Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation

TL;DR

This paper establishes a simple lower bound on the minimal Hilbert space dimension required to generate bipartite quantum correlations, aiding in understanding quantum resource requirements and ruling out finite-dimensional representations.

Contribution

It introduces an easy-to-compute lower bound on the Hilbert space dimension needed for bipartite quantum correlations, which is tight for many known cases and can detect non-convexity.

Findings

Bound is tight for many correlations

Bound is multiplicative under product correlations

Can witness non-convexity of certain quantum sets

Abstract

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the non-convexity of certain restricted-dimensional quantum correlations.