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Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes)
by Markus Müller
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Authors (as registered SciPost users):  Markus Müller 
Submission information  

Preprint Link:  https://arxiv.org/abs/2011.01286v2 (pdf) 
Date submitted:  20210210 15:18 
Submitted by:  Müller, Markus 
Submitted to:  SciPost Physics Lecture Notes 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
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 higherorder 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 threedimensional, and how one obtains the complex numbers and operators of the usual representation of QT.
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Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2021311 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2011.01286v2, delivered 20210311, doi: 10.21468/SciPost.Report.2684
Report
These lecture notes are about the framework of Generalized Probabilistic Theories (GPTs). This framework explores the set of possible physical theories (possibly going beyond quantum theory) using an operational approach. Notably, this framework allows to see what are (some of) the principles at the core of quantum theory. These possibilities make it an active field in quantum foundations.
The lecture notes start by motivating using such an approach that would go beyond quantum theory. This is followed by the introduction of the framework of GPTs, and finally by the reconstruction of quantum theory itself from L. Hardy's axioms.
While the topic is abstract and quite technical, the manuscript is very clear. The approach is rigorous and well motivated, and the mathematical proofs are given with the right amount of details for the physicist. This manuscript will also be an excellent entry point for the reader in the world of GPTs. Furthermore, the literature review exposed in the manuscript will allow the interested reader to deepen and go beyond the approach exposed in the manuscript. I therefore recommend its publication.
Requested changes
I have noted the following typos/minor mistakes/small questions during my reading:
1  p. 4: Einstein has shown us that two simple physical principle*s* single out […]
2  p. 12: Chribella and Yuan 2016 > Chiribella and Yuan 2016.
3  p. 15: at contains enough dimensions > it contains enough dimensions.
4  p. 15: I think that the given the definition 5, the intersection of A_+ and A_+ should be {0} and not the empty set.
5  p. 18: Since the normalized state*s* span […]
6  p. 30: When discussing Lemma 22, it is mentioned that the effects e^{(1)} and e^{(2)} are physically allowed. It is not clear in the text whether it is a consequence of the no restriction hypothesis or not.
7  p. 31: The justification of Eq. (6) does not appear obvious to me. In particular, without giving the details, what are the axioms/hypotheses that were used in order to obtain (6)?
8  p. 33: irreps > irreducible representations
9  It could be worth mentioning some examples of what is achievable with the GPT framework beyond quantum theory. I am for example thinking about [C. Lee and J. Selby, Proc. R. Soc. A 2018 474 20170732 (2019)], in which the authors show that given some assumptions on the decoherence processes from a GPT to quantum theory, there are some limitations on "super quantum" theories.