Bell violation by the vacuum at arbitrarily large distances

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.

Problem

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

\beta(L)=\tfrac12\sup\Big|\,\omega\big(A_1(B_1+B_2)+A_2(B_1-B_2)\big)\Big|,

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 $\sqrt2$ .

Question. Is $\beta(L)>1$ for arbitrarily large $L$ ? Or is there a distance beyond which the vacuum correlations satisfy the CHSH inequality exactly?

Background

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.

What is known

The vacuum is not separable at any distance, and an analogue of the positive-partial-transpose condition fails for regions at any separation [6].
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.
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.
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). Tangent diamonds touch and wedges are unbounded, so neither result controls $\beta(L)$ at large separation.

What would count as progress

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.
A proof that $\beta(L)=1$ for all $L$ beyond some $L_0$ , which would separate Bell nonlocality from distillability in quantum field theory.
Intermediate steps: rigorous violations for bounded, separated regions at distances of several Compton wavelengths, or the problem solved first in $1+1$ dimensions.

References

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
R. F. Werner and M. M. Wolf, "Bell inequalities and entanglement", review for Quantum Information and Computation (2001). arXiv:quant-ph/0107093
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
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
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
R. Verch (contact), "Bell inequalities for long range vacuum correlations", Open Quantum Problems, IQOQI Vienna, problem 12 (posted 2002). Archived page

Source

Imported from QIQCOP Zoo 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.