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## Introduction to Quantum Information Theory—PHYS 7895

This course introduces the subject of communication with quantum systems. Quantum information theory exploded in 1994 when Peter Shor published his algorithm that can break RSA encryption codes. Since then, physicists, mathematicians, computer scientists, and engineers have been determining the ultimate capabilities for quantum computation and quantum communication. In this course, we study the transmission of information over a noisy quantum communication channel. This course is intended for adventurous minds. There are many aspects of this new field that have not yet been explored. If you take this course, you could develop the mental discipline to contribute to this exciting field in the early stage of its development.

Course SyllabusCourse Flyer

Preprint of textbook on Quantum Shannon Theory

Quantum Information Theory textbook

Guideline for Final Presentations

Office hours from 10:30-11:30am on Mondays in 447 Nicholson Hall

### Homeworks

### Lectures

#### Lecture 18 (28 Oct 2013)

- Shannon entropy
- conditional entropy
- conditioning does not increase entropy
- mutual information
- relative entropy
- data processing inequality

#### Lecture 17 (23 Oct 2013)

- monotonicity of fidelity under quantum channels
- Fuchs-van-de-Graaf inequalities for trace distance and fidelity
- fidelity as the minimum Bhattacharya overlap
- gentle measurement lemma

#### Lecture 16 (21 Oct 2013)

- fidelity between pure states
- fidelity between mixed states
- Uhlmann's theorem

#### Lecture 15 (18 Oct 2013)

- trace norm and its properties
- trace distance
- variational characterization of trace distance
- monotonicity of trace distance
- operational interpretation of trace distance

#### Lecture 14 (16 Oct 2013)

- entanglement distribution for qudits
- super-dense coding for qudits
- teleportation for qudits

#### Lecture 13 (9 Oct 2013)

- generalized dephasing channels
- Naimark extension theorem
- teleportation and super-dense coding

#### Lecture 12 (7 Oct 2013)

- purified quantum theory
- isometric extensions of quantum channels
- example of qubit erasure channel and argument for why 1/2 erasure channel has zero quantum capacity

#### Lecture 11 (2 Oct 2013)

- finished the proof of the Choi-Kraus representation theorem
- interpretations of noisy quantum maps as arising from unitary interaction with an environment or from the loss of a measurement outcome

#### Lecture 10 (30 Sep 2013)

- general form of measurements
- positive operator-value measure (POVM) formalism
- partial trace
- completely positive, trace preserving, linear maps and discussion of why we need each property
- first part of the Choi-Kraus representation theorem

#### Lecture 9 (25 Sep 2013)

- introduction to the density operator
- density operator as the state
- reformulation of postulates with mixed states
- spectral decomposition
- purity

#### Lecture 8 (23 Sep 2013)

- finished off the Bell theorem / CHSH game
- Schmidt decomposition

#### Lecture 7 (18 Sep 2013)

- no cloning theorem
- how the ability to clone leads to the ability to signal superluminally
- introduction of the Bell theorem / CHSH game

#### Lecture 6 (16 Sep 2013)

- Homework 1 assigned
- finished off discussing the postulates of quantum mechanics
- several manipulations with quantum states
- composite quantum systems, tensor product
- matrix representations of X and Z operators in computational and diagonal bases
- Pauli matrices, Hadamard operator
- measurement of qubits

#### Lecture 5 (11 Sep 2013)

- finished off the proof of Shannon's channel capacity theorem
- started with the postulates of quantum mechanics

#### Lecture 4 (9 Sep 2013)

- example of an error correction code
- started the proof of Shannon's channel capacity theorem

#### Lecture 3 (4 Sep 2013)

- finished off the proof of Shannon's data compression theorem
- coding strategy—“keep the typical sequences”

No class on 2 Sep 2013 (Labor Day)

#### Lecture 2 (28 Aug 2013)

- data compression example
- Huffman code
- information content
- entropy as expected information content
- law of large numbers (sample entropy close to the true entropy)
- typicality
- introduction of Shannon's source coding theorem

#### Lecture 1 (26 Aug 2013)

- introduction of course
- outline of goals
- ideas for final presentation

Previous course on quantum information theory taught at McGill University

*Last modified: January 02, 2014.*