skip to content

Lecture: Foundations of Quantum Mechanics

Lecturer: David Gross. Exercises: Vahideh Eshaghian. Times and venue: Mo 12pm, Tue 10am, Seminarraum 0.03, Theoretical Physics Building.


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


  • Link to electronic white board used in lecture.
  • Some notes on the measurement problem.
  • Zoom link for remote participation.
  • Recordings (pw is "kwantum"):
    • 1 (Intro, contextuality)
    • 2 (contextuality inequalities)
    • 3 (CHSH, local hidden variable models, Fine's Theorem)
    • 4 (Loopholes)
    • 5 (Bertlmann, "elements of reality", impossible machines)
    • 6 (Generalized probabilistic theories: Definitions and examples)
    • 7 (Convexity)
    • 8 (All contextuality inequalities; all extremal non-signal correlations)
    • 9 (Quantum CHSH violations, Tsirelson's bound)
    • 10 (Klyachko's Pentagram)
    • 11 (Information Causality for PR boxes)
    • 12 (Information theory intro and general IC principle)
    • 13 (Density matrices and their geometry)
    • 14 (Multi-partite quantum systems, partial trace)
    • 15 (Steering) (Suboptimal video, as the usual recording equipment wasn't available)
    • 16 (General measurements and Naimark's Thm part 1)
    • 17 (Naimark's Thm part 2 and the measurement problem)
    • 18 (Measurements, decoherence)
    • 19 (Algebraic QM 1)
    • 20 (Guest lecture by Nikolai Miklin)
    • 21 (Algebraic QM 2)



  • The lecture is good for 6 ECTS points and will fit into the GR/QFT and the CM/CP areas of specialization.
  • Dates, time, venue: Mondays at 12pm, Tuesdays at 10am. Seminarraum 0.03, Theoretical Physics New Building, Zülpicher Str. 77a.
  • We will aim to provide a live Zoom stream and will also post recordings online.
  • Every second week, the Tuesday slot 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, Tisrelson's bound
  • 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 comfotable 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; The "too kinky to be real" lecture; not the information-theoretic details of information causality
  • Mixed states (where they come from and how they are described mathematically)
  • POVMs: definition, one or two examples, statement of Naimark's dilation theorem.
  • For perfect grade: Explain the geometry behind "steering"
  • The measurement problem and basics of decoherence.