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Introduction to Open Quantum Systems

Tue. + Thu., 10–12B322, Exactum building (Kumpula)Lecture + Exercise10 cr

The course addresses physicists and mathematicians at advanced undergraduate or graduate level. For physics students the course will provide an introduction to theory and examples of quantum systems interacting with their environment. For students of mathematics the course will provide an introduction into to an algebraic treatment of probability and Markov processes.


  • probabilistic foundations of quantum mechanics
  • composite systems, entanglement
  • quantum channels alias completely positive maps
  • quantum Markov processes
  • Lindblad equation
  • repeated interaction systems
  • stochastic Schrödinger equation


The course requires linear algebra and differential calculus as discussed in the basic courses. In addition, some very basic understanding of probability is needed. For physicists we recommend some pre-knowledge in quantum mechanics.


The main part of the course will rely on the following books:

  1. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems, Oxford University Press, 2002
  2. Wiseman, H. M., Milburn, G. J.: Quantum Measurement and ControlCambridge University Press, 2009

Course Organization


06.09.2016–15.12.2016, weeks 36–42 and 44–50
every Tuesday and Thursday, 10–12,
room B322 in Exactum (Kumpula)


every Tuesday, 09-10, room B321
every Thursday, 09-10, room B322

LecturersDmitry Golubev (Aalto),
Paolo Muratore-Ginanneschi (Helsinki),
Kay Schwieger (Helsinki) 


The course as a dedicated presemo room (web space). To access this web space just follow the foregoing link.  Students are encouraged to use the presemo room for any civilized activity pertaining the course.

Lecture Notes

The lecture notes are grouped not in correspondence to actual lectures but by topics

Lecture 01: The Postulates of Quantum Mechanics Lecture 11: Completely Positive Semigroups and the Lindblad Equation
Lecture 02: Probability & MeasurementsLecture 12: Dynamics and the time asymptotic behavior of the evolution operator
Lecture 03: Dynamics by examplesLecture 13: van Hove's weak coupling scaling limit
Lecture 04: Hilbert Spaces, Operators, and Tensor ProductsLecture 14: the Born approximation
Lecture 05: Subsystems, Commutants, IndependenceLecture 15: the Markov approximation and the Lindbladian
Lecture 06: Purification, EntanglementLecture 16: Linbladian dynamics of the Qubit
Lecture 07: Jaynes Cummings Hamiltonian ILecture 17: Linbladian dynamics of the Caldeira-Leggett model
Lecture 08: Jaynes Cummings Hamiltonian IILecture 18: Quantum jumps
Lecture 09: Completely Positive Maps, No-Cloning TheoremLecture 19: An overview of (Poisson) stochastic differential equations
Lecture 10: Kraus-Stinespring Representation, Indirect Measurement, Markov ProcessesLecture 20:




  • On 20.09 we will have exercise session as usual 09-10 (Kay Schwieger) and main lecture 10-12 (Paolo Muratore-Ginanneschi)
  • 22, 27, 29.09 lectures will be as follows: 09-11 main lecture (lecturer Kay Schwieger) 11-12 exercises and complements

Starting from the week of Monday 31/10 the exercises are on Tuesdays from 14:15-16:00 in room B119.

Course feedback

Course feedback can be given at any point during the course. Click here.


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