Constructor Theory of Probability

TL;DR

This paper recasts the problem of quantum probability within constructor theory, showing how stochasticity and decision-making can emerge in non-probabilistic superinformation theories.

Contribution

It generalizes the decision-theoretic approach to quantum probability using constructor theory, establishing conditions under which superinformation theories can support probabilistic decision-making.

Findings

Unpredictability of measurement outcomes arises in superinformation theories.

Stochasticity in repeated measurements can emerge under certain conditions.

Superinformation theories can support decision-making as if probabilistic, via a generalized Deutsch-Wallace argument.

Abstract

Unitary quantum theory, having no Born Rule, is non-probabilistic. Hence the notorious problem of reconciling it with the unpredictability and appearance of stochasticity in quantum measurements. Generalising and improving upon the so-called 'decision-theoretic approach' (Deutsch, 1999; Wallace, 2003, 2007, 2012), I shall recast that problem in the recently proposed constructor theory of information - where quantum theory is represented as one of a class of superinformation theories, which are local, non-probabilistic theories conforming to certain constructor-theoretic conditions. I prove that the unpredictability of measurement outcomes (to which I give an exact meaning via constructor theory), necessarily arises in superinformation theories. Then I explain how the appearance of stochasticity in (finitely many) repeated measurements can arise under superinformation theories. And I establish sufficient conditions for a superinformation theory to inform decisions (made under it) as if it were probabilistic, via a Deutsch-Wallace-type argument - thus defining a class of decision-supporting superinformation theories. This broadens the domain of applicability of that argument to cover constructor-theory compliant theories. In addition, in this version some of the argument's assumptions, previously construed as merely decision-theoretic, follow from physical properties expressed by constructor-theoretic principles.