A tight Tsirelson inequality for infinitely many outcomes

TL;DR

This paper derives a new tight quantum bound for the CGLMP Bell inequality with infinitely many outcomes, showing optimal violations occur with specific measurements and states, relevant for experimental quantum physics.

Contribution

It introduces a novel tight inequality for quantum violations of the CGLMP inequality with infinite outcomes, without assumptions on Hilbert space dimension.

Findings

Maximal violation achieved by conjectured optimal measurements

Optimal state is pure but not maximally entangled

Provides an approximate state approaching the optimal in the infinite limit

Abstract

We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.