Sidan finns bara på engelska

Quantum Foundations

PhD course on the foundations of Quantum Mechanics

Spring 2023

Quantum Theory is an enormously successful theory from a practical point of view. It correctly predicts both non-relativistic and relativistic phenomena to extraordinary precision and has driven major technological developments such as the laser, superconductivity and micro-circuitry. Recent experimental advances show coherence and entanglement of quantum systems routinely. And yet, even after more than a century, nobody seems to understand quantum mechanics. What are the properties that distinguish quantum systems from classical systems? Is the quantum-mechanical description complete? What does it describe?

This seminar series is not intended to explain quantum mechanics, but rather to expose the difficulties. We will start with some common background to prepare for the main goals: Bell nonlocality, and Kochen-Specker contextuality. The intent is to both cover the theory but also discuss experiments and how to overcome their shortcomings.

This iteration of the course will be adapted with people that have some knowledge of quantum mechanics, but I will attempt to explain concepts theory as we proceed. Some knowledge of probability theory, linear algebra, and complex numbers will be enough to follow most of the discussion.


The default timeslot is Wednesdays at 15:15 in Systemet, and aim for the usual 2x45-minute duration. Any changes in this will be announced. Let me know if you want to receive such announcements by email.

Content of the seminars

  1. (8 Feb) Examples of theories from classical physics, a discussion of locality as a property, a discussion of realism as a property, properties of quantum mechanics, Fourier transform of wave functions, the uncertainty relation, Einstein’s single-slit wave-function collapse problem and locality and Bohr’s response, Einstein’s photon-from-a-clocked-box problem and Bohr’s response, Ehrenfest’s discussion with Einstein, intro to EPR
  2. (17 Feb) EPR more in-depth, EPR elements of reality, EPR concept of locality, intro to nonlocality in Quantum Mechanics, Bohr’s response, von Neumann’s impossibility proof, Bohmian mechanics, Bell’s counterexample
  3. Nonlocality proper, Bell inequalities, quantum violations
  4. Problems in Bell inequality experimental tests and how to handle them
  5. Mach Zehnder interferometer, quantum bomb testing, two-photon interferometry, intro to the Franson interferometer
  6. The Franson interferometer. Postselection and the connection to the coincidence loophole. Energy-time entanglement versus time-bin entanglement. Elements of reality in the Franson setup. Three proposals to produce genuine energy-time entanglement.
  7. (12 Apr) Contextuality. The Peres-Mermin square. Common confusions on which measurements to perform. Spin-1 contextuality. The proof by Kochen-Specker. Gleason’s theorem. Intro to the graph representation of Kochen-Specker sets. Intro to contextuality inequalities.
  8. (19 Apr) Contextuality continued. Projectors of measurement eigenspaces. The Graph of orthogonality relations. Graph quantities: the Independence number and the Lovasz number. Criticism of contextuality results. Experimental limitations. Handling experimental limitations.

Course size

The nominal course size is 6 hp.

Examination assignments

The assignment is to read and present a paper, say a twenty-twentyfive-minute presentation followed by discussion, half an hour in total. Some suggestions (some are really good papers, some are not):

Earlier iterations of the course

There was an earlier iteration of the course in 2012 and also a different version spring 2022.