18. Fundamental limits for transmitting classical information
Communications scheme
Holevo information
Classical capacity
Entanglement breaking channels
19. Fundamental limits for transmitting private classical information
Communications scheme
Private information
Private capacity
20. Quantum key distribution (QKD) protocols
BB84
Six states protocol
SARG
Eckert
21. Classical post-processing in QKD
Parameter estimation
Min-entropy
Randomness extractors
Leftover hash lemma
Information reconciliation as error correction
22. CSS codes and the security of quantum key distribution
CSS codes
Reduction to prepare and measure QKD
23. Uncertainty relations and security
Entropic uncertainty relations
Security from uncertainty
Weekly homework (100%, 4 hours per week) in the form of written and programming exercises
Wilde, Mark. Quantum Information Theory, Second Edition. Cambridge University Press, 2017. https://markwilde.com/qit-notes.pdf
Wolf, Ramona. Quantum Key Distribution. Springer Lecture Notes in Physics, 2021. 978-3030739904