Computability in Basic Quantum Mechanics

TL;DR

This paper develops a framework for understanding computability in basic quantum mechanics using the category $ extbf{QCB}_0$, establishing effective representations for states and observables.

Contribution

It introduces a notion of computability for quantum data structures within the category $ extbf{QCB}_0$, connecting quantum measures and valuations to effective representations.

Findings

Effective version of von Neumann's Spectral Theorem established

Quantum states and observables are shown to reside within $ extbf{QCB}_0$ category

Provides a foundation for computability in quantum mechanics

Abstract

The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space $\mathcal{H}$. In terms of the Hilbert lattice $\mathcal{L}$ of closed linear subspaces of $\mathcal{H}$ the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch's Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category $\mathbf{QCB}_0$ which is equivalent to the category $\mathbf{AdmRep}$ of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann's Spectral Theorem.