A tight Tsirelson inequality for infinitely many outcomes

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.

5 pages, 2 figures; improved presentation, change in title, as published.

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• Bell nonlocality with a single shot
• Violations of Bell inequalities from random pure states
• Maximal Non-Classicality in Multi-Setting Bell Inequalities
• A proof of Bell's inequality in quantum mechanics using causal interactions
• Quantum Bell inequalities from Information Causality -- tight for Macroscopic Locality
• Single Bell inequality to detect genuine nonlocality in three-qubit genuinely entangled states
• Beating one bit of communication with and without quantum pseudo-telepathy