Long-range vacuum CHSH violation
- Field
- Topic
Problem
Does the vacuum of a massive free scalar Bose field violate the CHSH inequality between bounded localization regions at arbitrarily large spacelike separation? Fix a bounded region \(O\) in Minkowski spacetime, let \(O_L\) be a congruent spacelike translate at separation \(L\), and write \(\mathcal{A}(O)\) and \(\mathcal{A}(O_L)\) for the commuting local von Neumann algebras. For the vacuum state \(\omega_0\), define
where \(A,A'\in\mathcal{A}(O)\) and \(B,B'\in\mathcal{A}(O_L)\) range over self-adjoint contractions. With the normalization in Eq. (1), the local hidden-variable bound is \(1\). The open question is whether
Equation (2) asks for direct, unfiltered two-setting violation by the vacuum restrictions to the two local algebras.
Source
Werner and Wolf explicitly ask whether vacuum Bell violation persists at arbitrarily large separation; the same question is recorded in the open problem collection of Krüger and Werner [WW01], [KW05].
Progress
Summers and Werner proved vacuum Bell violation for suitable spacelike separated observables, including maximal violation in important short-range configurations. This does not control the fixed bounded-region geometry as \(L\) grows [SW85].
In a massive theory, clustering makes the excess \(\beta(L)-1\) decay with separation. Werner and Wolf explicitly noted that decay does not determine whether the excess remains positive at every finite distance or becomes exactly zero beyond a threshold [WW01].
Recent work closes part of the gap between algebraic existence arguments and explicit CHSH observables. Guimaraes, Roditi, and Sorella constructed bounded Hermitian field observables giving vacuum CHSH violation for a massive scalar field in \(1+1\) dimensions, but for tangent causal diamonds [GRS25]. Azevedo et al. obtained violations from unitary-deformed sign observables in complementary wedge algebras [AGG+26]. Tangent diamonds have zero gap and wedges are unbounded, so neither construction controls \(\beta(L)\) for congruent bounded regions as \(L\to\infty\).
Verch and Werner proved long-range nonclassicality through failure of the analogue of positive partial transpose and through distillability under broad algebraic hypotheses [VW05]. Reznik, Retzker, and Silman obtained Bell violations with localized probes coupled to the vacuum [RRS05]. Filtering, multi-copy distillation, or replacing the field algebras by detector systems is strictly weaker than establishing Eq. (2) for the unfiltered vacuum CHSH value.
Comment
The source problem asks precisely for the unbounded-distance alternative in Eq. (2) [KW05]. Persistent entanglement, distillability, generic Bell correlation, and detector-assisted violations do not decide the specified single-copy CHSH supremum.
References
- [SW85]
- S. J. Summers and R. F. Werner, “The Vacuum Violates Bell’s Inequalities,” Physics Letters A 110, 257–259 (1985).DOI
- [WW01]
- R. F. Werner and M. M. Wolf, “Bell Inequalities and Entanglement,” Quantum Information & Computation 1(3), 1–25 (2001).DOIarXiv
- [VW05]
- 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).DOIarXiv
- [RRS05]
- B. Reznik, A. Retzker, and J. Silman, “Violating Bell’s Inequalities in Vacuum,” Physical Review A 71, 042104 (2005).DOIarXiv
- [GRS25]
- 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).DOIarXiv