Quantum mechanics: The Bayesian theory generalised to the space of Hermitian matrices

TL;DR

This paper generalizes Bayesian probability theory to Hermitian matrices, deriving quantum mechanics' postulates from rational gambling rules, thus showing quantum mechanics as a self-consistent Bayesian framework in complex numbers.

Contribution

It introduces a Bayesian framework for quantum mechanics based on Hermitian matrices, deriving all four postulates from rationality principles.

Findings

Quantum mechanics' postulates are derived from Bayesian rationality rules.

Quantum operations are reinterpreted as probability rules within the Bayesian framework.

The theory confirms quantum mechanics as a self-consistent Bayesian probability theory in complex numbers.

Abstract

We consider the problem of gambling on a quantum experiment and enforce rational behaviour by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalised to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalised Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalisation (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.