Workshop on Completely Positive Maps
Duration: Jan 15, 2024 - Jun 04, 2024
Course Name: Workshop on Completely Positive Maps (RF/VS)
Course Overview:
This workshop centers around the theoretical exploration of completely positive maps and its associated domain. In the initial sessions, we will delve into the fundamentals of quantum theory, encompassing both mathematical principles and introductory physical aspects. Subsequently, we will pivot to the foundational concepts of quantum information, covering topics such as registers/states, quantum channels, and measurements. Following this groundwork, our focus will shift towards a comprehensive understanding of Completely Positive Maps, encompassing discussions on Dilation Theorems, Arveson's Extension Theorem, and the \(C^*\)-tensor product. The latter part of the workshop will introduce participants to Completely Bounded Maps and their interconnected areas. Additionally, we will delve into the abstract characterizations of Operator Systems and Operator Spaces, enriching the exploration of this intriguing field.
Lecturers: Indrajit Ghosh (SRF), Gahin Maiti (JRF) and Soumyashant Nayak (Asst. Prof.)
Lecture Timings: 10:00 AM - 04:15 PM with breaks (11:00-11:15 AM coffee, 01:00-02:00 PM lunch and 03:00-03:15 PM tea)
Venue: Phase I & III - left-side seminar room of 2nd Floor Auditorium. Phase II & IV - One of the Ground Floor Classrooms
References
- Vern Paulsen, Completely Bounded Maps and Operator Algebras.
- Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information.
- John Watrous, The Theory of Quantum Information.
- L. Susskind and A. Friedman, Quantum Mechanics: The Theoretical Minimum.
- Brian C. Hall, Quantum Theory for Mathematicians
- Gerald Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators
- F Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics
- Richard V. Kadison, John R. Ringrose, Fundamentals of the Theory of Operator Algebras. Volume II.
Schedule & Lecture Notes
| Phase | Date | Speaker | Topics | References & Notes |
|---|---|---|---|---|
| I | Jan 17 (Wed) | Soumyashant Nayak | Review of basic matrix analysis; | Chapter 1 of Watrous, Section 2.1 of Nielsen-Chuang |
| Jan 19 (Fri) | Indrajit Ghosh | Mathematical Aspects of Quantum Mechanics: States, Observables, Position Operator, Momentum Operator and Axioms (A1-A5) | Gerald Teschl and Brian C Hall; Lecture note | |
| Jan 22 (Mon) | Indrajit Ghosh | Hamiltonian (\(\hat{H}\)), Density Matrices in \(B(\mathcal{H})\), Modified Axioms (A6-A10) using Density, Composite System: States and Hamiltonians | Brian C Hall and F. Strochhi; Lecture note | |
| Jan 24 (Wed) | Gahin Maiti | Examples: Particle in a box, Harmonic oscillator, General measurements, Projective and POVM measurements, Entanglement, Examples | Section 2 of Neielsen-Chuang, Chapters 3, 6, 7 of Susskind-Friedman | |
| II | Mar 04 (Mon) | SN + GM | Basic notions of quantum information - Registers/states; | Chapter 2 of Watrous |
| Mar 08 (Fri) | Gahin Maiti | Quantum channels, Measurements; | Chapter 2 of Watrous | |
| Mar 11 (Mon) | Indrajit Ghosh | Operator Systems, Positive Maps, Fejer-Riesz Lemma, von Neumann Inequality, Russo-Dye Theorem etc. | Chapter 2 of Paulsen; Lecture note | |
| Mar 13 (Wed) | IG + GM | Problem-solving session (Positive Maps); Completely Positive Maps | Chapter 3 of Paulsen; Indrajit's note | |
| III | Apr 10 (Wed) | Indrajit Ghosh | Stinespring's Dilation Theorem, Sz.-Nagy Dilation, Naimark Dilation and Choi's Theorem on CP maps from \(\mathbb{M}_n\) to \(\mathbb{M}_k\) | Chapter 4 of Paulsen; Indrajit's note |
| Apr 12 (Fri) | Gahin Maiti | Completely Positive Maps into \(\mathbb{M}_n\); Problem-solving session | Chapter 6 of Paulsen | |
| Apr 17 (Wed) | Soumyashant Nayak | Tensor Products of \(C^*\)-algebras | Chapter 11 of Kadison-Ringrose | |
| Apr 19 (Fri) | Soumyashant Nayak | Tensor Products of \(C^*\)-algebras (Cont.); | Chapter 11 of Kadison-Ringrose | |
| IV | May 30 (Thu) | Indrajit Ghosh | BW topology on \(\mathcal{B}(X, Y^*)\), Arveson's Extension Theorems and \(\mathscr{C}\)-bimodule maps. | Chapter 7 of Paulsen; Indrajit's note |
| May 31 (Fri) | Gahin Maiti | Completely Bounded Maps | Chapter 8 of Paulsen | |
| Jun 01 (Sat) | Indrajit Ghosh | Injective \(C^*\)-algebras, Conditional Expectation: Tomiyama's Theorem, \(\mathscr{B}\)-dilation; Haagerup's Theorem, Kadison's Similarity Conjecture, Amenable Groups and Dixmier's Theorem | Chapter 7 and 9 of Paulsen; Indrajit's note | |
| Jun 04 (Mon) | IG + SN | Derivation problem on \(C^*\)-algebras; Abstract Characterizations of Operator Systems and Operator Spaces | Chapter 9 and 13 of Paulsen; Indrajit's note |
Last Updated: Jun 04, 2024