Cite this problem as **Problem 12**.

**Problem**

It is well known [1] that vacuum fluctuations maximally violate the CHSH-Bell inequalities for suitable spacelike separated observables, and that this violation goes to zero as the two localization regions are moved apart.

Decide whether some (necessarily small) violation of the inequalities is possible for regions arbitrarily far apart. For definiteness, consider a massive scalar free relativistic Bose field.

**Background**

It is known [2] that the vacuum is not separable at any distance. More recently [3], it has been shown that an analogue of the “positive partial transpose” condition fails for arbitrary regions at any distance. But the problem as stated above remains open.

**References**

[1] S. J. Summers and R. F. Werner, The vacuum violates Bell’s inequalities, Phys. Lett. A **110**, 257-259 (1985).

[2] H. Halvorson and R. Clifton, Generic Bell correlation between arbitrary local algebras in quantum field theory, J. Math. Phys. **41**, 1711-1717 (2000) and math-ph/9909013 (1999).

[3] R. Verch and R. F. Werner, Distillability and positivity of partial transposes in general quantum field systems, Rev. Math. Phys. **17**, 545-576 (2005) and quant-ph/0403089 (2004).