Computable Functions, the Church-Turing Thesis and the Quantum Measurement Problem

TL;DR

This paper explores how the quantum measurement problem and the finiteness of Hilbert space's granularity relate to the Church-Turing thesis, suggesting fundamental limits on physical computation.

Contribution

It proposes that the finiteness of Hilbert space's fine-graining reconciles quantum mechanics with the Church-Turing thesis, linking physical and computational principles.

Findings

Quantum observables could challenge the Church-Turing thesis if physically realized.

A finite, computational model of quantum measurement aligns with the Church-Turing thesis.

Finiteness of Hilbert space's granularity constrains physical realizability of certain quantum operations.

Abstract

It is possible in principle to construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolution, would contradict the Church-Turing thesis, which lies at the foundation of computer science. Elsewhere we have argued that the quantum measurement problem implies a finite, computational model of the measurement and evolution of quantum states. If correct, this approach helps to identify the key feature that can reconcile quantum mechanics with the Church-Turing thesis: finitude of the degree of fine-graining of Hilbert space. This suggests that the Church-Turing thesis constrains the physical universe and thereby highlights a surprising connection between purely logical and algorithmic considerations on the one hand and physical reality on the other.