Teaching plan for the course unit



Close imatge de maquetació




General information


Course unit name: Quantum Statistical Inference

Course unit code: 574644

Academic year: 2021-2022

Coordinator: Bruno Julia Diaz

Department: Department of Quantum Physics and Astrophysics

Credits: 3

Single program: S



Estimated learning time

Total number of hours 75


Face-to-face and/or online activities



-  Lecture

Face-to-face and online




-  Lecture with practical component

Face-to-face and online



Independent learning






Students are advised to have completed the mandatory subject Quantum Information Theory. Enrollment in the elective subjects Advanced Quantum Information and/or Quantum Communications is encouraged.



Competences to be gained during study


  1. Critical analysis of the rigor of theoretical developments and reliability of experimental measurements in the field of Quantum Science and Technology.  

  2. Analysis and resolution of quantum physics problems, using exact as well as approximate methods.

  3. Design and analysis of sensors leveraging quantum properties.

  4. Knowledge and application of quantum information theory.

  5. Knowledge and application of advanced concepts in quantum physics to multidisciplinary problems.





Learning objectives


Referring to knowledge

  1. To understand the differences between classical and quantum inference. 

  2. To develop a broad perspective of the array of inference tasks that are useful for quantum technologies, including both their theoretical and practical performance limits. 

  3. To become familiar with general-interest analytical and numerical analysis tools for quantum information, such as representation theory, semidefinite programming, matrix analysis, and quantum combs.



Teaching blocks


1. Quantum state discrimination.


  1. Single-shot protocols, with and without error margins.

  2. Asymptotic error rates. Chernoff, Stein, and Hoeffding error exponents.

  3. Multiple hypotheses and symmetric cases.

  4. Discrimination of quantum channels.

  5. Restricted strategies. Local vs global, adaptive, and entanglement assisted measurements.

  6. Practical applications. Quantum illumination, quantum reading.

2. Quantum parameter estimation


  1. Bayesian and point-wise formulations. 

  2. Ultimate limits. Fisher information and Cramer-Rao bound.

  3. Restricted strategies. Local vs global, adaptive, and entanglement assisted measurements.

  4. Quantum tomography of states and processes.

  5. Practical applications. Interferometry, magnetometry, atomic clocks.



Teaching methods and general organization


  1. Lectures where theoretical contents of the subject are presented. 

  2. Practical exercise classes in which students may participate. 

  3. Activities related to the subject suggested by the teaching staff. 



Official assessment of learning outcomes


  1. Two deliverable group exercises to be handed in as the course progresses. Each exercise is worth 2 points and will be announced a week before the deadline.

  2. A final written examination on the entire course content worth 6 points. 

  3. Members of the teaching staff may also consider students’ participation in class and in the optional tasks they suggest. Evaluation of competences.



Reading and study resources

Consulteu la disponibilitat a CERCABIB


Paris, M.; Rehacek, J. (Eds.), Quantum State Estimation, Lecture Notes in Physics 649 (Springer, Berlin Heidelberg 2004).

Helstrom, C. W., Quantum Detection and Estimation Theory (Academic Press, New York 1976).

Hayashi, M. (ed), Asymptotic Theory of Quantum Statistical Inference (World Scientific 2005).

Rexiti, M.; Mancini, S., Discriminating qubit amplitude damping channels, arXiv:2009.01000


  1. Barnett, S., Croke, S., Quantum state discrimination, Advances in Optics and Photonics 1, 238 (2009)

Audenaert, K., Calsamiglia, J., Munoz-Tapia, R., Bagan, E., Masanes, L., Acin, A., Verstraete, F., Discriminating states: the quantum Chernoff bound, Phys. Rev. Lett. 98, 160501 (2007)

Bisio, A.; Chiribella, G.; D’Ariano, G.M.; Facchini, S; Perinotti, P, Optimal quantum tomography, IEEE Journal of Selected Topics in Quantum Electronics, 15, 1646-1660 (2009)

Web page

Quantum Tomography tutorial in Qiskit: https://qiskit.org/documentation/tutorials/noise/8_tomography.html