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.

Place

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):

• Aspect’s experiment(s), Aspect et al, PRL 1981, PRL 1982a, and PRL 1982b.
• Testing noninvasive measurability, Leggett and Garg, PRL 1985
• How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Aharonov, Albert, and Vaidman, PRL 1988
• Bell’s theorem without inequalities, Greenberger, Horne and Zeilinger, Am J Phys 1990
• Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Bennett et al PRL 1993
• Joel Fisher (May 22nd): A Heisenberg microscope for correlated photons, Dopfer, PhD Thesis (in German), Ch. 4, 1998
• William Stenlund (May 17th): Wave-particle duality of C60 molecules, Arndt et al, Nature 1999
• Joakim Argillander (May 22nd): Measuring the speed of quantum information, Scarani et al, PLA 2000
• Christoffer Hindlycke (May 10th): Quantum computing and hidden variables, Aaronson, PRA 2005
• Single-Particle Diffraction and Interference at a Macroscopic Scale, Couder and Fort, PRL 2006
• Testing Born’s Rule in Quantum Mechanics with a Triple Slit Experiment, Sinha et al, Proceedings of FPP-5, 2009
• Information causality as a physical principle, Pawlowski et al, Nature 2009
• Utkarsh Singh (May 17th): On the reality of the quantum state, Pusey et al, Nature Physics 2012
• Hannah Helgesen (June 2nd): Graph-Theoretic Approach to Quantum Correlations, Cabello et al PRL 2014
• Daniel Spegel-Lexne (June 2nd): A strong no-go theorem on the Wigner’s friend paradox, Bong et al Nature Physics 2020
• Challenging local realism with human choices, Abellán. et al, Nature 2018.
• Efficient Contextual Ontological Model of n-Qubit Stabilizer Quantum Mechanics, Hindlycke and Larsson PRL 2022
• Conjugate Logic, Johansson et al The quantum-like revolution, Springer 2023
• Double-slit time diffraction at optical frequencies, Tirole et al, Nature Physics 2023

Earlier iterations of the course

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