Deriving the Generalised Born Rule from First Principles

TL;DR

This paper derives the generalised Born rule from basic principles in process theories, showing that the rule naturally emerges from the structure of quantum processes and noise introduction.

Contribution

It demonstrates that the generalised Born rule can be derived from minimal process-theoretic assumptions and explores how noise strengthens the scalar-probability identification.

Findings

Any process theory with basic structure is equivalent to one satisfying the Born rule.

Introducing noise strengthens the scalar-probability correspondence to semiring isomorphisms.

The derivation connects foundational principles to the formal structure of quantum probabilities.

Abstract

A basic postulate of modern compositional approaches to generalised physical theories is the generalised Born rule, in which probabilities are postulated to be computable from the composition of states and effects. In this paper we consider whether this postulate, and the strength of the identification between scalars and probabilities, can be argued from basic principles. To this end, we first consider the most naive possible process- theoretic interpretation of textbook quantum theory, in which physical processes (unitaries) along with states and effects (kets and bras) and a probability function from states and effects satisfying just some basic compatibility axioms are identified. We then show that any process theory equipped with such structure is equivalent to an alternative process theory in which the generalised Born rule holds. We proceed to consider introduction of noise into any such theory, and observe that the result of doing so is a strengthening of the identification between scalars and probabilities; from bare monoid homomorphisms to semiring isomorphisms.