Quantum complementarity and logical indeterminacy

TL;DR

This paper explores how quantum states and measurements relate to logical indeterminacy, showing that undecidable propositions correspond to measurements with completely random outcomes in quantum systems.

Contribution

It demonstrates encoding mathematical propositions in quantum states and links logical undecidability to measurement randomness in mutually unbiased bases.

Findings

Undecidable propositions correspond to random measurement outcomes.

Quantum states can encode logical propositions.

Measurement randomness indicates logical indeterminacy.

Abstract

Whenever a mathematical proposition to be proved requires more information than it is contained in an axiomatic system, it can neither be proved nor disproved, i.e. it is undecidable, or logically undetermined, within this axiomatic system. I will show that certain mathematical propositions on a d-valent function of a binary argument can be encoded in d-dimensional quantum states of mutually unbiased basis (MUB) sets, and truth values of the propositions can be tested in MUB measurements. I will then show that a proposition is undecidable within the system of axioms encoded in the state, if and only if the measurement associated with the proposition gives completely random outcomes.