# Gull’s Theorem Revisited

## Abstract

**:**

## 1. Introduction

#### Bell’s Theorem as a Theorem of Distributed Computing

`stackexchange`discussion [16]) also attempts to decode Gull’s proof, but in our opinion, is also incomplete, becoming stuck at the same point as we did.

## 2. Gull’s Theorem: A Distributed Computer Simulation of the Singlet Correlations Is Impossible

#### A Classical Distributed Computer Simulation of the Singlet Correlations Is Impossible

## 3. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bell, J.S. On the Einstein Podolsky Rosen paradox. Physics
**1964**, 1, 195–200. Available online: https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf (accessed on 5 May 2022). [CrossRef][Green Version] - Hensen, B.; Bernien, H.; Dréau, A.; Reiserer, A.; Kalb, N.; Blok, M.S.; Ruitenberg, J.; Vermeulen, R.F.; Schouten, R.N.; Abellán, C.; et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature
**2015**, 52, 682–686. [Google Scholar] [CrossRef] [PubMed] - Gill, R.D. Statistics, Causality and Bell’s Theorem. Stat. Sci.
**2014**, 29, 512–528. [Google Scholar] [CrossRef] - Gull, S. Quantum Acausality and Bell’s Theorem. 2016. Available online: http://www.mrao.cam.ac.uk/~steve/maxent2009/images/bell.pdf (accessed on 5 May 2022).
- Gill, R.D. Comment on “Dr. Bertlmann’s Socks in a Quaternionic World of Ambidextral Reality”. IEEE Access
**2021**, 9, 44592–44598. [Google Scholar] [CrossRef] - Gill, R.D. Comment on “Bell’s theorem versus local realism in a Quaternionic model of physical space”. IEEE Access
**2021**, 9, 154933–154937. [Google Scholar] [CrossRef] - Gill, R.D. Does Geometric Algebra provide a loophole to Bell’s Theorem? Entropy
**2020**, 22, 61. [Google Scholar] [CrossRef][Green Version] - Gill, R.D. Comment on “Quantum Correlations Are Weaved by the Spinors of the Euclidean Primitives”. R. Soc. Open Sci.
**2022**, 9, 201909. [Google Scholar] [CrossRef] - Tsirelson, B. Entanglement (Physics). 2010; Knowino Internet Encyclopedia, Originally Citizendium. Last Edited 2010. Available online: https://www.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Entanglement_(physics).html (accessed on 5 May 2022).
- Clauser, J.F.; Horne, M.A.; Shimony, A.; Holt, R.A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett.
**1969**, 23, 880–884. [Google Scholar] [CrossRef][Green Version] - Fine, A. Hidden variables, joint probabilities, and the Bell inequalities. Phys. Rev. Lett.
**1982**, 48, 291–295. [Google Scholar] [CrossRef] - Boole, G. An Investigation of the Laws of Thought, on Which Are Funded the Mathematical Theories of Logic and Probabilities; Macmillan: London, UK, 1853; Available online: https://www.gutenberg.org/files/15114/15114-pdf.pdf (accessed on 5 May 2022).
- Vorob’ev, N.N. Consistent families of measures and their extension. Theory Probab. Appl.
**1962**, 7, 147–163. [Google Scholar] [CrossRef] - Einstein, A.; Podolsky, B.; Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev.
**1935**, 16, 777–780. [Google Scholar] [CrossRef][Green Version] - Bohm, D.; Aharonov, Y. Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky. Phys. Rev.
**1957**, 108, 1070–1076. [Google Scholar] [CrossRef] - Anonymous. Question posted on “Stackexchange” by “Eran Medan”, Reply by “Chiral Anomaly”. 2020. Available online: https://physics.stackexchange.com/questions/547039/help-understanding-prof-steve-gulls-explanation-of-bells-theorem (accessed on 5 May 2022).
- Schoenstadt, A.L. An Introduction to Fourier Analysis. Fourier Series, Partial Differential Equations and Fourier Transforms; Naval Postgraduate School: Monterey, CA, USA, 1992; Available online: https://faculty.nps.edu/bneta/soln3139.pdf (accessed on 5 May 2022).
