Is Hilbert space discrete?

TL;DR

This paper argues that if spacetime is discrete due to quantum gravity, then Hilbert space may also be inherently discrete, impacting quantum information and gravity theories.

Contribution

It introduces the idea that spacetime discretization implies a natural discretization of Hilbert space, a novel perspective linking quantum gravity and quantum information.

Findings

Discretization of spacetime suggests Hilbert space may be discrete.

Experiments cannot exclude Hilbert space discreteness in a universe with minimal length.

Implications for quantum information and quantum gravity are discussed.

Abstract

We show that discretization of spacetime naturally suggests discretization of Hilbert space itself. Specifically, in a universe with a minimal length (for example, due to quantum gravity), no experiment can exclude the possibility that Hilbert space is discrete. We give some simple examples involving qubits and the Schrodinger wavefunction, and discuss implications for quantum information and quantum gravity.