Bell violation by the vacuum at arbitrarily large distances

Decide whether the vacuum of a free massive scalar field violates the CHSH inequality between bounded regions at arbitrarily large distances.

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1	# Bell violation by the vacuum at arbitrarily large distances
2
3	The vacuum of a relativistic quantum field is entangled across space: suitable measurements in two separated regions violate a Bell inequality. For a massive field the violation shrinks exponentially as the regions move apart. Nobody knows whether it survives at every distance or vanishes exactly beyond some finite separation.
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5	## Problem
6
7	Take the free massive scalar field in Minkowski spacetime and its vacuum state $\omega$. Fix a bounded spacetime region $O$, let $O_L$ be its translate by a spatial distance $L$, and let $\mathcal{A}(O)$ and $\mathcal{A}(O_L)$ be the algebras of observables localized there. Define the vacuum CHSH value
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9	$$\beta(L)=\tfrac12\sup\Big\|\,\omega\big(A_1(B_1+B_2)+A_2(B_1-B_2)\big)\Big\|,$$
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11	where the supremum runs over self-adjoint $A_1,A_2\in\mathcal{A}(O)$ and $B_1,B_2\in\mathcal{A}(O_L)$ of norm at most $1$. Local hidden-variable models give $\beta\le1$, and quantum theory allows up to $\sqrt2$.
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13	Question. Is $\beta(L)>1$ for arbitrarily large $L$? Or is there a distance beyond which the vacuum correlations satisfy the CHSH inequality exactly?
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15	## Background
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17	Summers and Werner proved in 1985 that the vacuum violates Bell's inequalities, maximally so for suitable nearby regions [1]. Werner and Wolf noted in 2001 that in a massive theory the violation decreases exponentially with separation on the scale of the Compton wavelength, and called its persistence at arbitrarily large distance an open problem [2]. It is No. 12 in the Open Quantum Problems collection of IQOQI Vienna, contact R. Verch, which fixes the free massive scalar field for definiteness [6]. The question asks whether entanglement that is known to exist at all distances also shows up in the simplest Bell test.
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19	## What is known
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21	- The vacuum is not separable at any distance, and an analogue of the positive-partial-transpose condition fails for regions at any separation [6].
22	- Verch and Werner showed that the vacuum, and many other states of any quantum field model, are distillable across arbitrary spacelike distances [3]. Distillation uses many copies and local filtering, so it does not decide the single-copy CHSH value.
23	- With two detectors that couple locally to the field, Reznik, Retzker and Silman showed that correlations between arbitrarily distant regions cannot be reproduced by a local hidden-variable model [4]. This probes the field through auxiliary systems, not through $\mathcal{A}(O)$ and $\mathcal{A}(O_L)$ directly.
24	- Explicit bounded field observables that violate CHSH in the vacuum have been built for a massive scalar field in $1+1$ dimensions, for tangent causal diamonds [5] and for complementary wedges ([Azevedo et al., 2026](https://doi.org/10.1140/epjc/s10052-026-15445-1)). Tangent diamonds touch and wedges are unbounded, so neither result controls $\beta(L)$ at large separation.
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26	## What would count as progress
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28	- A proof that $\beta(L)>1$ for all $L$, for instance a family of explicit observables (functions of smeared fields or Weyl operators) whose CHSH value exceeds 1 by a rigorously bounded, exponentially small amount.
29	- A proof that $\beta(L)=1$ for all $L$ beyond some $L_0$, which would separate Bell nonlocality from distillability in quantum field theory.
30	- Intermediate steps: rigorous violations for bounded, separated regions at distances of several Compton wavelengths, or the problem solved first in $1+1$ dimensions.
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32	## References
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34	1. S. J. Summers and R. Werner, "The vacuum violates Bell's inequalities", Physics Letters A 110, 257–259 (1985). [doi:10.1016/0375-9601(85)90093-3](https://doi.org/10.1016/0375-9601(85)90093-3)
35	2. R. F. Werner and M. M. Wolf, "Bell inequalities and entanglement", review for Quantum Information and Computation (2001). [arXiv:quant-ph/0107093](https://arxiv.org/abs/quant-ph/0107093)
36	3. R. Verch and R. F. Werner, "Distillability and positivity of partial transposes in general quantum field systems", Reviews in Mathematical Physics 17, 545–576 (2005). [doi:10.1142/S0129055X05002364](https://doi.org/10.1142/S0129055X05002364)
37	4. B. Reznik, A. Retzker and J. Silman, "Violating Bell's inequalities in vacuum", Physical Review A 71, 042104 (2005). [doi:10.1103/PhysRevA.71.042104](https://doi.org/10.1103/PhysRevA.71.042104)
38	5. M. S. Guimaraes, I. Roditi and S. P. Sorella, "Class of bounded Hermitian operators for the Bell–Clauser–Horne–Shimony–Holt inequality in quantum field theory", Physical Review D 112, 085009 (2025). [doi:10.1103/x939-vzq7](https://doi.org/10.1103/x939-vzq7)
39	6. R. Verch (contact), "Bell inequalities for long range vacuum correlations", Open Quantum Problems, IQOQI Vienna, problem 12 (posted 2002). [Archived page](https://web.archive.org/web/20231029013530/https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-for-long-range-vacuum-correlations)
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41	## Source
42
43	Imported from [QIQCOP Zoo op_660dbca2258b1a9b](https://qiqc-op.com/problem/op_660dbca2258b1a9b/), curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.