Classical and Quantum Probability

TL;DR

This paper surveys the evolution of probability theory from classical to quantum, highlighting key developments like Brownian motion, Ito calculus, and quantum stochastic calculus, and discusses their implications and differences.

Contribution

It introduces a comprehensive overview of classical and quantum probability, emphasizing the development of quantum stochastic calculus and its relation to classical probability.

Findings

Quantum probability extends classical concepts.

Quantum stochastic calculus provides new tools.

Bell's inequality explains limits of hidden variable theories.

Abstract

We survey the development of probability from 1900, starting with Bachelier's theory of speculation. Fisher information appears in the theory of estimation. We touch on Brownian motion, and the Wiener integral. The Ito calculus, and its relation to to the heat equation, is mentioned. Quantum theory is introduced as a generalisation of probability, rather than of mechanics. The weakness of attempts to describe quantum theory in terms of hidden variables is explained, by a simple proof of Bell's inequality. Quantum versions of the Langevin equation are discussed, and the theory of continuous tensor products is used to give a possible quantum version. The quantum stochastic calculus of Barnett, Wilde and the author, as well as that of Parthasarathy and Hudson, is introduced.