Mathematical Introduction to Quantum Information Processing [MA5057]
Summer semester 2019
News
- The re-exam will be on October 2 at 14:00-15:00 in HS 2 (5604.EG.011).
- The exam will be on August 14 at 14:00-15:00 in Interims HS 2 (5620.01.102).
- No notes, books, or electronic tools will be allowed in the exam. The exam will be 60min plus additional 5min of 'Einlesezeit' at the beginning, in which you can read but are not allowed to write.
- No exercise classes in the last week (July 22-26). Exercise sheet will be discussed in the lecture on Fri, July 26.
Content
Quantum computation, quantum communication, and quantum cryptography are all high-level forms of quantum information processing. This course will introduce and analyze the basic building blocks of quantum information processing from a mathematical perspective. Beginning with the abstract foundations of quantum theory, the course will deal with quantum measurement theory, the description, steering and application of quantum evolutions, quantum statistical inference, and quantum tomography. One of the main aims of the course is to develop a better understanding of the fundamental limits of quantum information processing concerning speed, disturbance, precision, heat production and the use of other resources.
After introducing the basic formalism, the course will occasionally depart from textbook-material and cover more advanced result.
| Date |
Content |
Further reading |
| Apr 29 |
Intro, Hilbert space crash course |
|
| May 03 |
B(H) and its ideals, positivity, trace |
|
| May 06 |
Dualities, operator topologies, density operators |
|
| May 10 |
Convergence of density operators, POVM's, Born's rule, Holevo-Nayak no-go theorem |
|
| May 13 |
Lipschitz bounds for probabilities, convex sets and extreme points |
|
| May 17 |
Mixtures of states, functional calculus, majorization, von Neumann entropy |
|
| May 20 |
Composite systems, direct sums and tensor products, Hilbert-Schmidt isomorphism, Schmidt-decomposition, partial trace |
|
| May 24 |
Tensor rank, reduced density operators, maximally entangled states |
|
| May 27 |
Purification, marginal distributions, Heisenberg-/Schrödinger picture, Kraus representation |
|
| May 31 |
Positive and completely positive maps, Choi matrices |
|
| Jun 07 |
Instruments, Naimark's theorem, commuting dilations |
|
| Jun 14 |
Uncertainty relations, joint measurability, Knill-Laflamme / no measurement without disturbance |
|
| Jun 17 |
Minimum uncertainty states, time-energy uncertainty relations, proof of Knill-Laflamme / no measurement without disturbance |
|
| Jun 21 |
Quantum error correction, moment-based quantum speed limits, unbounded Hamiltonians, condition for evolution to orthogonal states |
|
| Jul 01 |
Quantum steering and EPR |
notes |
| Jul 05 |
EPR, Local hidden variable models, Bell inequalities, CHSH |
notes, further reading  |
| Jul 08 |
Cirelson representation of correlation matrices, Grothendieck' inequality |
notes |
| Jul 12 |
Chain of impossible machines, mixed-state entanglement, entanglement witnesses and positive maps |
notes |
| Jul 15 |
Extendability characterization of separable states, decomposable positive maps, PPT-states |
notes |
| Jul 19 |
Entanglement-assisted teleportation, superdense coding, outlook on state transformation via LOCC |
teleportation |
| Jul 22 |
Quantum Zeno- and Anti-Zeno-effect, interaction-free hypothesis testing |
notes |
| Jul 26 |
Discussion of the last exercise sheet |
|
Lecture Notes
Growing lecture notes can be found
here (version from June 22). Please try not to print them - they are under construction!
| Date |
Content |
Exercises |
solution sketches |
| Apr 29 |
Hilbert space basics |
ex1 |
sol1 |
| May 03 |
Positivity |
ex2 |
sol2 |
| May 11 |
Density operators and POVMs |
ex3 |
sol3 |
| May 17 |
Density operators and convexity |
ex4 |
sol4 |
| May 24 |
Tensor products |
ex5 |
sol5 |
| May 31 |
Positive and completely positive maps |
ex6 |
sol6 |
| Jun 07 |
Dual maps and commuting dilations |
ex7 |
sol7 |
| Jun 15 |
Commutators, uncertainty relations, tensor powers |
ex8 |
sol8 |
| Jun 22 |
QECC, time-energy uncertainty relations |
ex9 |
|
| Jul 06 |
LHV models, Bell inequalities |
ex10 |
sol10 |
| Jul 12 |
Entanglement |
ex11 |
sol11 |
| Jul 19 |
Entanglement II |
ex12 |
sol12 |
Literature
- Functional analysis basics: almost any text book or lecture notes will do. For instance Analysis Now by Pedersen (Chap. 3).
- The mathematical framework for quantum theory is covered for instance in the book by Heinosaari and Ziman. An early online version can be found here .
