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. Generalizing and improving upon the so-called ‘decision-theoretic approach’, 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 constructor theory gives an exact meaning) 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.
Highlights
Quantum theory without the Born Rule is deterministic [1]
Principle I requires subsidiary theories to have two crucial properties: (i) they must support a topology on the set of physical processes they apply to, which gives a meaning to a sequence of approximate constructions, converging to an exact performance of T; (ii) they must be non-probabilistic—as they must be expressed exclusively as statements about possible/impossible tasks
I have reformulated the problem of reconciling unitary quantum theory, unpredictability and the appearance of stochasticity in quantum systems, within the constructor theory of information, where unitary quantum theory is a particular superinformation theory—a non-probabilistic theory obeying constructor-theoretic principles
Summary
Quantum theory without the Born Rule (hereinafter: unitary quantum theory) is deterministic [1]. This distinguishes it from (apparent) randomness, which, as I shall explain, requires a quantitative explanation Another key finding of this paper is a sufficient set of conditions for superinformation theories to support a generalization of the decision-theory approach to probability, thereby explaining the appearance of stochasticity, namely that repeated identical measurements have different unpredictable outcomes but are to all appearances, random. My version of the decision-theory argument explains how the numbers fx can inform decisions of a player satisfying non-probabilistic rationality axioms under certain superinformation theories (§7) Those theories would account for the appearance of stochasticity at least as adequately as unitary quantum theory.
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