A tight Tsirelson inequality for infinitely many outcomes

¹  Mathematical Institute, Leiden University - Niels Bohrweg 1, 2333 CA Leiden, The Netherlands, EU
²  Department of Statistics, São Paulo University - Rua do Matão 1010, 05508-090, São Paulo, Brazil
³  Blackett Laboratory, Imperial College London - Prince Consort Road, London SW7 2AZ, UK, EU
⁴  Institute for Theoretical Physics, Utrecht University - Leuvenlaan 4, 3584 CE Utrecht, The Netherlands, EU
⁵  CWI - Science Park 123, 1098 XG Amsterdam, The Netherlands, EU
⁶  Department of Mathematics, University of Iceland - Dunhaga 3, 107 Reykjavik, Iceland

Corresponding author: zohren@math.leidenuniv.nl

Received: 11 March 2010
Accepted: 29 March 2010

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.