COMP-761: Quantum Information Theory

Time: Tuesday and Thursday from 4:00 to 5:30 (Winter 2009)

Room: McConnell 320

Instructor:      Patrick Hayden

                        Office: ENGMC 108N

                        Phone: 398-5491

                        Email: patrick@cs.mcgill.ca

                        Office hours: By appointment

Course description:

This course will present the quantum analog of Shannon’s information theory. This area has seen an explosion of interest and a correspondingly rapid technical advance over the past ten years, largely in response to the development of quantum-mechanically based cryptographic protocols and Shor’s famous algorithm for factoring integers. The unavoidable presence of noise in any quantum-mechanical information processing device means that error-correction techniques will play a crucial role in any practical application of quantum cryptography or computing. This course will focus on asymptotic protocols for compression, communication, error correction and state distillation, identifying the absolute limits placed on those tasks by quantum mechanics.

Familiarity with quantum mechanics is recommended. The course content is very mathematical, but elementary. Students should be comfortable with basic probability theory, linear algebra and real analysis. The material will be covered through a combination of lectures and student presentations.

Course outline:

• Classical information theory:
• Compression: Shannon’s noiseless coding theorem
• Error correction: Shannon’s noisy coding theorem
• The birth of the qubit: Schumacher compression
• Tools for quantum information:
• Review of quantum-mechanical formalism (including Bell’s theorem)
• Inequalities for von Neumann entropy
• Strong subadditivity
• Three brilliant trivialities:
• Superdense coding
• Teleportation
• Coherent classical communication
• The family of quantum protocols
• The mother, father and fully quantum Slepian-Wolf protocols
• Consequences
• Entanglement distillation
• State merging
• Entanglement-assisted and quantum capacities
• Quantum reverse Shannon theorem
• The HSW theorem: classical data through a noisy quantum channel
• A noiseless postlude:
• Superdense coding of quantum states and its consequences
• Majorization and entanglement manipulation

Grading: 55% assignments, 35% presentations, 10% scribe notes.

Text: There is no text for the course. However, roughly the first month’s worth of material can be found in Quantum Computation and Quantum Information by Nielsen and Chuang, which is an excellent introduction to the field as a whole. Elements of Information Theory by Cover and Thomas should meet your classical information theory needs.

Assignments:

Assignment 1

Assignment 2

Assignment 3

Assignment 4

Assignment 5

Presentations: More information can be found here.

Scribe notes:

You will be expected to produce detailed lecture notes for three 1.5 hour classes.

Use scribe.sty, originally written by Jeff Erickson. 

Unedited student notes: use at your own risk...

Lecture 1: Uncertainty and compression [ pdf tex ]

Lecture 2: Properties of entropy and mutual information [ pdf tex ]

Lecture 3: Gambling, communication and mutual information [ pdf tex ]

Lecture 4: Shannon’s noisy coding theorem [ pdf tex ]

Lecture 5: Fano’s inequality and intro to quantum mechanics [ pdf tex ]

Lecture 6: A computer science perspective on nonlocality [ pdf tex ]

Lecture 7: Nonlocality (cont.d.) Density operators and the partial trace. Superdense coding. [ pdf tex ]

Lecture 8: Quantum operations [ pdf tex ]

Lecture 9: Teleportation and intro to resource inequalities [ pdf tex ]

Lecture 10: Quantum entropy [ pdf tex ]

Lecture 11: Coherent classical communication: the cobit. [ pdf tex ]

Lecture 12: From Lieb’s concavity theorem to monotonicity of relative entropy [ pdf ]

Lecture 13: Quantum data compression [ pdf tex ]

Lecture 14: Fidelity vs trace distance. Schumacher converse. Entanglement distillation. [ pdf tex ]

Lecture 15: State transfer and the decoupling argument [ pdf tex ]

Lecture 16: Comments on group integration. Decoupling calculation (part 1) [ pdf tex ]

Lecture 17: Decoupling calculation (part 2) [ pdf ]

Lecture 18: Optimality of decoupling. Entanglement-assisted capacities [ pdf tex ]

Lecture 19: Qubit erasure channel example. Quantum capacity [ pdf tex ]

Embarrassing obligatory inclusion:   

By the direction of Senate (January 29, 2003), all course outlines have to include the following statement:

McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).