A Non-Probabilistic Model of Relativised Predictability in Physics

Alastair Abbott,Karl Svozil,Cristian Calude

doi:10.3390/info6040773

Abstract

Unpredictability is an important concept throughout physics and plays a central role in quantum information theory. Despite this, little effort has been devoted to studying generalised notions or models of (un)predictability in physics. In this paper, we continue the programme of developing a general, non-probabilistic model of (un)predictability in physics. We present a more refined model that is capable of studying different degrees of “relativised” unpredictability. This model is based on the ability of an agent, acting via uniform, effective means, to predict correctly and reproducibly the outcome of an experiment using finite information extracted from the environment. We use this model to study the degree of unpredictability certified by different quantum phenomena further, showing that quantum complementarity guarantees a form of relativised unpredictability that is weaker than that guaranteed by Kochen–Specker-type value indefiniteness. We exemplify further the difference between certification by complementarity and value indefiniteness by showing that, unlike value indefiniteness, complementarity is compatible with the production of computable sequences of bits.

Highlights

Many physical processes and phenomena are intuitively thought of as unpredictable: the roll of a die, the evolution of weather systems and the outcomes of quantum measurements, to mention a few.While ad hoc definitions of unpredictability may exist within certain domains, little work has been done towards developing a more general understanding of the concept
By considering prediction agents with access to restricted sets of extractors with which to obtain information for prediction, this model allows various intermediate degrees of prediction to be formalised. Models of prediction such as this can be applied to arbitrary physical systems, we have discussed in detail their utility in helping to understand quantum unpredictability, which plays a key role in quantum information and cryptography
Unlike measurements certified by value indefiniteness, those certified by complementarity alone are not necessarily unpredictable: they are unpredictable relative to the ability of the predicting agent to access the values of complementary observables, a more epistemic, relativised notion of predictability