Lecture: Foundations of Quantum Mechanics

Lecturer: David Gross. Exercises: Mariami Gachechiladze, Lukas Franken

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

Language is English, despite what Ilias says (don't know how to change that...)

 

Organization

The lecture will take place online. Video recordings will be provided. The lecture is good for 6 ECTS points and will fit into the GR/QFT and the CM/CP areas of specialization.

There will be Zoom online meetings:

• Every Monday at 4pm with the lecturer David Gross
• Every Tuesday at 4pm with the exercise team Mariami Gachechiladze and Lukas Franken

Next lecture meeting: Monday, 6th of July, 4.00pm. Click here to join Zoom meeting.

The recap meeting: Wednesday, July 22nd, 4pm. Click here to join Zoom meeting. (Tuesday does not work for me!)

For exercises and general communication it is highly recommended to join the Slack-Channel, which is set up for organization of all master lectures' tutorials offered by the Institute for Theoretical Physics. Click here to join! Mail addresses associated with Universitites Cologne and Bonn can join automatically, for access with your private mail, please contact Lukas Franken.

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
• 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
• 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: Unambiguous state discrimination and POVMs
• Quantum operations: Problem statement, complete positivity, Kraus representation
• For perfect grade: Be able to discuss partial transpose map
• Not: Proofs of the dilation theorems, philosophical issues around the measurement problem
• Not: The long essay we read in Week 11

Material

Using this link, you can access the contents of the whiteboard. Videos can be accessed via direct links below, or via the Vimeo showcase. Password is "quantum", with "qu" replaced by "kw" (to discourage scrapping).

Exercises

Solutions to all discussed exercises.

• Exercise Sheet 1
• Exercise Sheet 2
• Exercise Sheet 3
• Exercise Sheet 4
• Exercise Sheet 5
• Exercise Sheet 6 
• Exercise Sheet 7 
• Exercise Sheet 8 
• Exercise Sheet 9   (Mathematica file)
• Exercise Sheet 10 (Due to July 13th)

Week 1

Lectures

• Introduction
• Contextuality
• CHSH inequality
• Local hidden variable models

Further Reading

• Introduction to contextuality

Week 2

Lectures

• Impossible machines
• Klyachko's pentagram (simple proof provided by Vahideh Eshaghian)
• CHSH violations

Further reading

• The paper that came up with the concept of "impossible machines" in this context
• Klyachko's pentagram (While the central statement is of course valid, there are some slightly exaggerated claims in the paper. Check out the footnote in the contextuality paper linked to under Week 1 or talk to us.)

Week 3

Lectures

• Elements of reality
• Bell inequalities without inequalities 1: Mermin Square and state-independent contextuality
• Bell inequalities without inequalities 2: GHZ. This has been made into exercise sheet number two. Aspects to discuss during exercise session:
  • Why does the GHZ scenario give rise to an LHV violation, while the Mermin Square - which looks mathematically similar - only leads to contextuality?
  • Getting rid of inequalities in favor of "deterministic violations" is conceptually nice. Is the distinction also relevant for experiments?
• Loopholes 1: "Free Will" and Locality Loophole
• Loopholes 2: Detector Loophole

Further Reading

• Super-determinism
  • Supportive: t'Hooft's book
  • Supportive: Hossenfelder and Palmer
  • Critical: Mateus' blog
• "Loophole-free" Bell tests
  • Heralded experiment in Delft
  • Purely optical experiment in Vienna
  • New York Times coverage
  • Thorough theory paper

 

Week 4

Announcement: Previous videos used 1k (or "full HD") resolution. All following videos are available in 4k. If you have sufficient bandwith, make sure to select 4k by clicking on the cogwheel.

• Tsirelson's bound [After uploading the video, I realized that in 4k (much more than in 1k, used before), you can see that my shirt *really* needs to be ironed. If people start complaining, I'll reduce the resolution again!]
• General Probabilistic Theories
  • 1 - The No-Signalling Principle and Popescu-Rohrlich Boxes
  • 2 - Convexity: generalities
  • 3 - Convexity: examples and overview of non-signalling set (cut from 2nd video, as it got too long)

Further Reading

• SageMath, CoCalc.com, polytopes in Sage

Week 5

• General Probabilistic Theories
  • 5 - All the Bell inequalities
    • [Sidebar: Correlation functions and Fourier analysis on the cube]
  • 6 - All non-signalling theories

Further Reading

• Analysis of non-signalling theories
• iPython notebook used in lecture (use local installation of SageMath, or cocalc.com to view and edit).

Week 6

• General Probabilistic Theories
  • 7 - Symmetries of convex polytopes
  • 8 - Boxworld: Too kinky to be real

Further Reading

• [Self-promotion] Symmetries of boxworld
• Books on convexity: Barvinok, Webster

 

Week 7

• Quantum Steering
  • 1 - Preamble
  • 2 - "States" revisited
  • 3 - Convex geometry of quantum states
  • 4 - Recall: Many-body Hilbert spaces and tensor product
  • 5 - Reduced and conditional states
  • 6 - Steering: Idea
  • 7 - Steering: Formal definition

Further Reading

• A review article.

Week 8

• Measurement Theory [There seems to be some bug with the Vimeo hosting platform - after the first video, all are uploaded to my personal account, rather than the Institute's one. I hope this doesn't cause any problems]
  • 1 - Axiomatic approach: Positive Operator-Valued Measures
  • 2.1 - Constructive approach: Projective Measurements
  • 2.2 - Constructive approach: Naimark's Dilation Theorem
  • 3 - Application: Unambiguous state discrimination

 

Week 9

• Measurement Theory
  • 4.1 - Quantum operations: Intro (Sorry for metallic sound. No idea what went wrong. Rebooted computer, then it worked better).
  • 4.2 - Quantum operations: Partial transpose
  • 4.3 - Quantum operations: Complete positivity
  • 4.4 - Quantum operations: Kraus operators and post-measurement states
  • 5.1 - The measurement problem: Overview
  • 5.2 - The measurement problem: Modeling the measurement device

 

Week 10

• Joint measurability
  • 1: Heisenberg revisited
  • 2: Criterion for joint measurability
  • 3: Busch's program
  • 4: Jointly measurable unsharp spin-1/2 POVMs

Further reading:

• Original paper, obituary.
• Talk by Paul Busch: https://www.youtube.com/watch?v=eN2aYgfIzow

 

Week 11

• Journal Club: The Ghost in the Quantum Turing Machine