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Quantum Statistical Inference [MA5438]

Sommersemester 2022

Prof. Dr. Michael Wolf

Lecturer: Prof. Dr. Michael Wolf
Assistant: Tim Möbus
Lecture: Tue 14:15, Thu 12:00, room 019@5123 register
Exercises: Tue 16:15, room 02.04.011@5604, Wed 10:15, room 01.5901.013@5901  



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 Pfeil
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 Pfeil
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)


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.


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 This usually improves the overall quality of the submitted homework and you will learn more by discussing your approaches with others.


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.