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Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes)
by Markus P. Müller
 Published as SciPost Phys. Lect. Notes 28 (2021)
Submission summary
As Contributors:  Markus Müller 
Arxiv Link:  https://arxiv.org/abs/2011.01286v3 (pdf) 
Date accepted:  20210324 
Date submitted:  20210316 08:22 
Submitted by:  Müller, Markus 
Submitted to:  SciPost Physics Lecture Notes 
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.
Published as SciPost Phys. Lect. Notes 28 (2021)
Author comments upon resubmission
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 norestriction 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 higherorder interference.