Quantum Statistical Inference [MA5438]
Sommersemester 2022
News
- The inspection of the second exam will take place on Oct. 25th, 9:00-10:00 in room 03.10.011@5610.
- The second exam will be written and take place on Oct. 14th, 9:00-10:00 in room MI HS2. No cheat-sheet will be allowed.
- The exam inspections will take place on Sep 1st, 9:00-10:00 and on Sep 2nd, 14:00-15:00 both in room 03.10.011@5610.
- The exam will be written and take place on Aug 4, 11:00-12:00 in room MI HS2. No cheat-sheet will be allowed.
- From now on, the Thursday lecture will start at 12:00 sharp.
- The tutorials will take place on Tuesday in room 02.04.011@5604 and on Wednesday in room 01.5901.013@5901 starting on 3rd of May.
- Submission in groups of up to three people is allowed and strongly recommended. Please make use of this option when possible by sending an e-mail with the names of the group members to tim.moebus@tum.de.
Content
The first part of the course is a mathematical introduction to the probabilistic and statistical structure of quantum theory. This will in particular cover density operators, positive operator-valued measures, and completely positive maps. The second part then applies the formalism and combines it with other techniques, especially from data analysis, in order to address problems of hypothesis testing, parameter estimation, tomography, learning, and predictive inference.
| notes/date |
content |
additional material |
| April 26 |
Intro, Hilbert spaces, Operators |
extended notes including some more advanced topics |
| April 28 |
Density operators, POVMs, Born's rule |
extended notes including some more advanced topics |
| May 3 |
trace-norm interpretation, (unambiguous) state discrimination, composite & reduced systems |
extended notes on tensor products including some more advanced topics |
video lecture on POVMs  |
| May 5 |
partial trace, Naimark dilation, quantum channels |
extended notes including some more advanced topics |
| Kraus/Choi/environment representation will be covered on May 12 |
video lecture on quantum channels |
| May 12 |
Kraus/Choi/environment representation, dual maps, instruments |
see previous extended notes |
| May 17 |
No-information-without-disturbance, optimality conditions for multiple hypothesis testing |
|
| May 19 |
Examples of optimal hyp. testing, bounds for pretty good and square-measurement |
|
| May 24 |
Fidelity bounds, finite-n discrimination, Quantum Chernoff bound |
|
| May 31 |
Quantum Chernoff theorem, comparing error rates of local and global strategies, asymmetric hypothesis testing |
|
| Jun 2 |
Relative entropy, Hiai-Petz theorem |
|
| Jun 9 |
Quantum Stein's Lemma, diamond norm |
chapter on quantum hypothesis testing for states, incl. references |
| Jun 14 |
Numerical range, Toeplitz-Hausdorff, spectral arc-length |
|
| Jun 21 |
Discrimmination of Holevo-Werner channels, isometries and unitaries (parallel scheme) |
updated notes on hypothesis testing of quantum channels |
| Jun 23 |
Discrimination of unitaries, sequential scheme |
|
| Jun 28 |
Perfect discrimination of arbitrary quantum channels (parallel schemes), entanglement-assisted disjointness |
|
| Jun 30 |
Perfect discrimination of arbitrary quantum channels (adaptive schemes) |
combined chapter on hyp. testing of quantum operations |
| Jul 12/14 |
Quantum tomography, notions of informational completeness, Minkowski dimension, generic measurements, SIC-POVMs |
|
| Jul 19 |
Informational completeness vs. topological embeddings |
|
| Jul 21 |
Projected least-squares estimator, Haar measure, t-designs |
|
| Jul 26 |
2-designs vs SIC-POVMs, statistical properties of PLS estimators |
|
| Jul 28 |
Q & A |
combined lecture notes (Chap.1 contains extra material) |
Tutorials
The tutorials will take place on Tuesday and Wednesday starting on 3rd of May and are roughly structured into preparation and after that there will be time to answer your questions: For preparation, we will discuss and solve preparing exercises for the homework. The tasks will be marked as such on the exercise sheet. The second half of the tutorial is for answering questions, reviewing key steps from the lecture, and repeating important solutions from previous exercise sheets.
Exercises
We upload the exercise sheet, which deepens the topics of the Tuesday and Thursday lecture, on Friday. The first exercise sheet will be uploaded on Friday, the 29th of April.
The exercise sheets consist of two parts: the tutorial exercises and homework. As the name suggests, the tutorial exercises are discussed in the tutorials and prepare the homework, which can be submitted on moodle (will be updated soon) to achieve a grade bonus. The deadline is always Friday at 6 PM, exactly one week after the upload of the exercise sheet.
To sum up: Friday: upload –> Tuesday and Wednesday: preparation in tutorials –> Friday: submission
Grade Bonus
As already mentioned, you can achieve a grade bonus to improve your passed exam grade by 0.3 (e.g. 2.3 -> 2.0, but 4.3 still means not passed). In order to get this, you must submit your exercises and at least 75% of your solutions must be reasonable. In practice, this means that your solutions will be skimmed (no correction, no feedback) and you will receive one point per exercise if the solution is reasonable. Submission in groups of up to three people is allowed and strongly recommended. Please make use of this option when possible by sending an e-mail with the names of the group members to
tim.moebus@tum.de. This usually improves the overall quality of the submitted homework and you will learn more by discussing your approaches with others.
Literature
There is no textbook on quantum statistical inference or a similar topic yet. Below, we will collect some useful resources on specific topics covered in the course. This list will be expanded as the course progresses.
- The mathematical framework of quantum theory is covered for instance in the book by Heinosaari and Ziman. An early online version can be found here .
- John Watrous' The Theory of Quantum Information is an online available book that covers what its title claims.
