A quantum mechanical bound for CHSH-type Bell inequalities

TL;DR

This paper derives a simple quantum mechanical Tsirelson bound for CHSH-type Bell inequalities using singular value decomposition, providing insights into optimal measurements, Hilbert space dimensions, and measurement setting adjustments.

Contribution

It introduces a straightforward mathematical method to determine Tsirelson bounds, enhancing understanding of quantum limits in Bell tests and their physical implications.

Findings

Derived a simple Tsirelson bound using singular value decomposition

Identified optimal measurement observables for Bell inequalities

Extended bounds to rotated measurement settings and dimension constraints

Abstract

Many typical Bell experiments can be described as follows. A source repeatedly distributes particles among two spacelike separated observers. Each of them makes a measurement, using an observable randomly chosen out of several possible ones, leading to one of two possible outcomes. After collecting a sufficient amount of data one calculates the value of a so-called Bell expression. An important question in this context is whether the result is compatible with bounds based on the assumptions of locality, realism and freedom of choice. Here we are interested in bounds on the obtained value derived from quantum theory, so-called Tsirelson bounds. We describe a simple Tsirelson bound, which is based on a singular value decomposition. This mathematical result leads to some physical insights. In particular the optimal observables can be obtained. Furthermore statements about the dimension of the underlying Hilbert space are possible. Finally, Bell inequalities can be modified to match rotated measurement settings, e.g. if the two parties do not share a common reference frame.