# 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

- (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
- (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
- Nonlocality proper, Bell inequalities, quantum violations
- Problems in Bell inequality experimental tests and how to handle them
- Mach Zehnder interferometer, quantum bomb testing, two-photon interferometry, intro to the Franson interferometer
- 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.
- (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.
- (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.