Operator Space theory: a natural framework for Bell inequalities

TL;DR

This paper demonstrates that Operator Space Theory offers a comprehensive mathematical framework for analyzing Bell inequalities, revealing how quantum violations scale with local Hilbert space dimension and impacting various quantum information applications.

Contribution

It introduces Operator Space Theory as a natural framework for Bell inequalities and quantifies violation scaling with local Hilbert space dimension.

Findings

Quantum states with local dimension n can violate Bell inequalities by approximately √n / log^2 n.

Applications include improved noise resistance and Hilbert space dimension estimation.

Provides insights into communication complexity in quantum systems.

Abstract

In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\frac{\sqrt{n}}{\log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.