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 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.
Highlights
It is true that the perhaps better known instance of this question asks whether quantum theory (QT) would somehow break down and become classical in the macroscopic regime: for example, spontaneous collapse models (Ghirardi, Rimini, and Weber (1986) [36], Bassi et al (2013) [12]) try to account for the emergence of a classical world from quantum mechanics via dynamical modifications of the Schödinger equation
A fascinating complementary development in Quantum Foundations research — the one that these lectures will be focusing on — is to explore the exact opposite: could nature be even “more crazy” than quantum? Could physics allow for even stronger-than-quantum-correlations, produce more involved interference patterns than allowed by QT, or enable even more magic technology than what we currently consider possible? If classical physics is an approximation of quantum physics, could quantum physics be an approximation of something even more general?
It tells us that QT is just one possible theory among many others that could potentially describe the statistical aspects of nature
Summary
Everybody can take an existing theory and modify it arbitrarily; but the art is to find a modification that is self-consistent, physically meaningful, and consistent with other things we know about the world That these desiderata are not so easy to satisfy is illustrated by Weinberg’s (1989) [85] attempt to introduce nonlinear corrections to quantum mechanics. Superluminal information transfer is in direct conflict with Special Relativity, showing that QT is in some sense a very “rigid” theory that cannot be so modified (see Simon et al, 2001 [77]) This suggests to search for modifications of QT not on a formal, but on an operational level: perhaps a more fruitful way forward is to abandon the strategy of direct modification of any of QT’s equations, and instead to reconsider the basic framework which we use to describe simple laboratory situations. To get an intuition for the basic assumptions of the GPT framework, let us first discuss two examples of potential phenomena that would transcend classical and quantum physics: superstrong nonlocality and higher-order interference
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