- Razmi, H. Is the Clauser–Horne model of Bell’s theorem completely stochastic? J. Phys. A
**2005**, 38, 3661–3664. [Google Scholar] [CrossRef] - Razmi, H. A New Proof of Bell’s Theorem Based on Fourier Series Analysis. Ann. Fond. Louis Broglie
**2007**, 32, 69–76. Available online: https://aflb.minesparis.psl.eu/AFLB-321/aflb321m555.pdf (accessed on 5 May 2022). - Tumulka, R. Comment on “A New Proof of Bell’s Theorem Based on Fourier Series Analysis”. Ann. Fond. Louis Broglie
**2009**, 34, 39–41. Available online: https://aflb.minesparis.psl.eu/AFLB-341/aflb341m641.pdf (accessed on 5 May 2022). - Razmi, H. Reply to “Comment on A New Proof of Bell’s Theorem Based on Fourier Series Analysis” by Roderich Tumulka. Ann. Fond. Louis Broglie
**2009**, 34, 43–44. Available online: https://aflb.minesparis.psl.eu/AFLB-341/aflb341m641b.pdf (accessed on 5 May 2022). - Gill, R.D. The Triangle Wave Versus the Cosine: How Classical Systems Can Optimally Approximate EPR-B Correlations. Entropy
**2020**, 22, 287. [Google Scholar] [CrossRef][Green Version] - Gill, R.D. Accardi contra Bell: The impossible coupling. In Mathematical Statistics and Applications: Festschrift for Constance van Eeden; Lecture Notes–Monograph Series; Moore, M., Leger, C., Froda, S., Eds.; Institute of Mathematical Statistics: Hayward, CA, USA, 2003; pp. 133–154. Available online: https://arxiv.org/abs/quant-ph/0110137 (accessed on 5 May 2022).
- Gill, R.D. Time, Finite Statistics, and Bell’s Fifth Position. arXiv
**2003**, arXiv:quant-ph/0301059. [Google Scholar] - Bierhorst, P. A robust mathematical model for a loophole-free Clauser–Horne experiment. J. Phys. A Math. Theor.
**2015**, 48, 195302. [Google Scholar] [CrossRef][Green Version] - Elkouss, D.; Wehner, S. (Nearly) optimal P values for all Bell inequalities. NPJ Quantum Inf.
**2016**, 2, 16026. [Google Scholar] [CrossRef][Green Version] - Kaszilowski, D.; Zukowski, M. Bell theorem involving all possible local measurements. Phys. Rev. A
**2000**, 61, 022114. [Google Scholar] [CrossRef][Green Version] - Van der Horst, V. Locality of Quantum-Related Measurement Statistics. Bachelor’s Thesis, Leiden University, Leiden, The Netherlands, 2021. Available online: https://studenttheses.universiteitleiden.nl/handle/1887/3196304 (accessed on 5 May 2022).
- Van Dam, W.; Grunwald, P.; Gill, R.D. The statistical strength of nonlocality proofs. IEEE Trans. Inf. Theory
**2005**, 51, 2812–2835. [Google Scholar] [CrossRef]

**Figure 1.**A possible correlation function allowed by quantum mechanics (blue) and one allowed by local realism (red). The asterisks mark the predictions for the angle pairs typically chosen in a Bell-type experiment. Alice uses angles ${0}^{\circ}$, ${90}^{\circ}$; Bob uses ${45}^{\circ}$ and ${135}^{\circ}$. The four differences (Bob minus Alice) are $45-0=45$, $45-90=-45$, $135-0=135$, $135-90=45$. One correlation is large and positive, three are large and negative.

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Gill, R.D.
Gull’s Theorem Revisited. *Entropy* **2022**, *24*, 679.
https://doi.org/10.3390/e24050679

**AMA Style**

Gill RD.
Gull’s Theorem Revisited. *Entropy*. 2022; 24(5):679.
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**Chicago/Turabian Style**

Gill, Richard D.
2022. "Gull’s Theorem Revisited" *Entropy* 24, no. 5: 679.
https://doi.org/10.3390/e24050679