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Lecture: Foundations of Quantum Mechanics

Lecturer: David Gross. Exercises: Lionel Dmello. Times and venue: Wed, Thur 12pm (alternating one / two lectures per week, 3SWS), Seminarraum 0.03, Theoretical Physics Building.

Announcements

  • The tutorial for sheet 5 will be held on Thursday, Jan 8th.
  • We have now set up a Slack channel for the course (actually we're reusing the channel we used for previous courses). Please sign up using smail.uni-koeln.de or thp.uni-koeln.de email addresses. Use this link to sign up  (If you participated in Mathematical Methods II in summer 2024 or in TP2: Quantum Mechanics in winter 2024/25, you're probably already in there). Please use the slack channel to ask questions and/or form groups for the exercises. If there are any issues, please contact Lionel Dmello.
  • Going forward, please submit your exercise sheet directly to your respective tutor -- Lionel Dmello (ldmello@thp.uni-koeln.de), Tanmay Singal (tsingal@thp.uni-koeln.de), Stephen Dsouza (sdsouza@thp.uni-koeln.de). If you are unsure who is your tutor, please write to Lionel Dmello (or ask on Slack).

Course description

The course will cover the properties of quantum mechanics that are fundamentally different from any classical theory. This contrasts with the usual QM classes, where the goal is to treat specific systems; and with Quantum Information Theory, where the focus lies on ways to exploit quantum behavior for computation and communication.

Topics are:

  • Contextuality, Bell inequalities, and "quantum non-locality"
  • Causality and its interaction with quantum probability
  • Uncertainty relations and their interpretation
  • The problem of joint measurability
  • No-cloning
  • Generalized Probabilistic Theories, post-quantum correlations and their convex geometry
  • Measurement theory and decoherence
  • Projective representations of symmetries

Resources

  • Work-in-progress lecture notes. (Disclaimer: lecture notes have not been proofread, reader discretion is advised.)
  • The Sage notebook (jyputer) for finding all the contextuality inequalities an the PR-boxes.
  • Derivation of optimal height for Klyachko's pentagram by Vahideh Eshaghian.

Exercises

  • Sheet 1, due October 22nd.
  • Sheet 2, due November 05th.
  • Sheet 3, due November 19th.
  • Sheet 4, due November 28th (Typo: In part (d) the product must be inside the expectation value).
  • Sheet 5, due January 7th (Typo: In sub-question (h) there should not be a (-1)^y in the matrix element. Other minor typos corrected. Total points on the sheet are now 25).
  • Sheet 6, due January 21st, 12:00 noon.

Organization

  • The lecture is good for 6 ECTS points.
  • Every second week, one of the slots will be an exercise class.
  • This lecture was held before.

Exam Standards

For those taking an exam, here's what we expect. We will not ask for topics listed as "not relevant", but won't mind talking about them if you bring them up.

  • Definition of non-contextual distributions, significance, examples, quantum violations
  • Everything about the CHSH inequality
  • The "impossible machines"
  • For a perfect grade: State-independent contextuality, loopholes, know about Tisrelson's bound (details of proof not required)
  • Not: Klyachko's Pentagram proof, the more philosophical aspects of the notion of "elements of reality"
  • Be able to qualitatively talk about the non-signalling polytope
  • For perfect grade: Be somewhat comfortable with the basic notions of convexity introduced; understand the basic ideas behind information causality
  • Not: The computer code to find dual descriptions of convex bodies; obviously don't have to memorize vertices / facets or such details
  • Mixed states (where they come from and how they are described mathematically)
  • POVMs: definition, one or two examples, statement of the dilation theorem.
  • For perfect grade: Explain the geometry behind "steering"
  • The measurement problem and basics of decoherence.