Tsirelson's Problem

TL;DR

This paper investigates Tsirelson's problem, exploring whether quantum correlations from commuting observables can always be approximated by finite-dimensional tensor product models, linking the problem to finite-dimensional approximation.

Contribution

The paper establishes the equivalence between Tsirelson's problem and the approximability of quantum correlations by finite-dimensional systems.

Findings

Tsirelson's problem is equivalent to the finite-dimensional approximation of quantum correlations.

Finite-dimensional models can approximate certain quantum correlation functions.

Physical examples satisfying the approximation condition are discussed.

Abstract

The situation of two independent observers conducting measurements on a joint quantum system is usually modelled using a Hilbert space of tensor product form, each factor associated to one observer. Correspondingly, the operators describing the observables are then acting non-trivially only on one of the tensor factors. However, the same situation can also be modelled by just using one joint Hilbert space, and requiring that all operators associated to different observers commute, i.e. are jointly measurable without causing disturbance. The problem of Tsirelson is now to decide the question whether all quantum correlation functions between two independent observers derived from commuting observables can also be expressed using observables defined on a Hilbert space of tensor product form. Tsirelson showed already that the distinction is irrelevant in the case that the ambient Hilbert space is of finite dimension. We show here that the problem is equivalent to the question whether all quantum correlation functions can be approximated by correlation function derived from finite-dimensional systems. We also discuss some physical examples which fulfill this requirement.