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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 Syllabus
Course 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

Homework 4

Homework 3

Homework 2

Homework 1

Lectures

Lecture 18 (28 Oct 2013)

  • Shannon entropy
  • conditional entropy
  • conditioning does not increase entropy
  • mutual information
  • relative entropy
  • data processing inequality
Reading Material: Chapter 10 of arXiv:1106.1445

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
Reading Material: Chapter 9 of arXiv:1106.1445

Lecture 16 (21 Oct 2013)

  • fidelity between pure states
  • fidelity between mixed states
  • Uhlmann's theorem
Reading Material: Chapter 9 of arXiv:1106.1445

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
Reading Material: Chapter 9 of arXiv:1106.1445

Lecture 14 (16 Oct 2013)

  • entanglement distribution for qudits
  • super-dense coding for qudits
  • teleportation for qudits
Reading Material: Chapter 6 of arXiv:1106.1445

Lecture 13 (9 Oct 2013)

  • generalized dephasing channels
  • Naimark extension theorem
  • teleportation and super-dense coding
Reading Material: Chapters 5 and 6 of arXiv:1106.1445

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
Reading Material: Chapter 5 of arXiv:1106.1445

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
Reading Material: Chapter 4 of arXiv:1106.1445, notes on the Choi-Kraus theorem, Supplementary viewing material: Lecture by Vern Paulsen on the Choi-Kraus theorem

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
Reading Material: Chapter 4 of arXiv:1106.1445, notes on the Choi-Kraus theorem, Supplementary viewing material: Lecture by Vern Paulsen on the Choi-Kraus 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
Reading Material: Chapter 4 of arXiv:1106.1445

Lecture 8 (23 Sep 2013)

  • finished off the Bell theorem / CHSH game
  • Schmidt decomposition
Reading Material: Scarani's The device-independent outlook on quantum physics, pages 107-108 of arXiv:1106.1445

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
Reading Material: Section 3.5.4 of arXiv:1106.1445, Peres' How the no-cloning theorem got its name, Scarani's The device-independent outlook on quantum physics

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
Reading Material: Chapter 3 of arXiv:1106.1445, also Hardy's Quantum theory from five reasonable axioms

Lecture 5 (11 Sep 2013)

  • finished off the proof of Shannon's channel capacity theorem
  • started with the postulates of quantum mechanics
Reading Material: Sections 2.2, 13.10, and Chapter 3 of arXiv:1106.1445, also Hardy's Quantum theory from five reasonable axioms

Lecture 4 (9 Sep 2013)

  • example of an error correction code
  • started the proof of Shannon's channel capacity theorem
Reading Material: Sections 2.2 and 13.10 of arXiv:1106.1445

Lecture 3 (4 Sep 2013)

  • finished off the proof of Shannon's data compression theorem
  • coding strategy—“keep the typical sequences”
Reading Material: Sections 2.1 and 13.1 - 13.4 of arXiv:1106.1445

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
Reading Material: Sections 2.1 and 13.1 - 13.4 of arXiv:1106.1445

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.