Quantum Statistical Inference [MA5438]
Sommersemester 2022
News
 The second exam will be written and take place on Oct. 14, 9:0010:00 in room MI HS2. No cheatsheet will be allowed.
 The exam inspections will take place on Sep 1st, 9:0010:00 and on Sep 2nd, 14:0015:00 both in room 03.10.011@5610.
 The exam will be written and take place on Aug 4, 11:0012:00 in room MI HS2. No cheatsheet 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 email 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 operatorvalued 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 
tracenorm 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 
Noinformationwithoutdisturbance, optimality conditions for multiple hypothesis testing 

May 19 
Examples of optimal hyp. testing, bounds for pretty good and squaremeasurement 

May 24 
Fidelity bounds, finiten discrimination, Quantum Chernoff bound 

May 31 
Quantum Chernoff theorem, comparing error rates of local and global strategies, asymmetric hypothesis testing 

Jun 2 
Relative entropy, HiaiPetz theorem 

Jun 9 
Quantum Stein's Lemma, diamond norm 
chapter on quantum hypothesis testing for states, incl. references 
Jun 14 
Numerical range, ToeplitzHausdorff, spectral arclength 

Jun 21 
Discrimmination of HolevoWerner 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), entanglementassisted 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, SICPOVMs 

Jul 19 
Informational completeness vs. topological embeddings 

Jul 21 
Projected leastsquares estimator, Haar measure, tdesigns 

Jul 26 
2designs vs SICPOVMs, 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 email 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.