Upper Bounding Hilbert Space Dimensions which can Realize all the Quantum Correlations

TL;DR

This paper establishes new upper bounds on the Hilbert space dimensions needed for quantum systems to realize all correlations in Bell scenarios, showing that small, finite dimensions suffice in many cases.

Contribution

The authors provide novel upper bounds on Hilbert space dimensions for realizing quantum correlations, including multipartite and nonstandard scenarios, expanding understanding of finite-dimensional quantum systems.

Findings

A bipartite system with one party limited to a qubit can realize all extremal correlations.

Both parties can be restricted to qubits to reproduce all quantum correlations in bipartite scenarios.

Upper bounds are extended to multipartite and communication-involved Bell scenarios.

Abstract

We introduce novel upper bounds on the Hilbert space dimensions required to realize quantum correlations in Bell scenarios. We start by considering bipartite cases wherein one of the two parties has two settings and two outcomes. Regardless of the number of measurements and outcomes of the other party, the Hilbert space dimension of the first party can be limited to two while still achieving all convexly extremal quantum correlations. We then leverage Schmidt decomposition to show that the remaining party can losslessly also be restricted to a qubit Hilbert space. We then extend this idea to multipartite scenarios. We also adapt our results to provide upper bounds of local Hilbert space dimensions to achieve any quantum correlation, including convexly non-extremal correlations, by utilizing Caratheodory's theorem. Finally, we generalize our results to nonstandard Bell scenarios with communication. Taken together, our results fill in several previously unresolved aspects of the problem of determining sufficient Hilbert space dimensionality, expanding the collection of scenarios for which finite-dimensional quantum systems are known to be sufficient to reproduce any quantum correlation.