---
title: "A readiness signal makes the Bell trial countable"
description: "Why does a herald before the setting choice differ from keeping only favourable detections afterwards?"
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language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-evidence/event-ready-bell-trials-finite-statistics/"
last_modified: "2026-10-24"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# A readiness signal makes the Bell trial countable

Sandbox Physics · 2026-10-16

Why does a herald before the setting choice differ from keeping only favourable detections afterwards?

## The herald decides which attempts count as trials

In 2015 Hensen and colleagues tested two electron spins separated by 1.3 kilometres. A spin is a quantum system with two possible readout results along a chosen direction. Before the two stations chose those directions, a photon signal at a central station announced that an entangled pair of spins was ready. Efficient spin readout then supplied an outcome on every accepted trial.

Compare two ways of selecting data. In the first, the ready flag is set before new inputs are generated, and every output after that is kept. In the second, inputs and outputs already exist and only the favourable coincidences are kept. Both select data, but the order of events gives a local model very different openings.

Read the evidence: B. Hensen et al. (2015), Experimental Loophole-Free Violation of a Bell Inequality Using Entangled Electron Spins Separated by 1.3 km — author manuscript, arXiv:1508.05949

## A simple game shows the local limit

The CHSH game gives the comparison a simple rule. Each side receives either zero or one and replies with either zero or one. They win when the replies agree, except when both inputs are one, when they must disagree. In the expression, a and b are the inputs, u and v are the replies, and the circled plus means zero for matching replies and one for different replies. The fresh inputs are uniform and independent of any shared instructions.

There are only four input pairs. A fixed local instruction gives each side an answer for each of its two inputs. Meeting the parity rule on any three pairs forces a loss on the remaining one. Randomly mixing shared instructions cannot push the average above three wins in four.

$$
\begin{aligned}u\oplus v&=a b,\\ P_{\mathrm{win}}^{\mathrm{local}}&\leq\frac34\end{aligned}
$$

Read the evidence: D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026

## The reported CHSH value and the game score are different numbers

The 2015 spin experiment recorded 245 trials and 196 game wins. It also reported a CHSH correlation value of 2.42, with a conventional standard deviation of 0.20. The settings did not occur exactly equally often, so the correlation value and a value converted from the overall win rate need not match.

As a quick check, the win fraction is four fifths. Converting it to a CHSH value, assuming exactly equal input frequencies, gives 2.4 instead of 2.42. That gap does not make the report inconsistent. It shows why a statistic should be quoted as it was defined, without swapping in a convenient rounded value.

Read the evidence: B. Hensen et al. (2015), Experimental Loophole-Free Violation of a Bell Inequality Using Entangled Electron Spins Separated by 1.3 km — author manuscript, arXiv:1508.05949 D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026

## Devices with memory still face the same bound

As long as each trial’s inputs are fresh and independent, a local device that remembers every earlier input and outcome still faces the same upper bound on its chance of winning that trial. A finite-sample analysis can apply the bound trial by trial, without assuming the apparatus behaves identically every time.

With a fixed number of trials and ideal uniform inputs, the bound on the game score comes from the upper tail of a binomial distribution with success probability three quarters. Predictable inputs can change that bound, and choosing when to stop can change which test applies. A reported p-value therefore has to come with an account of how the experiment handled its imperfections.

Hensen’s analysis, which allowed for device memory and the measured imperfections in the settings, reported a p-value of about 0.039. It says how extreme the observed statistic would be if a specified local model were true. It does not say there is a 3.9 percent chance that locality is true.

Before quoting a significance number, find out which trials were accepted, how they were scored, what was assumed about the settings and when the experiment stopped. A summary based on normal errors and a test that allows adversarial memory answer different statistical questions. Both still need the assumption that the settings are chosen independently.

Read the evidence: D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026 B. Hensen et al. (2015), Experimental Loophole-Free Violation of a Bell Inequality Using Entangled Electron Spins Separated by 1.3 km — author manuscript, arXiv:1508.05949

## Different platforms solve the loss problem differently

Another route used photons. Giustina and colleagues’ 2015 experiment combined high-efficiency detectors with an inequality that explicitly counts nondetections. The spin and optical platforms therefore solve the same loss problem with different trial definitions, scores and timing designs.

What the readiness signal buys is a definite order: preparation succeeds, new settings are chosen, and the accepted outcomes are all retained. That order lets inefficient preparation coexist with a meaningful Bell test, as long as the later readout and independence requirements are met.

Read the evidence: Giustina et al. (2015), Significant-loophole-free test of Bell’s theorem with entangled photons

## Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

- [See when the herald arrives and check every accepted outcome window](https://sandboxphysics.com/labs/bell-chsh/)
- [Compare conditional protocol runs](https://sandboxphysics.com/labs/quantum-teleportation/)

## Sources and reading notes

We read Hensen and colleagues’ 2015 arXiv manuscript, including the 245-trial result, and Elkouss and Wehner’s statistical analysis. We also read Giustina and colleagues’ 2015 photon paper to compare how the two experiments handled trials. The game enumeration and the ideal binomial tail are our own explanatory calculations; they do not reanalyze the original event log. The linked model assumes an ideal nondestructive photon-pair readiness flag, retains every accepted miss and timing failure, and does not reproduce the historical spin platform or its 245 trials.

1. [B. Hensen et al. (2015), Experimental Loophole-Free Violation of a Bell Inequality Using Entangled Electron Spins Separated by 1.3 km — author manuscript, arXiv:1508.05949](https://arxiv.org/pdf/1508.05949)
2. [D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026](https://arxiv.org/pdf/1510.07233)
3. [Giustina et al. (2015), Significant-loophole-free test of Bell’s theorem with entangled photons](https://arxiv.org/abs/1511.03190)

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