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

TL;DR

This paper introduces generalized probabilistic theories (GPTs), explores phenomena beyond quantum theory, and reconstructs quantum theory from operational principles, explaining why quantum bits are represented by a three-dimensional Bloch ball.

Contribution

It provides an overview of GPTs, discusses potential phenomena beyond quantum mechanics, and offers a reconstruction of quantum theory from fundamental principles.

Findings

Quantum bits are represented by a three-dimensional Bloch ball.

Quantum theory can be reconstructed from principles like Tomographic Locality and Continuous Reversibility.

Potential phenomena beyond quantum include superstrong nonlocality and higher-order interference.

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.