On infinite-dimensional state spaces

TL;DR

This paper connects the infinite-dimensionality of quantum systems to operational relations between unitaries, providing a new combinatorial group theory proof and bounds for approximate relations.

Contribution

It introduces an operational criterion involving unitary relations to distinguish finite- and infinite-dimensional quantum systems, with a new proof and bounds for approximate cases.

Findings

Finite-dimensional systems satisfy a specific unitary relation implying commutation.

Infinite-dimensional realizations can violate this relation, indicating infinite-dimensionality.

Provides bounds on dimension when the relation holds approximately.

Abstract

It is well-known that the canonical commutation relation $[x,p]=i$ can be realized only on an infinite-dimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finite-dimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinite-dimensional model in which $[x,p]=i$ holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinite-dimensionality. One drawback of using the canonical commutation relation for this purpose is that it has unclear operational meaning. Here, we identify an operationally well-defined context from which an analogous conclusion can be drawn: if two unitary transformations $U,V$ on a quantum system satisfy the relation $V^{-1}U^2V=U^3$, then finite-dimensionality entails the relation $UV^{-1}UV=V^{-1}UVU$; this implication strongly fails in some infinite-dimensional realizations. This is a result from combinatorial group theory for which we give a new proof. This proof adapts to the consideration of cases where the assumed relation $V^{-1}U^2V=U^3$ holds only up to $\eps$ and then yields a lower bound on the dimension.