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Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes)

by Markus P. Müller

Submission summary

As Contributors: Markus Müller
Arxiv Link: (pdf)
Date accepted: 2021-03-24
Date submitted: 2021-03-16 08:22
Submitted by: Müller, Markus
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. In particular, I show why a quantum bit is described by a Bloch ball, why it is three-dimensional, and how one obtains the complex numbers and operators of the usual representation of QT.

Published as SciPost Phys. Lect. Notes 28 (2021)

Author comments upon resubmission

I would like to thank the referee for their careful check of the manuscript and for their helpful comments. I have corrected all the typos, and I have implemented the following clarifications (the numbers are those given by the referee’s comments):

List of changes

6: After the half sentence on page 30, saying that e^{(1)} is a valid effect, I have aded the following comment in brackets: “(recall that we assume the no-restriction hypothesis in all of these lecture notes; otherwise, we would need an additional argument to show that e^{(1)} is physically allowed).”

7: I have expanded the explanation where Eq. (6) comes from; see now the top of page 32.
Namely, it follows from two other lemmas: multiplicativity of the maximally mixed state, and representation of the maximally mixed state as a mixture of perfectly distinguishable pure states. All postulates are used to prove those. Since the lecture notes only intend to give a summary or sketch of the representation, I refer to our paper for the details.

9: I have added a reference to the paper by Lee and Selby on page 12. Note that, in the corresponding paragraph, I am also mentioning some other things that can be done with the GPT framework; in particular, to formulate consistent theories of higher-order interference.

Submission & Refereeing History

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Resubmission 2011.01286v3 on 16 March 2021

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