# Tight Bounds for the Pearle-Braunstein-Caves Chained Inequality Without the Fair-Coincidence Assumption

## TL;DR

This paper derives tight bounds on the coincidence probability for the Pearle-Braunstein-Caves chained Bell inequality, eliminating the coincidence-time loophole without assuming fair coincidences, which enhances the reliability of Bell test experiments.

## Contribution

It provides the minimum coincidence probability bounds for the PBC inequality for any number of settings, without relying on the fair-coincidence assumption.

## Key findings

- Derived tight bounds for the PBC inequality's coincidence probability.
- Identified critical coincidence probability thresholds to close the loophole.
- Applicable to quantum communication protocols like Quantum Key Distribution.

## Abstract

In any Bell test, loopholes can cause issues in the interpretation of the results, since an apparent violation of the inequality may not correspond to a violation of local realism. An important example is the coincidence-time loophole that arises when detector settings might influence the time when detection will occur. This effect can be observed in many experiments where measurement outcomes are to be compared between remote stations because the interpretation of an ostensible Bell violation strongly depends on the method used to decide coincidence. The coincidence-time loophole has previously been studied for the Clauser-Horne-Shimony-Holt (CHSH) and Clauser-Horne (CH) inequalities, but recent experiments have shown the need for a generalization. Here, we study the generalized "chained" inequality by Pearle-Braunstein-Caves (PBC) with two or more settings per observer. This inequality has applications in, for instance, Quantum Key Distribution where it has been used to re-establish security. In this paper we give the minimum coincidence probability for the PBC inequality for all N and show that this bound is tight for a violation free of the fair-coincidence assumption. Thus, if an experiment has a coincidence probability exceeding the critical value derived here, the coincidence-time loophole is eliminated.